iterative observer-based estimation algorithms for...

Iterative Observer-based Estimation Algorithms for

Steady-State Elliptic Partial Differential Equation Systems

Dissertation by

Muhammad Usman Majeed

In Partial Fulfillment of the Requirements

For the Degree of

Doctor of Philosophy

King Abdullah University of Science and Technology

Thuwal, Kingdom of Saudi Arabia

May, 2017

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EXAMINATION COMMITTEE PAGE

The dissertation of Muhammad Usman Majeed is approved by the examination com-

mittee

Committee Chairperson: Taous Meriem Laleg-Kirati - Associate Professor

Committee Members: Ralph C. Smith - Professor, Jeff S. Shamma - Professor, David

E. Keyes - Professor, Ying Wu - Associate Professor

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©May, 2017


All Rights Reserved

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ABSTRACT

Iterative Observer-based Estimation Algorithms for Steady-State

Elliptic Partial Differential Equation Systems


Steady-state elliptic partial differential equations (PDEs) are frequently used to

model a diverse range of physical phenomena. The source and boundary data estima-

tion problems for such PDE systems are of prime interest in various engineering dis-

ciplines including biomedical engineering, mechanics of materials and earth sciences.

Almost all existing solution strategies for such problems can be broadly classified

as optimization-based techniques, which are computationally heavy especially when

the problems are formulated on higher dimensional space domains. However, in this

dissertation, feedback based state estimation algorithms, known as state observers,

are developed to solve such steady-state problems using one of the space variables

as time-like. In this regard, first, an iterative observer algorithm is developed that

sweeps over regular-shaped domains and solves boundary estimation problems for

steady-state Laplace equation. It is well-known that source and boundary estimation

problems for the elliptic PDEs are highly sensitive to noise in the data. For this, an

optimal iterative observer algorithm, which is a robust counterpart of the iterative

observer, is presented to tackle the ill-posedness due to noise. The iterative observer

algorithm and the optimal iterative algorithm are then used to solve source localiza-

tion and estimation problems for Poisson equation for noise-free and noisy data cases

respectively. Next, a divide and conquer approach is developed for three-dimensional

domains with two congruent parallel surfaces to solve the boundary and the source

data estimation problems for the steady-state Laplace and Poisson kind of systems

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respectively. Theoretical results are shown using a functional analysis framework,

and consistent numerical simulation results are presented for several test cases using

finite difference discretization schemes.

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ACKNOWLEDGEMENTS

This thesis document not only represents my research work but is a milestone

that is achieved after several years of dedicated work at Estimation Modeling and

Analysis Group (EMANG) at King Abdullah University of Science and Technology

(KAUST). Ever since I arrived at KAUST, I have felt it like my home. I have come

across several amazing people and I have had countless memorable experiences, some

of those I would love to recall in the following.

First and foremost, I would like to thank my research supervisor Prof. Dr. Taous

Meriem Laleg-Kirati for her consistent support and encouragement during my Ph.D.

work. Her guidance helped me in all the time of research and writing of this thesis.

The joy and enthusiasm that she has for research were contagious and motivational.

I could not have imagined having a better advisor and mentor for my Ph.D. studies.

Members of EMANG have contributed a lot to my personal and professional

growth. I am highly indebted to all of my colleagues here who bore me with patience

and helped me learn and grow over the years. I shall always cherish the fruitful time

spent with Fadi Eleiwi and Ayman Karam in our office desk space. I also like to thank

Abeer Aldoghaither, Sharefa Asiri, Zehor Belkhatir and Shahrazed Elmetennani for

very useful academic discussions and for their consistent feedback on my research. I

would like to acknowledge EMANG postdocs, Dayan Liu, Chadia Zayane, Ibrahima

N’Doye and Sarah Mechhoud for their benevolent help and encouragement during

rough times in my Ph.D. I am also grateful to a number of renowned researchers who

visited EMANG and provided valuable feedback on my research. I would certainly

like to thank my friends at KAUST particularly Tamour Javed, Bilal Janjua, Furrukh

Sana and Awad Alquaity, who always helped and motivated me along the ebbs and

flows of the journey and without whom my KAUST story would be incomplete.

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Many thanks to Prof. Rabia Djellouli at Department of Mathematics, California

State University Northridge (CSUN) for giving me an opportunity to work in his group

for about two months during the summer of 2012. His rigorous supervision helped

me develop a strong interest in my area of research. I am thankful to Prof. Miroslav

Krstic from University of California San Diego (UCSD) for fruitful discussions during

his couple of visits at KAUST and for hosting me for a talk in his research group at

UCSD. I am also indebted to Prof. Alexandre Bayen from University of California

Berkeley (UC Berkeley) to host me for a couple of months as a visiting scholar to

work on some of the most interesting and challenging problems in control and partial

differential equations. I am also thankful to Prof. Ralph Smith from North Carolina

State University (NCSU) for taking out time to review my thesis work and to provide

constructive feedback.

A very heartily thanks to my whole family. Words cannot express how grateful I

am to my parents and my late uncles. They have raised me with love and discipline

and supported me in all my pursuits. I am also thankful to my brothers and only sister

who guided me through years and left no stone unturned to help me succeed. And

my loving wife Saira, whose help and support during my Ph.D. is highly appreciated.

Thank you all.

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TABLE OF CONTENTS

Examination Committee Page 2

Copyright 3

Abstract 4

Acknowledgements 6

List of Abbreviations 12

List of Symbols 13

List of Figures 15

1 Introduction 20

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.3 Proposed Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.4 Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 State Observer Theory and Inverse Problems: Two Worlds Apart 26

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2 Observation Theory for Dynamical Systems . . . . . . . . . . . . . . 26

2.2.1 Finite-Dimensional Linear Time-Invariant Systems . . . . . . 27

2.2.2 Infinite-Dimensional Linear Partial Differential Equation (PDE)

Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3 Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.3.1 Forward and Inverse Problems . . . . . . . . . . . . . . . . . . 37

2.3.2 Ill-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.3.3 Some Solution Strategies for the Inverse Problems . . . . . . . 39

2.3.4 Ill-posedness of Boundary Data Estimation Problems for Laplace

Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

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3 Iterative Observers for Boundary Estimation Problems for Laplace

Equation 44

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2 Iterative Observer Design for Boundary Estimation . . . . . . . . . . 45

3.2.1 Problem Statment on a Rectangular Domain . . . . . . . . . . 45

3.2.2 Notations and Definitions . . . . . . . . . . . . . . . . . . . . 46

3.2.3 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2.4 Preliminary Analysis . . . . . . . . . . . . . . . . . . . . . . . 51

3.2.5 Convergence Analysis . . . . . . . . . . . . . . . . . . . . . . . 56

3.2.6 Numerical Implementation . . . . . . . . . . . . . . . . . . . . 62

3.2.7 Results and Simulations . . . . . . . . . . . . . . . . . . . . . 66

3.3 Robust Iterative Algorithm for Boundary Estimation . . . . . . . . . 71

3.3.1 Problem Statement on an Annulus Domain . . . . . . . . . . . 71

3.3.2 Problem Reformulation . . . . . . . . . . . . . . . . . . . . . . 71

3.3.3 Derivation of Optimal MSE Minimizer Algorithm . . . . . . . 73

3.3.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 82

3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4 Iterative Observer-based Approach for Source Localization and Es-

timation for Poisson Equation 88

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.2 Iterative Observer-based Strategy for Point Source Localization . . . 89

4.2.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . 90

4.2.2 Preliminary Analysis and Results . . . . . . . . . . . . . . . . 91

4.2.3 Point Source Localization Strategy . . . . . . . . . . . . . . . 94

4.2.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . 96

4.2.5 Further Simulation Results . . . . . . . . . . . . . . . . . . . . 98

4.3 Robust Iterative Algorithm-based Strategy for Inverse Source Local-

ization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 103

4.3.2 Robust Iterative Observer Design . . . . . . . . . . . . . . . . 104

4.3.3 Two-step Process for Source Localization . . . . . . . . . . . . 112


4.4 Distributed Potential Field Estimation for Poisson Equation . . . . . 117



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4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5 Dimension Decomposition Approach for 3D Domains 121

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.2 Boundary Estimation Problem for Laplace Equation in 3D . . . . . . 121


5.2.2 Theoretical Analysis and Results . . . . . . . . . . . . . . . . 125

5.2.3 Dimension Decomposition . . . . . . . . . . . . . . . . . . . . 131

5.2.4 Observer Design for the Subproblem . . . . . . . . . . . . . . 133

5.2.5 Numerical Implementation and Simulation Results . . . . . . 134

5.3 Point Source Localization Problem for Poisson Equation in 3D . . . . 135


5.3.2 Point Source Localization as Boundary Estimation Problem . 137

5.3.3 Two-step Process for Source Localization: . . . . . . . . . . . 140

5.3.4 Boundary Estimation Problem for Laplace Equation . . . . . 140

5.3.5 Preliminary Theoretical Results . . . . . . . . . . . . . . . . . 140

5.3.6 Observability Result . . . . . . . . . . . . . . . . . . . . . . . 143

5.3.7 Dimension Decomposition . . . . . . . . . . . . . . . . . . . . 143

5.3.8 Observer Design for the Subproblem . . . . . . . . . . . . . . 145

5.3.9 Numerical Implementation and Results . . . . . . . . . . . . . 146

5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6 Concluding Remarks 151

6.1 Summary of the Thesis Work . . . . . . . . . . . . . . . . . . . . . . 151

6.2 Future Research Directions . . . . . . . . . . . . . . . . . . . . . . . . 152

6.2.1 Extensions to Arbitrary Shaped Domains . . . . . . . . . . . . 152

6.2.2 Iterative Observer Applications to Other Steady-State PDE

Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

6.2.3 Steady-State Energy Field Imaging Technique . . . . . . . . . 153

References 159

Appendices 164

A.1 An Example of Exactly Observable System based on String Equation [11]165

A.1.1 Semigroup Generated by A . . . . . . . . . . . . . . . . . . . 166

A.1.2 Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

B.1 Journal Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

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B.2 Reviewed Conference Papers & Proceedings . . . . . . . . . . . . . . 169

B.3 Talks & Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . 170

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LIST OF ABBREVIATIONS

DPS Distributed Parameter Systems

ECG Electrocardiography

EEG Electroencephalography

EMANG Estimation Modeling and Analysis Group

KAUST King Abdullah University of Science and

Technology

MEG Magnetoencephalography

NCSU North Carolina State University

ODE Ordinary Differential Equation

PDE Partial Differential Equation

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LIST OF SYMBOLS

A′ State operator matrix after full system dis-

cretization

C Discrete observation operator matrix

K Gain operator matrix

λmn Infinite set of eigenvalues corresponding to

Φmn

λn Infinite set of eigenvalues corresponding to Φn

C Observation operator matrix

Csub Observation operator matrix for the sub prob-

lem

Ω1 Annulus domain without boundaries

Ω2 Rectangular domain without boundaries

Ω3 3D domain with two congruent parallel sur-

faces without boundaries

Ω4 3D rectangular prism without boundaries

A Differential operator matrix in proper Hilbert

space

Asub Differential operator matrix in proper Hilbert

space defined over the rectangular cross-

section of Ω3 or Ω4

M Linear or nonlinear mapping operator between

two normed spaces

Ψ Data to output map operator

P Discrete error covariance matrix

Φmn Infinite set of orthonormal basis functions for

all m, n ∈ Z?

Φn Infinite set of orthonormal basis functions for

all n ∈ Z?

Q Discrete process noise covariance matrix

R Discrete measurement noise covariance matrix

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T Exponential of A under certain conditions

S Exponential of A − KC under certain condi-

tions

W Exponential of an unbounded operator under

certain conditions

T Discrete observability matrix

X Complex Hilbert space with proper inner

product

Y Complex Hilbert space with proper inner

product

X1 Part of complex Hilbert space X that satisfies

the special conditions given in Theorem 4

Z? Non zero set of integers

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LIST OF FIGURES

2.1 Concept of a general Luenberger observer, dashed lines represent the

auxiliary observer structure for the linear finite-dimensional time vary-

ing systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.1 Rectangular domain Ω2. . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2 Idea of iterations over rectangular domain Ω2. . . . . . . . . . . . . . 49

3.3 Domain Ω2 after discretization and fictitious points outside ΓB, index

i = 0 represents fictitious points. . . . . . . . . . . . . . . . . . . . . . 63

3.4 Two dimensional rectangle domain with homogeneous Neumann side

boundaries, Example 1. . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.5 Comparison of exact and observer constructed solution on the bottom

boundary ΓB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.6 Two dimensional rectangle domain with homogeneous Dirichlet side

boundaries, Example 2. . . . . . . . . . . . . . . . . . . . . . . . . . . 68


boundary ΓB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68


boundary ΓB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.9 Annulus domain Ω1 with inner boundary Γin and outer boundary Γout. 72

3.10 Fictitious points close to inner boundary Γin, with index i = 0. . . . . 76

3.11 True solution or solution obtained by solving the problem (3.70) with

h = sin(θ) + sin(3θ) and g = 0 over Ω1 . . . . . . . . . . . . . . . . . 83

3.12 Solution obtained from optimal iterative algorithm over Ω1, after a

number of iterations in the direction of time-like variable θ . . . . . . 83

3.13 Difference between the true and optimal observer algorithm solutions

over Ω1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.14 On Γin, comparison of true boundary h = sin(θ) + sin(3θ) to the one

recovered by optimal iterative algorithm using Cauchy data from Γout 83

3.15 Numerical solution over Ω1 obtained by solving the problem (3.70) with

pulse shaped h on Γin and g = 0 on Γout . . . . . . . . . . . . . . . . 84

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3.16 Solution obtained from optimal iterative algorithm over Ω1, after a

number of iterations in the direction of time-like variable θ . . . . . . 84

3.17 Difference between the true and the recovered solutions over Ω1 . . . 84

3.18 On Γin, comparison of the true pulse-shaped boundary signal to the

one recovered by optimal iterative algorithm using only the Cauchy

data from Γout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.19 Noisy data on Γout . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.20 On Γin: Comparison of true h = sin(θ) to the one recovered by robust

iterative algorithm using noisy Cauchy data with measurement noise

variance σ2 = 10−3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.21 Percentage relative error in the Dirichlet measurement data on Γout vs.

percentage relative error in the recovered solution on Γin . . . . . . . 87

4.1 Left: Rectangular domain Ω2 with ∂Ω2 = Γ1 ∪ Γ2 ∪ Γ3 ∪ Γ4, Right:

Co-ordinate axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.2 Numerical simulation result for the Poisson equation over domain Ω2

with a point source in the middle using homogeneous Neumann bound-

ary data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.3 Dirichlet data g1 on bottom boundary Γ1. Because of symmetry g2|Γ2, g3|Γ3

and g4|Γ4 would be similar. . . . . . . . . . . . . . . . . . . . . . . . . 98

4.4 Dirichlet data g1|Γ1 and estimated S1 = ξ1 on opposite boundary Γ2

using iterative observer. Because of symmetry, qualitatively similar

profiles for other three cases. . . . . . . . . . . . . . . . . . . . . . . . 98

4.5 Top plot: Solution profile S1 over Ω2 obtained by solving problem (4.7)

with g1 on Γ1 and corresponding S1 = ξ1 on opposite boundary Γ2 and

insulated (homogeneous Neumann) side boundaries.

From 2nd to 4th: Plots for S2, S3 and S4 obtained using similar procedure. 99

4.6 From top to bottom: Weight profiles w1, w2, w3 and w4 over Ω2 cor-

responding to solution profiles S1, S2, S3, S4 respectively, as shown in

Figure 4.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.7 Weighted sum as given in equation (4.16) over Ω2. Marker in the middle

represents the minima, where the negative point source is located. . . 100

4.8 Top: Non-centered point source inside Ω2. Bottom: Weighted sum

with marker representing minimum point and the location of point

source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

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4.9 Top: Two opposite polarity well seperated point sources in Ω2. Bot-

tom: Weighted sum with markers representing minimum and maxi-

mum points and locations of two point sources. . . . . . . . . . . . . 102

4.10 Top: Three closely located point sources in Ω2. Bottom: Weighted

sum with minima locating approximate position of point sources. . . 102

4.11 Left: Domain Ω2 after discretization and fictitious points outside Γ2,

index i = 0 represents fictitious points (in blue). . . . . . . . . . . . . 107

4.12 Square domain Ω2 with a point source in the middle using homogeneous

Neumann boundary data. . . . . . . . . . . . . . . . . . . . . . . . . 114

4.13 Noisy measurement data g with σ1 = 5 × 10−4, because of symmetry

of the special case under consideration (one point source in the middle

of the domain), qualitatively similar profiles on all parts of ∂Ω2. . . . 114

4.14 Comparison of noisy measurement data g on Γi and robust iterative

observer solution on the opposite boundary. σ1 = 5 × 10−4 and σ2 =

1 × 10−1. Because of symmetry, qualitatively similar profiles on all

parts of ∂Ω2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.15 Top figure: Solution profile S1 with measurement data g1 on Γ1, in-

sulated side boundaries and boundary estimate obtained using robust

iterative observer on Γ2. From 2nd to 4th: Plots for S2, S3 and S4

respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.16 From top to bottom: Weight profiles w1, w2, w3 and w4 respectively,

obtained by solving boundary value problem (4.15) for i ∈ 1, 2, 3, 4. 116

4.17 Weighted sum obtained using equation (4.16). Minimum point repre-

sented with white marker in the middle provides the location of point

source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4.18 Top: Solution of problem (4.27),(4.28) with two point sources of op-

posite polarity and homogeneous Neumann boundary on ∂Ω2. Below:

Weighted sum obtained using equation (4.16). Minimum and maxi-

mum points represented with white markers provide locations of point

sources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

4.19 Steady-state potential field u over Ω, obtained by numerically solving

Poisson equation (4.67) for various f with homogeneous Neumann side

boundaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.20 Weighted sum obtained from equation (4.16), which provides estimate

to the steady-state potential field u over Ω2 for various test cases shown

in Figure 4.19. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

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5.1 Left: Domain Ω3 with two congruent parallel surfaces ΓB and ΓT (ΓB

and ΓT Lipschitz continuous); Right: Plane containing time-like co-

ordinate x1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.2 Left: Cross-sectional plane ω of Ω3; Right: x2x3 plane orientation (in

gray); . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.3 Left: Cross-sectional plane ω of Ω3 parallel to x2x3 plane; Right: x2x3

plane orientation (in gray); . . . . . . . . . . . . . . . . . . . . . . . . 127

5.4 Analytical solution u on Γ1 . . . . . . . . . . . . . . . . . . . . . . . . 135

5.5 Analytical solution u on Γ2 . . . . . . . . . . . . . . . . . . . . . . . . 135

5.6 Recovered solution on Γ2 using observer algorithm . . . . . . . . . . . 135

5.7 Difference of the analytical solution and the one recovered by observer

algorithm on Γ2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.8 Rectangular prism Ω4 with six boundary surfaces, ∂Ω4 = ∪6i=1Γi. Sur-

faces Γ1, . . . ,Γ6 represent bottom, top, and side surfaces respectively. 137

5.9 Rectangular cross-section ω at a particular value of x in yz-plane inside

Ω4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

5.10 Idea of iterations over rectangular cross-section ω. . . . . . . . . . . . 146

5.11 Domain Ω4 with three orthogonal cross-sectional planes. . . . . . . . 147

5.12 A single point source in the middle of a 1 × 1 × 1 cube on a uniform

20× 20× 20 grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

5.13 Cross-sectional plane ω from Figure 5.12 on a 200× 200 uniform grid. 148

5.14 Iterative observer solution from equation (5.71) with h1,sub = 0 and

g1,sub extracted from Figure 5.13 on Γb and recovered boundary data

on opposite boundary. Similarly iterative observer solution can be

computed on cross-sectional planes parallel to ω. . . . . . . . . . . . . 149

5.15 Solution profile S2 on a 20× 20× 20 uniform grid, obtained by solving

boundary estimation problem (5.43) for i = 2 using iterative observer. 149

5.16 Weight profile v2 obtained by solving boundary estimation problem

(5.44) for i = 2 on a uniform 20× 20× 20 grid. (only two orthogonal

cross-sectional planes displayed) . . . . . . . . . . . . . . . . . . . . . 149

5.17 Weighted sum computed using equation (5.45). Local maxima in the

center locate the position of the point source. . . . . . . . . . . . . . 150

6.1 Domains under consideration for the boundary estimation problem for

Laplace equation with Γ∗, the unknown data boundary. . . . . . . . . 153

19

6.2 Domains under consideration for the source localization and estimation

problems for Poisson equation. . . . . . . . . . . . . . . . . . . . . . . 153

6.3 Numerical solution of the Poisson equation 4u = f over Ω4 with f =

exp(−(2.5(x−0.5))2−(2.5(y−0.5))2−(2.5(z−0.5))2) and homogeneous

Neumann boundary data at ∂Ω4. The solution is computed over a 50×50× 50 uniform grid using 2nd order accurate centered finite difference

discretization schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . 156

6.4 A particular cross-sectional view of the numerical solution presented

in Fig. 6.3, at x = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . 156

6.5 (In blue): Dirichlet data, extracted from Fig. 6.4 at Γb = ω ∩ Γ1,

corrupted with added white Gaussian noise with η1 = 0 and σ1 =

5×10−3; (In black) Numerical solution of boundary estimation problem

(5.66), obtained by using optimal iterative algorithm (4.49), (4.50) at

cross-section ω|x=0.5 at Γt = ω ∩ Γ2. . . . . . . . . . . . . . . . . . . . 156

6.6 Full solution profile over cross-section ω|x=0.5 obtained by using fi-

nite difference discretization schemes and the estimated boundary data

from Fig. 6.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

6.7 Tomographic image S1 over Ω4 obtained by solving boundary estima-

tion problem for Laplace equation (5.43) using dimension decomposi-

tion approach and optimal iterative algorithm. . . . . . . . . . . . . . 157

6.8 Weight profile v1 obtained by numerically solving boundary value prob-

lem (5.44) for index i = 1. . . . . . . . . . . . . . . . . . . . . . . . . 157

6.9 Weighted sum obtained from equation (5.45) by combining tomographic

image profiles S1, . . . , S6 and corresponding weights v1, . . . , v6. The

simulation result shows recovery of the distributed potential field pre-

sented in Fig. 6.3, using only the noise corrupted boundary data. . . . 158

6.10 A cross-sectional view of weighted sum over ω|x=0.5. The result above

to be compared with the cross-sectional view given in Fig. 6.4. . . . . 158

20

Chapter 1

Introduction

1.1 Motivation

Partial Differential Equations (PDEs) are widely used to model physical phenomena

including sound, heat, wave, fluid dynamics and quantum mechanics. Just as Or-

dinary Differential Equations (ODEs) are used to model one-dimensional space or

time evolving physical phenomena, PDEs are used to model multidimensional sys-

tems [1,2]. Over the last decades, there have been a lot of efforts to develop feedback

based strategies to control and observe dynamical PDE\ODE systems. The objec-

tives of these techniques are to drive the system in a particular way to achieve some

desired goals and to observe some system states and parameters respectively [3,4,5].

However, there is a subclass of PDEs, known as elliptic PDEs, that represents the

steady-state physical phenomena that do not evolve with time. Such phenomena

arise in mechanics of materials, bio-medical engineering and earth science applica-

tions [1, 6, 7, 8].

Customarily, the control and observation design techniques are only developed

for time varying PDE systems, broadly classified as hyperbolic and parabolic PDE

systems, with inherent time component [3,4,5,9,10,11,12]. Recently, there has been a

lot of work done on the feedback based system state estimation algorithms, commonly

know as state observer algorithms, for dynamical PDE systems. In dynamical systems

theory, the state observer is an algorithm that provides estimates of internal states

of a given real system from the measurements of inputs and outputs [13]. Various

21

kinds of state observer algorithms have been proposed for dynamical PDE systems in

literature. A limited introduction to the observation design for PDE systems can be

found in [11,14,15,16,17,18,19].

On the one hand state observers are only developed for time varying PDE sys-

tems, and on the other hand the unknown source and boundary data estimation

problems for steady-state elliptic PDEs belong to the class of ill-posed problems.

The ill-posedness is in the sense of stability, that is, a small amount of discrepancy

in the observed data destroys the estimated solution [20, 21]. In other words, the

mathematical problem is well-posed only for smooth data. Whereas, the real experi-

mental observations are always prone to measurement noise like sensor related issues

and other unavoidable artifacts. Nearly all existing solution techniques for estima-

tion problems for elliptic PDEs can be broadly classified as optimization based algo-

rithms [8,20,22,23,24]. Such techniques generally solve a series of similar well-posed

problems to achieve the convergence based on some preset criteria. These optimiza-

tion approaches are numerically costly, particularly, when the problem is posed on

a higher dimensional domain [25, 26]. In this scenario, the efforts to develop robust

state observer like algorithms, possibly using one of the space variables as time-like,

to achieve lower numerical complexity and robustness, becomes highly interesting.

This strongly motivates the study of observer design techniques for the steady-state

elliptic PDE systems.

The source and boundary data estimation problems for elliptic PDE systems have

been fundamental problems of interest in several areas of science and engineering

including corrosion problem and non-destructive testing in mechanics of materials,

steady-state reservoir modeling in earth science, mineral exploration in geo-science

and Electrocardiography (ECG) and Electroencephalography (EEG) applications in

biomedical discipline [27, 28,29,30,31,32].

22

1.2 Background

A vast literature is available for the observation design problems for dynamical sys-

tems including [4,11,13,14,15,33]. The pioneering works of Luenberger on the design

of feedback based system state estimation algorithms paved the way for the field of

state observer design for dynamical systems [13]. Early observer designs were pro-

posed for state estimation for finite dimensional lumped parameter systems governed

by ODEs. However, over the years, the concepts of observer design have been ex-

tended to infinite dimensional Distributed Parameter Systemss (DPSs) modeled by

time varying PDEs [11,14,16,34].

Traditionally for DPSs early or late lumping techniques are considered [35]. Early

lumping techniques transform DPS to a finite dimensional system of ODEs using

some approximation and discretization techniques [9, 10]. The resultant system of

ODEs is an approximation to the DPS and unknown states recovered by the state

observers may not be the estimate of true states [17]. On the other hand, late lump-

ing techniques exploit mathematical properties of underlying PDEs to develop ob-

server design. The potential challenges include the mathematical justification of

system observability and the design of observer gain. The system observability

ensures the existence of the solution. Various design techniques based on semi-

group theory, spectral theory, Lyapunov based design, backstepping approaches are

available [3, 11, 15, 16, 18, 19, 34, 36, 37, 38]. Further, some state-of-the-art inverse

and ill-posed source and boundary value problems for elliptic PDEs can be found

in [20, 39, 40]. Some of the existing optimization based strategies to solve inverse

source and boundary value problems for Laplace and Poisson equations can be found

in [24,25,26,30,32,41,42,43,44,45]. A state of the art discussion on inverse problems

is also provided in Chapter 2.

23

1.3 Proposed Approaches

In this thesis work, state observer design techniques are developed for steady-state

elliptic PDE systems. The idea is to use space as a time-like variable to develop

iterative observer algorithms to solve source and boundary data estimation problems

for elliptic PDE systems. Both early and late lumping techniques are used to develop

different variants of the proposed iterative observer algorithms.

As stated before, almost all the existing observer design techniques are focussed

on time-varying PDE systems [3, 11, 16, 18, 34, 36, 37, 38]. There has been very little

effort to develop observer-like algorithms for systems governed by steady-state elliptic

PDEs. One such example is [46], where an extra time variable is introduced to

solve steady-state heat conduction problem modeled by elliptic PDE as a parabolic

problem. However, the focus of this thesis work is to develop state observers that

sweep over the entire domain in a particular time-like direction to solve boundary

and source data estimation problems for linear elliptic equations namely Laplace and

Poisson equations. The theoretical framework is developed using functional analysis

tools and semigroup theory. The idea of using a space variable as time-like allows the

development of numerically efficient algorithms. The various algorithms, presented in

this thesis, are numerically implemented using finite difference discretization schemes.

The algorithms tackle 2D regular shaped annulus and rectangle domains and 3D

domains with two congruent parallel surfaces. The main contributions of this thesis

are also listed in the following section.

1.4 Thesis Contributions

Key contributions of this thesis report are listed below,

• Developed Iterative Observer for Boundary Estimation Problem for

Laplace Equation:

24

Iterative observer is developed using space as time-like to estimate the unknown

boundary data. The functional analysis framework and convergence of the

iterative observer are proved. The algorithm is implemented on a 2D rectangular

and annulus shaped domains.

• Provided Source Estimation and Localization Strategy for Poisson

Problem:

Source localization and estimation strategy is presented using iterative observer

algorithm for the boundary value problem for Poisson equation on regular

shaped domains.

• Designed Robust Iterative Observer for Ill-posed Boundary Estima-

tion Problem for Laplace Equation with Noisy Data:

Robust counterpart of the iterative observer is developed to tackle the inverse

and ill-posed boundary data estimation problem for Laplace equation on reg-

ular shaped domain. The proof of convergence of the developed algorithm is

provided.

• Extended Source Localization and Estimation Strategy to Inverse

Source Problems for Poisson Equation:

The source localization and estimation strategy is extended to tackle inverse

and ill-posed source problems for Poisson equation. The strategy and results

are presented on rectangular domains.

• Developed Extensions to 3D-Shaped Domains and Introduced Steady-

State Energy Field Imaging Technique:

Boundary estimation and source localization strategies are extended to 3D

shaped domains with two congruent parallel surfaces. The technique involve

dimension decomposition approach that divides down the 3D problem to a set

25

of 2D subproblems. Finally the iterative observer algorithm is implemented on

the 2D subproblems.

The various chapters are organized as follows. Some state-of-the-art literature

on the state observers and inverse and ill-posed problems is provided in Chapter 2.

An iterative observer is designed in Chapter 3 that sweeps over the regular shaped

annulus and rectangle domains to solve the boundary data estimation problems for

the Laplace equation. As discussed earlier, such boundary value problems are ill-

posed in the sense of stability. This makes the use of standard iterative observer

algorithms impossible without further investigations. Thus, a robust counterpart

of this iterative observer is developed in the second half of this chapter. Next, a

source estimation and localization strategy is developed in Chapter 4 to solve source

localization and estimation problems for the Poison equation in the 2D rectangular

domain. The strategy works well for the cases of smooth and noisy measurement

data. A dimension decomposition approach, based on iterative observer design, is

presented in Chapter 5 to solve the source and boundary data estimation problems

for 3D domains with two congruent parallel surfaces and insulated side boundary

surfaces. Theoretical and numerical results are consistently presented throughout the

document.

26

Chapter 2

State Observer Theory and Inverse Problems: Two Worlds

Apart

We must not cease from exploration and the end of all our exploring will

be to arrive where we began and to know the place for the first time.

(—T.S. Eliot)

2.1 Introduction

In the first half of this chapter, some state of the art literature on observation and

control design theory for dynamical systems is presented. Key definitions along with

some theoretical results are recalled. In the second half, some fundamental concepts

on inverse problems are discussed. The notion of ill-posedness is explained. The

literature study provided in this chapter will help to develop the framework for the

upcoming analysis and results in later chapters.

In the following, some dynamical systems concepts are presented for time evolving

finite and infinite dimensional systems.

2.2 Observation Theory for Dynamical Systems

In the last few decades, there has been a lot of focus on interdisciplinary research to

develop novel kind of solution strategies for challenging problems. With combined

efforts from diverse areas of research, new versatile and robust solution techniques

are emerging for various types of mathematical problems. One such joint venture

is the study of initial boundary value problems modeled by PDEs using dynamical

27

system techniques. In order to develop the understanding for observation problems

for linear PDE systems, it is natural to look into the theory developed for analogous

finite dimensional time varying systems.

2.2.1 Finite-Dimensional Linear Time-Invariant Systems

Given n ∈ N∗ and A ∈ Cn×n, consider the linear time invariant dynamical system,

ξ(t) = Aξ(t) +Bu(t) ∀ t ≥ 0, (2.1)

where ξ(t) ∈ Cn is the state of the system, ξ(t) represents system dynamics, u(t) is the

input and B is the input operator. For a pure observation problem, let u(t) = 0. The

above representation is called state-space representation. A state-space representation

for a dynamical system is a mathematical model of a physical system as a set of input,

output and state variables related by first-order evolution equations. “State-space”

refers to the space whose axes are the state variables. The state of the system can be

represented as a vector within that space. Now, suppose partial measurements y(t)

of the system are available through an observation operator C ∈ Cm×n, with m ∈ N∗

such that,

y(t) = Cξ(t) = CetAξ(0) ∀ t ≥ 0, (2.2)

here a natural question arises whether the observation y(t) is good enough to solve

problem (2.1) or not. Or in other words, whether the partial measurements y(t)

are good enough to provide full information about systems states ξ(t) for all t ≥

0. Answer to this question lies in the concept of observability of finite dimensional

systems.

Definition 1. Let (A,C) be the system defined by equations (2.1) and (2.2). Also

28

define the initial data to output map Ψτ ∈ L(Cn, L2([0,∞);Cm)) by setting,

(Ψτξ0) =

CetAξ0 t ∈ (0, τ),

0 t > τ.

(2.3)

System (A,C) is said to be observable if for some τ > 0, KerΨτ = 0. A necessary

and sufficient condition for observability of finite dimensional system (2.1)-(2.2) is the

so-called Kalman rank condition.

Theorem 1. [3, 11] We have,

ker Ψτ = ker

C

CA

CA2

...

CAn−1

(2.4)

In particular, system (A,C) is observable if and only if

rank

C

CA

CA2

...

CAn−1

= n. (2.5)

Proof. Proof of the above theorem is also recalled in appendix A.

Note that Kalman condition (2.5) is independent of τ . This shows in particular that

a finite-dimensional linear system (A,C) is observable in arbitrarily small time. As

will be seen later this is not true for infinite dimensional systems.

Definition 2. A matrix A ∈ Cn×n is called Hurwitz matrix if its eigenvalues are

29

located in the open left half-plane:

σ(A) ⊂ λ ∈ C|Re(λ) < 0 . (2.6)

Note that a system of the form (2.1) associated to a Hurwitz matrix A generates

stable trajectories:

limt→+∞

z(t) = 0. (2.7)

More precisely it can be proved that there exists 0 < ω < minλ∈σ(A) Re |λ| and M > 0

(depending on ω) such that

‖ξ(t)‖ ≤Me−ωt‖ξ0‖, ∀t > 0, ∀ξ0 ∈ Cn. (2.8)

2.2.1.1 State Observer

In dynamical systems theory, a state observer is a mathematical system that provides

an estimate of the internal states of a given physical system, from measurements of

the inputs and outputs of the real system.

As stated earlier, state of a linear discrete time system is assumed to satisfy equations,

ξ = Aξ,

ξ(0) = 0,

y(t) = Cξ(t).

(2.9)

The observer model of the physical system is typically derived from these equations.

Additional terms may be included in order to ensure that, on receiving successive

measurements, the model’s state converges to that of the real system. In particular,

the output of the observer may be subtracted from the real system output and then

multiplied by a matrix K, this is then added to the equations for the state of the

30

ξ

(ξ − ξ)→ 0as t→∞

y

y

+

−

System

ξ = Aξ

Model

˙ξ = Aξ +K

(y − Cξ

)K

Figure 2.1: Concept of a general Luenberger observer, dashed lines represent theauxiliary observer structure for the linear finite-dimensional time varying systems

observer to have Luenberger observer, defined by the equations below,

˙ξ = Aξ +K

(y − Cξ

). (2.10)

The above equation contains the copy of the real system plus a correction term. The

state estimation error dynamics are given by,

e = (A−KC)e, (2.11)

where e = ξ − ξ. Whenever pair (A,C) satisfies Kalman rank condition, (A −

KC) can be made Hurwitz such that continuous time error dynamics converges to

zero asymptotically. The general structure of a Luenberger observer for linear finite-

dimensional time varying systems is also shown in Figure 2.1. It is important to note

that, for time varying dynamical systems, various kinds of state observers have been

developed in literature [13,14,15,16,18,34,47,48].

31

2.2.2 Infinite-Dimensional Linear PDE Systems

At present, dynamical systems theory is one the most interdisciplinary areas of re-

search. The need for control design arises in most of the present day industrial

applications. In fact, the overlap of dynamical systems concepts applied to complex

physical systems modeled by deterministic and stochastic PDEs has produced a new

and highly enriched branch of modern mathematics. Readers can refer to some state-

of-the-art literature available in this domain [3, 4, 6, 11, 12, 38, 49, 50, 51, 52]. In the

following, only observation theory concepts for dynamical systems modeled by linear

time varying PDEs are discussed.

2.2.2.1 State-Space Representation for Second-Order Linear

PDE Systems

Many time varying physical phenomena including heat transfer and wave propagation

are modeled by second-order PDEs. Linear second-order PDEs on a two-dimensional

xt plane can be represented as,

a∂2v

∂t2+ b

∂2v

∂t∂x+ c

∂2v

∂x2+ d

∂v

∂t+ e

∂v

∂x+ fv = 0, (2.12)

where a, b, c, d, e and f are constants and a, b and c are not all zero. The above

equation can be represented as a first-order state equation in one of the variables as:

∂ξ

∂t= ξ(t) = Aξ, (2.13)

with t as time variable and state vector ξ. The state vector of such a system is an

element in an infinite-dimensional normed space. The state operator matrix A is

32

given as:

ξ(t; .) =

ξ1(t; .)

ξ2(t; .)

, A =

0 1

c

a

∂2

∂x2+e

a

∂

∂x+ f

b

a

∂

∂x+d

a

, (2.14)

with ξ1 = v and ξ2 =∂v

∂t. A dynamical system modeled by 2nd order linear PDEs

can be written as a first order system along with the initial condition as follows,

ξ = Aξ,

ξ(0) = ξ0.

(2.15)

with some boundary conditions, where “ ˙ ” represents partial derivative with respect

to variable t and A is a differential operator matrix.The available information about

the problem can be written by using an observation operator C acting on state variable

ξ such that,

y(t) = Cξ(t), (2.16)

here y is the output or observation. System dynamics are described by the partial

differential operator matrix A. Let us pose three questions before trying to find a

solution strategy to solve mathematical problem given by equations (2.15) and (2.16).

Q 1 : Solution of the first order state equation, given in equation (2.15), involves

exponential of the differential operator matrix A. What does it mean by the

exponential of A?

Q 2 : Does there exist a solution to the problem (2.15) and (2.16)?

Q 3 : What can be a particular algorithm to solve problem (2.16)?

Answers to above questions along with the design of solution algorithms is discussed

in infinite dimensional setting in the following subsections. The exponential of op-

33

erator matrix A leads to the study of semigroup generated by operator A. Naively,

a semigroup is the generalization of an exponential function to infinite dimensional

spaces. Answer to Q 2 leads to the study of system observability. First we study

observability concept for finite dimensional systems as given below. For more details,

readers are referred to works of Curtain and Zwart [3], review paper of Zuazua [5]

and Tucsnak and Weiss [11].

2.2.2.2 Exponential of Operator A

Let X be a complex Hilbert space (with inner product (., .) and corresponding norm

‖.‖) and A : D(A) → X be an unbounded operator on X. The objective is to find

the solution of the evolution problem of the form (2.15). Note that if A ∈ L(A) is a

bounded operator on X then the unique solution of (2.15) is given by,

ξ(t) = etAξ0. (2.17)

Typically A is a partial differential operator matrix and hence unbounded, the expo-

nential etA is no more defined. However, there is a class for which this notion can be

defined, leading to the concept of semigroups. For the introduction to the semigroups

readers are referred to the textbooks of Pazy [38] and Brezis [53]. In the following

strongly continuous semigroup is defined.

2.2.2.3 C0-Semigroup

C0-semigroup, also known as a strongly continuous one-parameter semigroup, is a

generalization of the exponential function. Just as exponential functions provide so-

lutions of scalar linear constant coefficient ordinary differential equations, strongly

continuous semigroups provide solutions of linear constant coefficient ordinary differ-

ential equations in Banach spaces.

34

Definition 3. [11, 38] A family T= (Tθ)θ≥0 of operators in L(X) is a strongly

continuous semigroup on X if

1. Tθ=0 = I. (identity property)

2. Tθ+φ =TθTφ ∀θ, φ > 0. (semigroup property)

3. limθ→0,θ>0 Tθz = z ∀z ∈ X. (strong continuity property)

Intuitively, T = (Tt)t≥0 models the time evolution of the state of a process, the

state at time t ≥ 0 being described by ξ(t) =Ttξ0. The simplest but rather limited

class of C0 semigroups is given by, Tt = etA for A∈ L(X).

Definition 4. [11, 38] A strongly continuous semigroup T is called a contraction

semigroup if ‖Tt‖ ≤ 1 for all t ≥ 0.

Definition 5. [11,38] Let T be a strongly continuous semigroup. Then, the operator

A : D(A)→X defined by,

D(A) =

ξ ∈ X : lim

t→0+

Ttξ − ξt

exists

, (2.18)

Aξ = limt→0+

Ttξ − ξt

, ∀ξ ∈ D(A). (2.19)

is called infinitesimal generator (or the generator) of the semigroup. Semigroup gen-

erated by an operator A is often denoted by T= (etA)t≥0.

Definition 6. [11, 38] Let A be the generator of a strongly continuous semigroup

T= (etA)t≥0. Then for ξ ∈ D(A) and t ≥ 0, we have etAξ ∈ D(A) and

d

dtetAξ = AetAξ = etAAξ. (2.20)

Definition 7. [11, 38]

35

• An operator A: D(A)→ X is called dissipative if,

Re(Aξ, ξ) ≤ 0, ∀ξ ∈ D(A). (2.21)

• An operator A: D(A)→ X is called m-dissipative (or maximal dissipative) if it

is dissipative and I −A is surjective.

Theorem 2. [38] (Lumer-Phillips Theorem) Let A : D(A) → X be an unbounded

operator on a Hilbert space X. Then the following two assertions are equivalent.

1. A is maximally dissipative.

2. A is the generator of a contraction semigroup (Tx)x≥0, i.e. ‖Tt‖ ≤ 1 for all

t > 0.

2.2.2.4 Observability for Infinite-Dimensional PDE Systems

For infinite-dimensional PDE systems, depending on the density argument in infinite-

dimensional spaces, there are at least three observability concepts as explained in the

following. Assume that Y is a complex Hilbert space and that C∈ L(X,Y ) is an

admissible observation operator for T. Let τ > 0, and let Ψτ be the output operator

associated with (A, C) given by,

(Ψτξ0) =

CTtξ0 t ∈ (0, τ),

0 t > τ.

(2.22)

These operators are elements of L(X,L2([0,∞);Y )).

Definition 8. [11] Let time τ > 0.

• The pair (A, C) is exactly observable in time τ if Ψτ is bounded from below.

• (A, C) is approximately observable in time τ if ker Ψτ = 0.

36

• The pair (A, C) is final state observable in time τ if there exists a kτ > 0 such

that ‖Ψτξ0‖ ≥ kτ‖Tτξ0‖ for all ξ0 ∈X.

It can be seen, using density of D(A∞) in X, that the exact observability of (A, C)

in time τ is equivalent to the fact that there exists kτ > 0 such that,

∫ τ

0

‖CTtξ0‖2dt ≥ k2τ‖ξ0‖2 ∀ ξ0 ∈ D(A∞). (2.23)

A simple example of exactly observable system based on the string equation is pro-

vided in Appendix A. The Luenberger type of observer design has a similar structure

as given,

˙ξ = Aξ +K

(y − Cξ

). (2.24)

The main challenge is to design the observer gain K such that (A−KC) is dissipative.

The observer design for infinite dimensional steady-state elliptic PDE systems will be

discussed in details in chapter 3. On the other hand, for time varying PDE systems

in infinite dimensional setting, various works can be visited [15,16,34,36,37].

In the following section some state of the art literature for inverse problems is

discussed.

37

2.3 Inverse Problems

Most people, if you describe a train of events to them, will tell you what

the result would be. They can put those events together in their minds,

and argue from them that something will come to pass. There are few

people, however, who, if you told them a result, would be able to evolve

from their own inner counsciousness what the steps were which led up to

that result. This power is what I mean when I talk of reasoning backward

or analytically. (—Arthur Conan Doyle, A Study in Scarlet)

An inverse problem is the one in which cause behind a certain physical phe-

nomenon is investigated with a given mathematical model for the system. Real phys-

ical phenomena are governed by laws of nature with infinite artifacts at different

scales. Some of these phenomena like diffusion and wave propagation etc. can be

modeled fairly accurately using PDEs. However, no mathematical model for a par-

ticular physical phenomenon is perfect, further, there are always some measurement

errors due to noise and other unavoidable circumstances, all these factors make inverse

problems a challenging field of research. In the last half century, with the advent of

highly remarkable computational devices, it has been possible to model complex ini-

tial boundary value problems for PDEs using efficient numerical techniques. This has

provided a center stage to study PDE based model problems numerically, as well as

their theoretical analysis of uniqueness and stability. Almost always these problems

are ill-posed in terms of existence, uniqueness or stability. Thus, their uniqueness and

stability studies are at the heart of the development of new solution methodologies.

2.3.1 Forward and Inverse Problems

In the jargon of the theory of inverse problems, there always exists a forward or

direct problem and vice versa. A forward problem is usually a well-posed problem in

contrast to the inverse problem which is ill-posed. This chapter describes ill-posedness

of inverse problems and some state of the art solution strategies.

38

In the remaining part of this chapter, letM(x) = y be the mathematical represen-

tation of a given physical system withM be the mathematical model, x be the input

or model parameters and y be the output data measurements. The forward problem

is to find the output y from the given mathematical modelM and input parameters x.

Forward Problem:

Mathematical

Model M(x)Input x Data y

In general, the forward problem is well-posed. This means with given input x, a

stable M(x) can be found uniquely. On the other hand inverse problem is to find

out the hidden model parameters x from given mathematical model of the systemM

and output data measurements y. In terms of functional analysis, an inverse problem

is represented as a mapping between metric spaces. Many of the inverse problems

are ill-posed in terms of stability, that is, the forward map M(x) does not have a

continuous inverse.

Inverse Problem:

Mathematical

ModelData Input?

2.3.2 Ill-posedness

Definition 9. A problem is called ill-posed in the sense of Hadamard [54] if it fails

to satisfy any of the following properties,

1. There exists at least one solution to the problem. (existence)

2. There exists at most one solution to the problem. (uniqueness)

39

3. The solution continuously depends on the the data. (stability)

Definition 10. Let X and Y be normed spaces, M : X → Y be any linear or

nonlinear mapping. Equation Mx = y is called properly posed or well-posed, if the

following holds [39]:

1. For every y ∈ Y , there exists, at least one x ∈ X such thatMx = y. (existence)

2. For every y ∈ Y , there exists, at most one x ∈ X such that Mx = y. (unique-

ness)

3. The solution of the inverse problem, x continuously depends on the data; that

is, for every sequence (xn) ⊂ X with Mxn → Mx as n → ∞, it follows that

xn → x as n→∞. (stability)

2.3.3 Some Solution Strategies for the Inverse Problems

In practical applications output data measurements are always corrupted with some

noise i.e. yδ = y + ε (ε be an additive noise). Further as stated earlier if the forward

map M(x) does not have a continuous inverse then small errors due to noise can

totally destory the solution of the inverse problem. Therefore, for constructing a stable

approximation of the solution a regularization strategy is required. Regularization

means constructing an approximate continous map Πα : Y → X that inverts M

approximately. Design of such a continous reverse map Πα always require some

apriori information on the system. In a broader sense solution strategies for inverse

problems lie in following two catagories.

1. Deterministic Regularization Techniques.

2. Stochastic or Bayesian Inversion.

40

2.3.3.1 Deterministic Regularization Techniques [39]

These regularization techniques are based on acquiring approximately continuous

reverse map using some a priori information on the system in a deterministic way. A

priori information usually comes from the physics of the problem and is exterior to

mathematics. This added information assures the robustness of the new approximate

solution against the measurement errors in the data. Following are various standard

regularization techniques used,

• Tikhonov’s Regularization, the most well-known method of regularization works

on the standard least square linear regression plus an added regularization func-

tional. The functional added is capable of taking into account the a priori

information about the system [39].

• Reduction of the number of parameters, to reduce the sensibility of the criteria

to data noise.

• Introduction of constraints, equalities and inequalities inside the functional just

to reduce the solution domain to physically acceptable values.

• Filtering of the trial data using signal processing techniques (frequency filters,

modified Fourier transforms, time frequency transformations etc.) [55].

• Quasireversibility method well suited for some Cauchy problems for elliptic

partial differential equations. This method works by changing the order of

derivative of the partial differential operator in a way to obtain a well-posed

problem [56].

2.3.3.2 Stochastic or Bayesian Inversion

In this class of solution strategies, all variables are considered to be random in order

to present every uncertainty. Thus one is interested in probability density functions.

41

This function is associated with the unknowns and with the data of the problem from

which one looks for the characteristic values: average values, correlations, value of

the largest probability [57].

An important thing to note is that often during the solution process an inverse

problem boils down to a numerical optimization problem in which a numerically com-

puted solution of the inverse problem is compared to some real experimental data in

the form of a cost function. Thus the study of the numerical optimization techniques

is also at the heart of solving an inverse problem [58]. There are various numerical

optimization techniques like linear or non-linear least squares, maximum plausibility,

Monte-Carlo method, linear programming, simulated annealing, genetic or evolu-

tionary algorithms, optimal control methods etc. Detailed study of the numerical

optimization techniques can be found in standard literature [12,57,58].

2.3.4 Ill-posedness of Boundary Data Estimation Problems

for Laplace Equation

In general, source and boundary data estimation problems for elliptic PDEs are highly

sensitive to noise and are ill-posed in the sense of stability. Two simple examples are

presented in the following to highlight this fact.

2.3.4.1 On the Annulus Domain Ω1 [25]

Let Ω1 be the annulusz = reιθ : r0 < r < 1, 0 ≤ θ < 2π

such that,

4u = 0 0 ≤ θ < 2π, r0 < r < 1,

u(eiθ) = γ(eiθ) 0 ≤ θ < 2π,

∂ru(eiθ) = 0 0 ≤ θ < 2π,

(2.25)

42

for analytical solution, let us use the Fourier series expansion of solution u and the

boundary values u|r=1 = γ,

u(r, θ) =∞∑

k=−∞

uk(r)eikθ, and, γ(eiθ) =

∞∑k=−∞

γkeikθ (2.26)

Writing the Laplacian in polar co-ordinates, a family of ordinarily differential equa-

tions is obtained, and for the sequence of Fourier coefficients uk,

r∂

∂r

(r∂uk∂r

)− k2uk = 0, r0 < r < 1,

uk(1) = γk,

∂∂ruk(1) = 0.

(2.27)

It is easy to see that the solution of (2.25) can be written as Fourier series,

u(reiθ) =∞∑

k=−∞

(rk + r−k

2

)γke

ikθ, (2.28)

since r0 < r < 1, the factor r−k grows exponentially as k → ∞, similarly rk grows

exponentially as k → −∞. This implies the analytical solution is highly sensitive

to noise or high frequency components in the available data. The solution is highly

ill-posed in terms of Hadamard’s stability criteria.

2.3.4.2 On the Rectangular Domain Ω2 [59]

Let Ω2 be a rectangular domain of base length π, such that,

4u = 0 x ∈ [0, π], y > 0,

u(x, 0) = 0 x ∈ [0, π],

∂u

∂y(x, 0) = α sin(nx) x ∈ [0, π].

(2.29)

43

Above Cauchy problem has a solution,

u(x, y) =α

nsinh(ny) sin(nx), (2.30)

for any ε > 0, c > 0 and y > 0 it is possible to find α and n such that,

‖α sin(nx)‖ < ε and∥∥∥αn

sinh(ny) sin(nx)∥∥∥ > c.

Clearly the above solution does not continuously follow the data.

2.4 Conclusion

After a quick literature review of observation theory for the finite- and infinite-

dimensional linear dynamical systems, along with the notion of ill-posedness, we are

all set to develop estimation algorithms for infinite-dimensional linear steady-state

systems modeled by the elliptic equations. In the following chapter, different from

traditional optimization based techniques, we develop iterative observer and robust

iterative observer algorithm to solve ill-posed boundary data estimation problems for

the steady-state Laplace equation.

44

Chapter 3

Iterative Observers for Boundary Estimation Problems for

Laplace Equation

Every solution to every problem is simple. It’s the distance between the

two where the mystery lies. (—Derek Landy, Skulduggery Pleasant)

3.1 Introduction

In this chapter, iterative observer algorithms are introduced for boundary estimation

problems for the steady-state Laplace equation. The mathematical problems are in-

troduced on regular-shaped domains with available data on the parts of the boundary.

First, an iterative observer algorithm is developed that sweeps over the whole domain

using space as time-like. The theoretical framework using semigroup theory is devel-

oped. Then, a robust optimal iterative algorithm is presented to tackle the ill-posed

boundary estimation problems with noisy boundary data. Two types of boundaries,

rectangular and annulus, are considered to develop the framework. Further types of

domains are considered in the later on chapters.

The boundary estimation problem for Laplace equation, also known as the Cauchy

problem for the Laplace equation has been a fundamental problem of interest in many

diverse areas of science and engineering. For example non-destructive testing appli-

cations in mechanics, where we are interested in finding inside cracks from boundary

measurements [27]. Biomedical applications in finding the actual heart potential

from electrocardiogram (ECG) data collected on the body torso. Finding the ac-

tual heart potential is vital to understand the functionality of heart valves [28, 29].

45

Readers may refer to a number of existing numerical solution techniques for elliptic

Cauchy problems for further understanding of nature of the mathematical problem,

e.g. [24, 25,26,41,42,43,44,45].

3.2 Iterative Observer Design for Boundary Estimation

In this section, first, a boundary estimation problem for steady-state elliptic Laplace

equation is mathematically formulated, on a rectangle domain, and then an iterative

observer algorithm is presented to solve this problem. The theoretical results are

presented using functional analysis framework.

3.2.1 Problem Statment on a Rectangular Domain

Let Ω2 be a rectangle domain in R2 with ΓB,ΓT ,ΓL,ΓR be top, bottom, left and right

boundaries respectively as shown in Figure 3.1 such that Ω2 = Ω2∪ΓB∪ΓT ∪ΓL∪ΓR,

Ω2 = (0, a) × (0, b) and ΓB ∩ ΓT ∩ ΓL ∩ ΓR = ∅. Boundary estimation problem for

Laplace equation is defined as,

Find u(x) on ΓB:

4u =∂2u

∂x2+∂2u

∂y2= 0 in Ω2,

u = f(x) on ΓT ,

∂u

∂n= g(x) on ΓT ,

(3.1)

with homogeneous Dirichlet or Neumann side boundaries, f and g are given suffi-

ciently smooth and∂

∂nrepresents the normal derivative to the top boundary ΓT .

In the following subsection, an iterative observer is developed to solve boundary

estimation problem for Laplace equation posed on the rectangular domain. The

theoretical concepts can be extended to annulus domain as well. However, to avoid

repetition, only the rectangular case is discussed.

46

Ω2 = (0, a)× (0, b)

ΓB

ΓT

ΓL

(0, 0)

(0, b)

(a, 0)

(a, b)

ΓR

Figure 3.1: Rectangular domain Ω2.

3.2.2 Notations and Definitions

In this section, let X be a Hilbert space with inner product 〈., .〉 and corresponding

norm ‖.‖. If X and Y are two Hilbert spaces then L(X, Y ) denotes the space of linear

operators from X to Y with induced norm. Further L(X) = L(X,X). Let an infinite

dimensional linear dynamical system be presented in state space representation as,

ξ(x) = Aξ(x); y(x) = Cξ(x); (3.2)

such that “ ˙ ” represents partial derivative with respect to time-like variable x, ξ be

a state vector, A : D(A) → X be the state operator matrix, C ∈ L(X, Y ) be the

observation operator with observation space Y .

Definition 11. Let x be a time-like variable, a family T = (Tx)x≥0 of operators in

L(X) defines a strongly continuous semigroup (C0-Semigroup) on X if,

1. T0 = I, (identity property)

2. Tx+w = TxTw, ∀x,w ≥ 0, (semigroup property)

3. limx→0+ ‖Txξ − ξ‖ = 0 ∀ξ ∈ X. (strong continuity property)

Definition 12. Let x be a time-like variable, C ∈ L(X, Y ) be the observation oper-

ator. For all x > 0, let Ψx ∈ L(X,L2 ([0, x];Y )) be the output map operator for the

47

system (3.2) such that,

(Ψxξ(0)) (x) =

CTxξ(0) ∀ x ∈ [0, x],

0 ∀ x > x.

(3.3)

Definition 13. Let time-like variable x > 0 and T be the strongly continuous semi-

group on space X with the generator A : D(A) → X and C ∈ L(X, Y ) be the

observation operator. The pair (C,A) is exactly observable in x if Ψx is bounded

from below.

The above definition of exact observability of the pair (C,A) is equivalent to the fact

that there exists kx > 0 such that,

∫ x

0

‖Ψxξ(0)‖2 dx ≥ k2x ‖ξ(0)‖2 ∀ ξ(0) ∈ X. (3.4)

Definition 14. Pair (C,A) as defined above is final state observable in time-like

interval x if there exists a constant kx > 0 such that,

‖Ψxξ(0)‖ ≥ kx‖Txξ(0)‖ ∀ ξ(0) ∈ X. (3.5)

Note 1: For x → 0 and given that k0 > 0, then using strong continuity of operator

semgigroup T we can see that definitions in equation (3.4) and (3.5) converge.

Lumer-Phillips Theorem:

Let A : D(A) → X be an unbounded operator on a Hilbert space X. Then the

following two assertions are equivalent.

1. A is maximally dissipative.

2. A is the generator of a contraction semigroup (Tx)x≥0, i.e. ‖Tx‖ ≤ 1 for all

x > 0.

48

Definition 15. Let x ∈ [c, d) for all c, d ∈ R and d > c then xm, for all m ∈ Z =

0 ∪ Z+, represents x over mth iteration over the interval [c, d) .

The idea of iteration is also illustrated in Figure. 3.2 for a rectangular domain Ω2.

Further, without loss of generality, let s ∈ [0, π/4], A : D(A)→ X be an unbounded

differential operator matrix given as,

A =

0 1

− ∂2

∂s20

, (3.6)

such that,

X = H1ΓT

(0,π

4

)× L2

(0,π

4

), (3.7)

D(A) =

[f ∈ H2

(0,π

4

)∩H1

ΓT

(0,π

4

)| dfds

(0) = c2

]×H1

ΓT

(0,π

4

), (3.8)

where,

H1ΓT

(0,π

4

)=f ∈ H1

(0,π

4

)| f(0) = c1

, (3.9)

and c1, c2 are constants (coming from Cauchy data at a particular point on ΓT ) and

X is a Hilbert space with scalar product given by,

⟨ q1

q2

,

p1

p2

⟩ =

∫ π4

0

dq1

ds(s)

dp1

ds(s)ds

+

∫ π4

0

q1(s)p1(s)ds+

∫ π4

0

q2(s)p2(s)ds. (3.10)

It can be seen that D(A∞) is dense in X.

Note 2: The state operator matrix A has two positive definite operators on anti-

diagonal, this indicates that A has both positive and negative eigenvalues. As given

above, the state operator matrix does not generate a strongly continuous semigroup.

The existence of exponential of such an operator, under certain conditions, is studied

49

a1

a2

a3

a4

a1

a2

a3

a4

Figure 3.2: Idea of iterations over rectangular domain Ω2.

in the following subsections.

3.2.2.1 Change of Variables

We propose to write down the Laplace equation in rectangular coordinates as given in

system (5.1) as a first order state equation by introducing two new auxiliary variables

ξ1, ξ2 as follows, ξ1(x, y) = u(x, y),

ξ2(x, y) =∂u

∂x,

(3.11)

and the resulting equation can now be written as,

∂ξ

∂x= Aξ, (3.12)

where,

ξ =

ξ1(x, y)

ξ2(x, y)

, A =

0 1

− ∂2

∂y20

. (3.13)

Using the new state variables ξ1 and ξ2 problem (5.1) can be written in an equivalent

form as,

50

Find ξ1(x, y) on ΓB:

∂ξ

∂x= Aξ in Ω2,

Cξ(x) = ξ1(x) = f(x) on ΓT ,

∂ξ1

∂y= g(x) on ΓT ,

(3.14)

with homogeneous Dirichlet/Neumann side boundaries.

3.2.3 Observer Design

Boundary value problem as given in system of equations (3.14) has a first-order state

equation in variable x and overdetermined data is available on ΓT . Before the intro-

duction of iterative observer equations, let us assume that left-hand boundary ΓL is

connected to right-hand boundary ΓR to have the notion of infinite time-like variable

x over the rectangular domain. The reason for having such an assumption is that

we are trying to develop an observer using space as time-like and hoping that this

observer will converge asymptotically in variable x. Let m be a non-negative integer

index of iteration over the domain Ω in horizontal direction. Let xm, as given in Def-

inition 15, represents x ∈ [0, a) for the m-th iteration over the interval [0, a). After

introducing iteration index m, now an observer-like algorithm can be developed as

follows,

Main result

Theorem 3. For consistent Cauchy data, boundary value problem given in (3.15)

asymptotically (m = 1, · · · ,∞) converges to the true solution of boundary value

51

problem (3.14).

∂

∂xξ(xm, y) = Aξ(xm, y)−KC(ξ(xm, y)− ξ) in Ω2,

∂

∂yξ1(xm, y) = g(x) on ΓT ,(∂2

∂x2+

∂2

∂y2

)ξ1(xm, y) = −KC(ξ(xm, y)− ξ) on ΓB,

ξ(xm, y) |initial= ξ(xm−1, y) in Ω2,

(3.15)

where “ ˆ ” represents estimated quantity and ξ(xm, y) |initial represents a bounded

estimate over the whole domain Ω2 at the start of m-th iteration. The algorithm starts

at index m = 1, which represents first iteration. ξ(x0, y) is initial guess at the start of

the first iteration over the whole domain Ω2. Any bounded initial guess ξ(x0, y) can

be chosen. For each subsequent iteration, the result of the previous iteration is used

as initial estimate as given in the last equation in (3.15). Third equation in (3.15)

is the assumption that Laplace equation is valid on the bottom boundary and this

provides necessary boundary condition required on ΓB. C is the observation operator

such that Cξ = ξ1 |ΓT . K is the correction operator chosen in such a way that state

estimation error on ΓT given by (Cξ(xm, y) − Cξ) converges to zero asymptotically

(m = 1, · · · ,∞).

3.2.4 Preliminary Analysis

Before moving to the proof of theorem (9), we note that the solution of the first-order

equation in system (3.14) leads to the concept of semigroup generated by unbounded

differential operator matrix A. We study the exponential of A using the functional

analysis framework from section 3.2.2.

52

3.2.4.1 Existence of Exponential of A

Theorem 4. Let n ∈ Z? (set of non zero integers), for A : D(A) → X (as given in

(3.13), (3.7) and (3.8)) there exists an infinite set of orthonormal eigenvectors (Φn)

and corresponding eigenvalues (λn). Furthermore, A generates a strongly continu-

ous semigroup for vectors

p1

p2

∈ X, if and only if, the decay rate of sequence

⟨ p1

p2

,Φn

⟩is greater than the growth rate of sequence eλnx for all n ∈ Z?.

Proof. Let,

Φn(y) = ρn

αnϕn(y)

βnϕn(y)

, (3.16)

be the orthonormal set of eigenvectors of operator A and λn be the eigenvalues such

that,

AΦn = λnΦn, (3.17) βnϕn

− ∂2

∂y2(αnϕn)

= λn

αnϕn

βnϕn

.

Assuming that αn, βn do not depend on y, second equation above suggests that we

are interested in finding the eigenfunctions of Laplacian operator−∂2

∂y2. This signifies

that unknown eigenfunctions ϕn ∈ C∞. Solving two equations in (3.17) gives,

λn =βnαn, (3.18)

ϕn(y) = C1 cos (λny) , (3.19)

where αn and βn depend on n. C1 and λn are chosen such that ϕn(y) in (3.19) forms

an orthonormal basis in L2(0, π

4

), with C1 = −

√8π, αn = 1 and βn = λn = 6 − 8n.

53

Finally an orthonormal set of eigenvectors can be formed in X with respect to norm

defined by (3.10) as,

Φn(y) = ρnφn(y) = ρn

αnϕn(y)

βnϕn(y)

, (3.20)

where, |ρn| =1∣∣√2βn∣∣ > 0, is a normalization factor. Now let us try to write semigroup

generated by operator matrix A can be written as an infinite series,

∑n∈Z?

eλnx

⟨ p1(y)

p2(y)

,Φn(y)

⟩Φn(y), ∀

p1

p2

∈ X. (3.21)

For x = 0 the above infinite series is clearly convergent, whereas for x → 0+ the

limit does not exist. Further, we note that above series expression (3.21) satisfies

identity and semigroup properties as given in Definition 11. However, it lacks strong

continuity, except if we assume that the projection terms in angle brackets above

decay faster than the growth rate of eλnx. This condition true for a wide range of

analytical functions that have a finite number of non-zero projections on the basis Φn.

This also reveals a historical fact about solving Cauchy problems for steady-state heat

equation that unique and stable solutions does not exist for non-smooth data [54].

Let the Hilbert space X, as given in equations (3.7) and (3.10), be composed of two

mutually exclusive parts as

X = X1 ⊕X2, (3.22)

where X1 satisfy conditions as stated above such that A forms a strongly continuous

semigroup and X1 and X2 both make the full space X. Thus with this additional

smoothness assumption equation (3.23) represents the strongly continuous semigroup

54

generated by operator matrix A.

Tx

p1(y)

p2(y)

=∑n∈Z?

eλnx

⟨ p1(y)

p2(y)

,Φn(y)

⟩Φn(y),∀

p1

p2

∈ X1.

This implies,

Tx

p1(y)

p2(y)

=∑n∈Z?

eλnxρn

(αn

⟨dp1

dy,dϕndy

⟩L2(0,π

4 )+ αn 〈p1, ϕn〉L2(0,π

4 )

+βn 〈p2, ϕn〉L2(0,π4 )

)Φn, ∀

p1

p2

∈ X1.

(3.23)

3.2.4.2 System Observability

Proposition 1. Let T be the strongly continuous semigroup generated by operator

matrix A under the assumptions as given in theorem 4. For any arbitrarily small

ε > 0 such that if |x− x| < ε, the pair (C,A) is final state observable (and further

exactly observable using Note 1 from section 3.2.2) in time-like interval |x− x| > 0

at a particular x, where C ∈ L(X, Y ) and Y = R.

Proof. Let ξ(0) ∈ X1 be the initial guess at x = 0, given by,

ξ(0) =

ξ1(0)

ξ2(0)

=

p1(y)

p2(y)

. (3.24)

Φn(y) for n ∈ Z? be an orthonormal basis in X. Let us first prove the final state

observability condition for a general mode Φn′ with corresponding eigenvalue λn′ as

follows,

55

For all Φn′ ∈ X and n′ ∈ Z?,

‖TxΦn‖X =

∥∥∥∥∥∑n∈Z?

eλnx 〈Φn′ ,Φn〉Φn

∥∥∥∥∥X

,

= eλnx ‖Φn‖X ,

= eλnx, (3.25)

also,

‖CTxΦn‖Y =

∥∥∥∥∥∑n∈Z?

eλnx 〈Φn′ ,Φn〉 CΦn

∥∥∥∥∥Y

,

= eλnx ‖CΦn‖Y ,

= eλnx|ρn|. (3.26)

Comparing equations (3.25) and (3.26) implies,

‖CTxΦn‖Y ≥ k?1 ‖TxΦn‖X , (3.27)

where k?1 > 0, if and only if,

k?1 ≤ |ρn|, (3.28)

for a particular choice of Φn there always exists k?1 such that final state observability

condition (3.5) is satisfied.

C ∈ L(X, Y ) is a linear boundary observation operator. Now let ξ(0) =∑

n∈Z? γnΦn

where γn are projection terms whose decay rate is greater than the growth rate of

eλnx with λn as eigenvalues of A corresponding to eigenvectors Φn. Clearly∑

n∈Z? γn

and∑

n∈Z? ρn are bounded from above, hence,

‖CTxξ(0)‖Y ≥ k?2 ‖Txξ(0)‖X ∀ ξ(0) ∈ X1, (3.29)

where k?1, k?2 both are independent of x. Further, using Note 1, for arbitrarily small

56

time-like interval ε pair (C,A) is exactly observable.

3.2.5 Convergence Analysis

After establishing the concept of exponential of A, under certain conditions, and the

fact that pair (C,A) is final state and exact observable, we are all set to prove the

main result.

3.2.5.1 Proof of the Main Result

Proof. Let us define state estimation error e(xm, y) as the difference of true state

ξ(x, y) from the one estimated ξ(xm, y),

e = ξ − ξ =

e1(xm, y)

e2(xm, y)

=

ξ1(xm, y)− ξ1(x, y)

ξ2(xm, y)− ξ2(x, y)

. (3.30)

Solution of the boundary value problem (3.14) with consistent boundary data provides

u = ξ1 over the whole domain Ω2. Boundary value problem for the state estimation

error can be given by subtracting problem (3.14) from the state observer equations

(3.15) as follows,

For m ≥ 1, find e(xm, y) = (ξ(xm, y)− ξ(x, y)) ∈ Ω2:

∂

∂xe(xm, y) = (A−KC)e(xm, y) in Ω2,

∂

∂ye1(xm, y) = 0 on ΓT ,(∂2

∂x2+

∂2

∂y2

)(ξ1(xm)− h(x)

)= −KCe(xm) on ΓB,

e(xm, y) |initial= e(xm−1, y) in Ω2.

(3.31)

57

Here h(x) is the true analytical solution on ΓB using consistent Cauchy data. Further

using the assumption that Laplace equation is valid on ΓB, the above system of error

dynamic equation can also be written in an equivalent form as,

For m ≥ 1, find e(xm, y) ∈ Ω2:

∂

∂xe(xm, y) = (A−KC)e(xm, y) in Ω2\ΓT ,

∂

∂ye(xm, y) = 0 on ΓT ,

e(xm, y) |initial= e(xm−1, y) in Ω2.

(3.32)

First equation in (3.32) is a system of ODEs in variable x and solution to this system

has to do with the exponential or the semigroup generated by operator matrixA−KC.

Let us denote this semigroup with S. Then solution to above system of ODEs can be

written as,

e(xm, .) = Sxm e(x0, .) m ≥ 1, (3.33)

for a particular iteration index m, xm is x ∈ [0, a) over m-th iteration. Given that un-

der certain conditions semigroup generated by A is strongly continuous, the observer

gain K can be chosen in a way that A−KC is dissipative. Then Sxm will decay expo-

nentially and state estimation error e(xm, .), for a number of iterations over the whole

domain, asymptotically converges to zero for any bounded initial value of e(x0, .).

3.2.5.2 Existence of Observer Gain K

Let the Hilbert space X as given in equations (3.7) and (3.10) be composed of two

mutually exclusive parts as,

X = X1 ⊕X2, (3.34)

58

where X1 satisfy conditions as stated in Theorem 4 such that A forms a strongly

continuous semigroup, and X1 and X2 both make the full space X. Following theorem

provides conditions on the existence of operator gain K.

Theorem 5. Under conditions as stated in theorem 4, let A as given in equation

(3.13) be the generator of a strongly continuous semigroup, C ∈ L(X, Y ) be an

observation operator and Y = R), then the following assertions are equivalent.

1. There exists a positive definite self-adjoint operator product KC ∈ L(X) where

K ∈ L(Y,X) such that A−KC generates a maximally dissipative semigroup.

2. There exists arbitrarily small ε > 0 such that if ‖x− x‖ < ε then pair (C,A) is

exactly observable in time-like interval ε.

Proof. Given self-adjoint positive definite operator product KC ∈ L(X), let us denote

by S and T the semigroups generated, under certain conditions, by A − KC and A

respectively.

1⇒ 2 :

Assume S is dissipative, let us show the following observability inequality, that is,

there exists x, kx > 0 such that,

∫ x

0

‖CTxe0‖2 dx ≥ k2x ‖e0‖2 ∀e0 ∈ X1, (3.35)

A is densely defined so the above inequality is enough to prove exact observability

for e0 ∈ D(A). Given e0 ∈ D(A), e(x) = Sxe0 presents the unique solution of,

∂e

∂x= (A−KC)e(x),

e(0) = e0.

(3.36)

59

Multiplying first equation in (3.36) by e(x),

1

2

d

dx‖e(x)‖2 = Re

⟨∂e

∂x, e(x)

⟩,

= Re 〈(A−KC)e(x), e(x)〉 , (3.37)

in this part we assume that A−KC is m-dissipative,

d

dx‖e(x)‖2 ≤ 0, (3.38)

Let e(x) = γ(x) + ζ(x) such that γ = Txe0 is the solution of,

∂γ

∂x= Aγ(x),

γ(0) = e0,

(3.39)

and ζ is the solution of,

∂ζ

∂x= Aζ(x)−KCe(x),

ζ(0) = 0.

(3.40)

Further we have that KC is positive definite,

0 ≤ Re 〈KCγ,KCγ〉 ≤ ‖KCγ(x)‖2X . (3.41)

Combining equations (3.38) and (3.41),

d

dx‖e(x)‖2 ≤ ‖KCγ(x)‖2

X , (3.42)

60

integrating both sides,

1

2

∫ x

0

d

dx‖e(x)‖2 dx ≤ 1

2

∫ x

0

‖KCγ(x)‖2X dx,

2(‖e(x)‖2 − ‖e0‖2) ≤ ∫ x

0

‖KCγ(x)‖2X dx,

(3.43)

finally we have,

k ‖e(x)‖2X1≤

∫ x

0

‖Cγ(x)‖2Y dx, (3.44)

where k =2

‖K‖2> 0 is independent of x. For arbitrarily small time-like interval

x− 0 = x = ε, above inequality is same as observability inequality (3.35).

2⇒ 1 :

We have e(x) = γ(x) + ζ(x), where γ is the solution of open loop system (3.39)

and e(x) is the solution of closed loop feedback system (3.36), we have that,

〈(A−KC)γ(x), γ(x)〉 ≥ 〈(A−KC)e(x), e(x)〉 , (3.45)

multiplying with −1 and using inequality (3.37),

〈(KC)γ(x), γ(x)〉 ≤ −1

2

d

dx‖e(x)‖2 + 〈(A)γ(x), γ(x)〉 , (3.46)

integrating both sides from 0 to x and with a positive α > 1 such that,

∫ x

0

〈(KC)γ(x), γ(x)〉 dx ≤ α(‖e0‖2 − ‖e(x)‖2)+

∫ x

0

〈Aγ(x), γ(x)〉 dx, (3.47)

61

Further we can write,

Re 〈Aγ, γ〉 = Re

⟨∂γ

∂x, γ

⟩,

=1

2

d

dx‖e(x)− ζ(x)‖2 ,

≤ 1

2

d

dx‖e(x)‖2 +

1

2

d

dx‖ζ(x)‖2 ,

≤ d

dx‖e(x)‖2 +

1

2

d

dx‖γ(x)‖2 , (3.48)

given that, under certain conditions, A generates a strongly continuous semigroup,

there exists β > 1 such that,

Re 〈Aγ, γ〉 ≤ βd

dx‖e(x)‖2 , (3.49)

integrating both sides,

∫ x

0

Re 〈Aγ, γ〉 dx ≤ −β(‖e0‖2 − ‖e(x)‖2) , (3.50)

combining above inequality with (3.51),

∫ x

0

〈(KC)γ(x), γ(x)〉 dx ≤ (α− β)(‖e0‖2 − ‖e(x)‖2) , (3.51)

α and β can be chosen appropriately large, let us take α− β = 2, we have

∫ x

0

〈KCγ(x), γ(x)〉 dx ≤ 2(‖e0‖2 − ‖e(x)‖2) ,∫ x

0

〈Cγ(x), Cγ(x)〉 dx ≤ 2

‖K‖(‖e0‖2 − ‖e(x)‖2) ,∫ x

0

‖Cγ(x)‖2Y dx ≤

2

‖K‖(‖e0‖2 − ‖e(x)‖2) , (3.52)

62

now using observability inequality (3.35) we have,

k2 ‖e0‖2 ≤ 2

‖K‖(‖e0‖2 − ‖e(x)‖2) ,

‖e(x)‖2 ≤(

1− 1

‖K‖

)‖e0‖2 , (3.53)

where

(1− 1

‖K‖

)< 1 if ‖K‖ > 1.

In the following section, the iterative observer is implemented numerically us-

ing fictitious points on the estimated solution boundary. Numerical results are also

presented.

3.2.6 Numerical Implementation

For numerical implementation, first order state equation given in (3.14) can be dis-

cretized in variable x using forward Euler as follows,

ξ = Aξ,ξn+1 − ξn

4x= Aξn,

ξn+1 = (I + (4x)A)ξn, (3.54)

here I is the identity matrix, n is the discrete index for variable x and 4x is the step

size along x after discretization. Further, equation (3.54) is discretized in variable y

using second-order accurate centered finite difference schemes to discretize the first-

and second-order derivative terms. The Cauchy data is available on the top bound-

ary, however, for the bottom boundary ΓB there is no data available and we assume

that the Laplace equation is valid on this boundary as given in problem (5.71). Nu-

merically, this condition can be implemented using fictitious points along the inner

boundary, as explained in following section.

63

3.2.6.1 Boundary Condition on ΓB

As stated above, a first-order state equation can be thought of as an ODE with respect

to variable x. Solution of this ODE is the state ξ over the whole vertical line, that is,

(y |ΓB , y |ΓT ). This can be thought of as a 2D Laplace equation that has been split

into a series of 1D state equations. To solve this 1D state equation in variable x, an

initial condition over the whole interval (y |ΓB , y |ΓT ) and boundary conditions on ΓB

and ΓT are required. Any initial guess can be chosen as (A−KC) will be dissipative

and any initial guess dies out. Two bounday conditions on ΓT are available, that is,

the measurement data and the available Neumann boundary data on ΓT . However,

on ΓB, it is assumed that Laplace equation is satisfied. That is,

∂2u

∂x2= −∂

2u

∂y2on ΓB. (3.55)

Equation (3.55) contains a second-order derivative in variable y. To discretize this

second derivative using a second-order accurate centered finite difference discretiza-

tion scheme on ΓB, there needs to be a fictitious point further outside the boundary

ΓB as shown in figure 3.3 [60].

i=0;n

i=1;n

i=2;n

i=0;n− 1

i=1;n− 1

i=2;n− 1

i=0;n + 1

i=1;n + 1

i=2;n + 1

n

i

ΓB

ΓT

Figure 3.3: Domain Ω2 after discretization and fictitious points outside ΓB, indexi = 0 represents fictitious points.

The second equation in (3.54), after full discretization, can be written as,

(ξ2)n+1i − (ξ2)ni4x

= −(ξ1)ni+1 − 2(ξ1)ni + (ξ1)ni−1

(4y)2, (3.56)

64

here i is the discrete index and (4y) is the step size along variable y such that i = 1

on the bottom boundary ΓB. Equation (3.56) on the bottom boundary ΓB can be

written as,

(ξ2)n+11 − (ξ2)n14x

= −(ξ1)n2 − 2(ξ1)n1 + (ξ1)n0(4y)2

, (3.57)

index i = 0 represents fictitious point and taking out this fictitious point gives,

(ξ1)n0 = 2(ξ1)n1 − (ξ1)n2 −(4y)2

4x

(ξ2)n+11 − (ξ2)n1

. (3.58)

(ξ1)n1 , (ξ1)n2 , (ξ2)n1 and (ξ2)n+11 are given by the initial guess of the states over the whole

domain. The algorithm is run for a number of iterations along x by using the solution

of the previous iteration as a guess for the next until the final convergence is achieved.

In the following subsection observer is presented in semi-discrete form and fictitious

points method is used to tackle the boundary condition on ΓB.

3.2.6.2 Observer in Semi-Discrete Form

In the following, state observer presented in system of equations (5.71) is discretized

only in variable x for simplicity.

ξn+1,m = (I + (4x)A)ξn,m −KC(ξn,m − ξn) in Ω2,

∂

∂yξn,m1 = gn(x) on ΓT ,

1

24x

(ξn+1,m

1 − ξn−1,m1

)= − ∂2

∂y2ξn,m1

−KC(ξn,m − ξn) on ΓB,

ξn,m |initial= ξn,m−1 in Ω2,

(3.59)

again, here ˆ represents estimated quantity and m is the index of iteration over the

rectangular domain. ξm,n |initial represents estimate for particular value of index n at

the start of mth iteration. The algorithm starts at m = 1 and index m = 0 represents

65

the raw data over the whole mesh before start of the algorithm. At the start of the

algorithm, any initial guess can be chosen over the whole domain and then solution

of the previous iteration is used as a guess for subsequent iterations. Fictitious points

are computed using formula given in (3.58) and are used in third equation in (3.59)

to discretize the second-order derivative with respect to y on ΓB. An important point

to note here is that the true Neumann boundary condition is applied on the outer

boundary and operators A,K and C are continuous in variable y.

3.2.6.3 Algorithm Step-by-Step

• Step 1 : Initialize mesh over the whole domain Ω2 with ξm=0 = ξ0.

• Step 2 : For m = 1, start at a particular value of x,

– Compute the fictitious point value (ξ1)n,m=10 for particular value of n using

equation (3.58).

– Solve system of equations (3.59) to find estimate ξn+1,m over a particular

vertical line.

– Repeat the process of finding fictitious point from equation (3.58) and

solving system of equations (3.59) for all n until one iteration on interval

of length a on the rectangular domain shown in Figure. 3.1 is complete.

• Step 3 : Repeat Step 2 for m ≥ 2 using result of (m− 1)th iteration as a guess

for mth iteration until convergence is achieved. That is, ‖ξ1 − ξm1 ‖ΓT < ε.

66

3.2.6.4 State Estimation Error and Computation of the Ob-

server Gain

State error boundary value problem in semi-discrete form can be written as,

For m ≥ 1, find en,m = (ξn,m − ξn) ∈ Ω2:

en+1,m = (I + (4x)A−KC) (ξn,m − ξn) in Ω2\ΓT ,

∂

∂yen,m1 =

∂

∂y

(ξn,m1 − ξn1

)= 0 on ΓT ,

en,m |initial= en,m−1 in Ω2.

(3.60)

Finally the state error difference equation after full discretization can be written as,

en+1,m = (I + (4x)A−KC) en,m for m ≥ 1, (3.61)

here A,K and C are discrete versions of operators A,K and C respectively and e is

the state estimation error after full discretization. Given (I + (4x)A) and observa-

tion matrix C, gain matrix K can be computed using Ackermann’s formula for pole

placement in Matlab such that eigenvalues of (I + (4x)A−KC) are inside the unit

circle on the complex plane [61].

3.2.7 Results and Simulations

For all numerical and analytical solutions in this section, a rectangle domain Ω =

(0, a)× (0, b) with a = 2π and b = 12

is considered. To validate the observer approach

a number of examples are presented as follows.

3.2.7.1 Example 1: Homogeneous Neumann Side Boundaries

Consider the boundary value problem in a rectangular domain with homogeneous

Neumann side boundaries as shown in Figure 3.4. This problem can be solved using

67

Ω2

u = cos(2x)

uy = 0

ux = 0 ux = 0

Figure 3.4: Two dimensional rectangle domain with homogeneous Neumann sideboundaries, Example 1.

separation of variables and solution is given as,

u(x, y) =cosh(4π(y − b)/a)

cosh(4πb/a)cos(4πx/a), (3.62)

To validate the observer-based approach this analytical solution given in (3.62) along

with homogeneous Neumann boundary condition is used as Cauchy data on the top

boundary ΓT . Using this Cauchy data state observer algorithm is run for a number

of iterations to recover the unknown boundary data on the bottom boundary ΓB.

Figure 3.5 shows the comparison of exact solution and the one recovered by using

Cauchy data on ΓB and observer algorithm.

0 1 2 3 4 5 6−1

−0.5

0

0.5

1

x−axis

Solu

tion u

Comparison of exact and observer solution on ΓB

Exact solution

Observer soluton

Figure 3.5: Comparison of exact and observer constructed solution on the bottomboundary ΓB.

68

3.2.7.2 Example 2: Homogeneous Dirichlet Side Boundaries

Ω2

u = sin(2x)

uy = 0

u = 0 u = 0

Figure 3.6: Two dimensional rectangle domain with homogeneous Dirichlet sideboundaries, Example 2.

Consider the boundary value problem with homogeneous Dirichlet side boundaries

as shown in Figure 3.6. Analytical solution is given as,

u(x, y) =cosh(4π(y − b)/a)

cosh(4πb/a)sin(4πx/a), (3.63)

Now using this analytical solution along with homogeneous Neumann boundary con-

dition on the top boundary ΓB, observer algorithm is run for a number of iterations to

recover the unknown Dirichlet boundary data on ΓB. Figure 3.7 shows the comparison

of the exact and observer constructed solution on ΓB.

0 1 2 3 4 5 6−1

−0.5

0

0.5

1

x−axis

Solu

tion u


Exact solution

Observer soluton


69

3.2.7.3 Example 3: Linear Combinations of Example 1 and

2

It is easy to see that any linear combination of above two example problems can be

solved using the observer-based technique. In other words any Dirichlet boundary

data on ΓB that can be represented as a trigonometric Fourier series can be recov-

ered using the observer-based approach given homogeneous Dirichlet, Neumann or

Robin kind of side boundaries. The requirement of such homogeneous side bound-

aries suggest that there are no active sources on the side boundaries which is indeed

the case for many applications like electrocardiography (ECG) where the objective is

to find a heart electric potential which is deep inside the body from the ECG data

available only on a limited part of body torso [28, 29]. The observer-based approach

is the preferred technique in cases where there is no information available on the side

boundaries. Figure 3.8 compares the exact solutions in different test cases to the one

obtained by using the observer. Numerical solution was achieved using homogeneous

Neumann boundaries on ΓL,ΓR and ΓT and non zero Dirichlet data on ΓB. The

observer solution was constructed using only the Cauchy data on ΓT .

The design of a dynamical systems inspired technique like observer for a steady-

state boundary value problem is challenging and the idea to use one of the space

variables as a time-like variable has not been considered before. Different from stan-

dard approaches to tackle this problem, an iterative observer is constructed in infinite

dimensional setting on a rectangle domain without introducing an extra time vari-

able. Laplace equation is presented as a first-order state equation with state operator

matrix. Conditions for the existence of exponential generated by this state operator

matrix are provided. Further the conditions for the existence of observer gain are

detailed. Numerical results are provided for different example test cases.

In the following section, first, it is established that boundary data estimation

problems for Laplace equation are highly sensitive to noise in the available boundary

70

0 1 2 3 4 5 6

−1.5

−1

−0.5

0

0.5

1

1.5

x−axis

Solu

tion u


Exact solution

Observer soluton

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

x−axis

Solu

tion u


Exact solution

Observer soluton

0 1 2 3 4 5 6

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

x−axis

So

lution u


Exact solution

Observer soluton

0 1 2 3 4 5 6

−1

−0.5

0

0.5

1

1.5

x−axis

So

lution u


Exact solution

Observer soluton


71

data. Further, to tackle such ill-posed problems, a robust iterative counterpart of the

iterative observer is developed.

3.3 Robust Iterative Algorithm for Boundary Estimation

In this section, a robust iterative observer is developed to solve ill-posed boundary

data estimation problem for Laplace equation, on an annulus domain, with noisy

boundary data. The problem is formulated on an annulus domain and an optimal

iterative algorithm is developed to solve the problem.

3.3.1 Problem Statement on an Annulus Domain

Let Ω1 be an annulus domain in R2 with two boundaries (Γin,Γout) and a hole as

shown in Figure 3.9 such that Ω1 = Ω1 ∪Γin ∪Γout and Γin ∩Γout = ∅. The boundary

estimation problem for Laplace equation in polar co-ordinates over Ω1 is given as,

Find u(θ) ∈ Γin :

4u = r2∂

2u

∂r2+ r

∂u

∂r+∂2u

∂θ2= 0 in Ω1,

∂u

∂r= g(θ) on Γout,

u(θ) = f(θ) on Γout,

(3.64)

where g = g + ω and f = f + υ represent perturbed Cauchy data with ω(η1, σ1)

and υ(η2, σ2) as additive white Gaussian noise with mean values η1, η2 and variances

σ1, σ2 respectively.

3.3.2 Problem Reformulation

In the subsection, we propose to rewrite this ill-posed boundary value problem in

state-space-like representation using one of the space variables as a time-like variable.

72

Ω1

Γin

Γout

Figure 3.9: Annulus domain Ω1 with inner boundary Γin and outer boundary Γout.

Let us introduce two auxiliary variables ξ1(r, θ) and ξ2(r, θ) such that,

ξ1(r, θ) = u(r, θ),

ξ2(r, θ) =∂u

∂θ.

(3.65)

Further, rewriting the Laplace equation as a first-order state equation gives,

ξ = Aξ, (3.66)

with,

ξ(r, θ) =

ξ1(r, θ)

ξ2(r, θ)

, (3.67)

A =

0 1

−r2 ∂2

∂r2− r ∂

∂r0

, (3.68)

where “ ˙ ” represents partial derivative with respect to θ. Now problem (3.64) can

be reformulated as,

73

Find ξ1(r, θ) ∈ Γin : ξ = Aξ in Ω1,

∂ξ1

∂r= g(r, θ) on Γout,

ξ1(r, θ) = f(r, θ) on Γout.

(3.69)

It is important to reemphasize that an analytical solution of the above problem exists

for smooth Cauchy data (f, g) only, and arbitrarily small noise in the Cauchy data

can destroy the solution [21,25].

In the following, a well-posed forward problem is presented. Solution of this well-

posed forward problem will be used to test the accuracy of the proposed optimal

algorithm.

Find ξ1 |Γout= f(r, θ) :

ξ = Aξ in Ω1,

∂ξ1

∂r= g(r, θ) on Γout,

ξ1(r, θ) = h(r, θ) on Γin,

(3.70)

where g and h are noise free smooth boundary conditions. Solution of the above

forward problem provides Cauchy data (f, g) on Γout. Now the inverse problem is

to estimate h(r, θ) using noise-free (f, g) and noisy (f , g) Cauchy data. In the

following sections problem (3.69) is discretized using finite difference discretization

and an optimal estimator is developed.

3.3.3 Derivation of Optimal MSE Minimizer Algorithm

Before proper algorithm formulation, let us discretize problem (3.69) using finite

difference discretization.

74

3.3.3.1 Numerical Discretization

Forward Euler-like approximation is used to discretize first derivative along θ as,

∂ξ

∂θ≈ 1

4θ[ξn+1 − ξn

], (3.71)

where n is discrete positive integer index for variable θ and 4θ is a small step size.

State equation, after discretization in variable θ, can be written as,

ξn+1 = (I + (4θ)A) ξn, (3.72)

here I is 2× 2 identity matrix. Second equation in (3.72) can be written as,

(ξ2)n+1 = (ξ2)n +4θ(−r2 ∂

2

∂r2− r ∂

∂r

)(ξ1)n , (3.73)

Next to discretize in variable r, following second-order accurate finite difference dis-

cretization approximations are used to discretize first and second order derivative in

above equation.

∂

∂r(ξ1)n ≈ 1

2(4r)[(ξ1)ni+1 − (ξ1)ni−1

], (3.74)

∂2

∂r2(ξ1)n ≈ 1

(4r)2

[(ξ1)ni+1 − 2 (ξ1)ni + (ξ1)ni−1

], (3.75)

i is the discrete index for variable r, i = 1 on Γin and (4r) is a reasonably small step

size along r. Now from (3.73),

1

4θ[(ξ2)n+1

i − (ξ2)ni]

=−r2

i

(4r)2

[(ξ1)ni+1 − 2(ξ1)ni + (ξ1)ni−1

]+−ri

2 (4r)[(ξ1)ni+1 − (ξ1)ni−1

],

(3.76)

75

rewriting above equation on inner boundary Γin, that is, at i = 1 gives,

1

4θ[(ξ2)n+1

1 − (ξ2)n1]

=−r2

1

(4r)2 [(ξ1)n2 − 2(ξ1)n1 + (ξ1)n0 ] +−r1

2 (4r)[(ξ1)n2 − (ξ1)n0 ] .

(3.77)

As state earlier, index i = 1 represents point on Γin and thus i = 0 represents fictitious

points outside domain Ω1 and close to inner boundary Γin, as shown in Figure 3.10.

Taking out fictitious points (ξ1)n0 from the above equation gives,

(ξ1)n0 =1

r21 −

4r2r1

[−(r2

1 +(4r)

2r1

)(ξ1)n2 + 2r2

1(ξ1)n1

]− 1

r21 −

4r2r1

[(4r)2

4θ

(ξ2)n+11 − (ξ2)n1

].

(3.78)

It is important to reemphasize here that our objective is to develop an optimal al-

gorithm which runs iteratively using θ as a time-like variable. Right-hand side of

the equation (3.73) only depends on the variable r. Thus fictitious point (ξ1)n0 for a

particular n can be found uniquely as it depends on (ξ2)n+11 . The idea of assuming

fictitious points on the inner boundary can be understood as an assumption that

Laplace equation is satisfied on the inner boundary. That is, we are trying to solve a

2D problem line by line assuming that,

∂2ξ1

∂θ2= −

(r∂ξ1

∂r+ r2∂

2ξ1

∂r2

)on Γin, (3.79)

which is an applicable boundary condition on Γin in the current state space-like

setup. At the start of the algorithm, (ξ1)n1 , (ξ1)n2 , (ξ2)n1 and (ξ2)n+11 come from the

initial guess over the domain Ω1. These fictitious points are used in optimal estimator

equations to tackle the boundary condition on the inner boundary, whereas Neumann

boundary condition g is used to tackle the boundary condition on outer boundary.

76

Γout

Γini = 0, n

i = 0, n + 1

i = 0, n− 1

n

i

Figure 3.10: Fictitious points close to inner boundary Γin, with index i = 0.

3.3.3.2 Optimal Estimator Derivation

After full numerical discretization the state equation can be written as,

ξn+1 = (I + (4θ)A) ξn in Ω1, (3.80)

along with Neumann boundary condition∂ξ1

∂r= g(r, θ) on Γout and boundary condi-

tion given by equation (3.79) on Γin. A is the discrete version of A and I is an identity

matrix. Both A and I are square matrices of the same size with dimension depending

on number of discretization points along r. Next Dirichlet boundary condition on

Γout can be written as a measurement or observation equation in discrete form as,

yn = Cξn + υn = fn + υn, (3.81)

where C is a discrete observation matrix such that Cξn = (ξ1)n |Γout , fn is the Dirichlet

boundary condition at θn = n × 4θ along with measurement noise υn. An impor-

tant condition to be satisfied for a system in state-space-like representation (3.80)

and (3.81) is the system observability. Observability is equivalent to the existence of

solution in linear dynamical theory [6, 62]. The Kalman rank condition for observ-

77

ability is given in Theorem 1 and also highlighted in the following. Observability is

the measure of how well the internal states can be recovered from the measurements

of output (observations). Let α be a positive integer constant, numerically system

(C,A′), with C as discrete observation matrix and A′ as discrete state operator matrix

of size α× α, is called observable if,

rank(T ) = rank

C

CA′

CA′2

...

CA′α−1

= α. (3.82)

In our case A′ = (I + (4θ)A). Let the covariance matrices of process and measure-

ment noise be stationary over θ and are given by,

Q = E[ωn(ωn)T

], (3.83)

R = E[υn(υn)T

], (3.84)

where Q is of the size of (I + (4θ)A) and R is a 1 × 1 matrix. Let ξn be the

estimate of true state ξn. The difference of estimated and true state can be written

as en =(ξn − ξn

). It is important to consider the ability of the estimator to predict

the states over a period of time-like variable θ, hence a feasible metric is the expected

value of positive definite error functional(ξn − ξn

)2

, given by,

ε(θ) = E[(en)2

]= E

[en(en)T

]= P n, (3.85)

where P n is the error covariance matrix at θ = n×4θ.

Proposition 1. Let pair (C,A′), with A′ = (I + (4θ)A), be observable as given

78

in equation (3.82) and n be a discrete non-negative integer index for variable θ. As

n → ∞ (that is, as θ → ∞, representing iterations over annulus domain), follow-

ing two step algorithm minimizes the mean square error functional given in equation

(3.85).

Prediction-step

¯ξn = A′ξn−1 in Ω1,

P n = A′P n−1A′T +Q,

¯ξn=0 = P n=0 = 0,

(3.86)

with Neumann boundary condition g on Γout and fictitious points

close to Γin given by equation (3.78), taking care of boundary

conditions.

Correction-step ξn =

¯ξn +Kn

(yn − C ¯

ξn),

Kn = P nCT (CP nCT +R)−1,

P n = (I −KnC)P n,

(3.87)

where ξn is an estimate of ξn,¯ξn is the prior-estimate of ξn, P n is the prior-estimate

of state error covariance matrix P n given by equation (3.85). Q and R as given in

equations (3.83) and (3.84) and Kn is the gain matrix.

Remark 1. Before going to the proof, it is important to remark that the idea of

iterations (n→∞) replaces the asymptotic time in standard Kalman filter algorithm.

79

Proof. Let us re-write equation (3.85) as,

P n = E[(ξn − ξn)(ξn − ξn)T

]. (3.88)

Let us assume the prior estimate of ξn is called¯ξn and was obtained by the knowledge

of the system (prediction). The correction equation can be written using the prior

estimate with measurement data as follows,

ξn =¯ξn +Kn

(yn − C ¯

ξn). (3.89)

Kn is the gain matrix which will be derived in a moment. The term(yn − C ¯

ξn)

is

the innovation or measurement residual. Substituting equation (3.81) into (3.89),

ξn =¯ξn +Kn

(Cξn + υn − C ¯

ξn). (3.90)

Now substituting equation (3.90) into (3.88) gives,

P n = E[ (

(I −KnC)(ξn − ¯ξn)−Knυn

) ((I −KnC)(ξn − ¯

ξn)−Knυn)T]

, (3.91)

taking the error of prior estimate(ξn − ¯

ξn)

as uncorrelated to measurement noise

gives,

P n = (I −KnC) E[(ξn − ¯

ξn)(ξn − ¯ξn)T

](I −KnC)T +Kn E

[υn(υn)T

](Kn)T ,

(3.92)

now substituting equations (3.84) and (3.88) into above equation yields,

P n = (I −KnC)P n(I −KnC)T +KnR(Kn)T . (3.93)

80

Above equation is the error covariance update equation and the diagonal contains

mean-square error as shown,

P nn =

E[en−1(en−1)T ] E[en(en−1)T ] E[en+1(en−1)T ]

E[en−1(en)T ] E[en(en)T ] E[en+1(en)T ]

E[en−1(en+1)T ] E[en(en+1)T ] E[en+1(en+1)T ]

, (3.94)

the trace of above matrix is the mean-square error and it is to be minimized with

respect to Kn. Rewriting equation (3.93) gives,

P n = P n −KnCP n − P nCT (Kn)T +Kn(CP nCT +R)(Kn)T , (3.95)

using the fact that trace of a matrix is equal to trace of its transpose, it can be seen

that,

T [P n] = T [P n]− 2T [KnCP n] + T [Kn(CP nCT +R)(Kn)T ], (3.96)

where T [P n] is the trace of covariance matrix P n. Differentiating with respect to Kn,

d

dKnT [P n] = −2CP n + 2Kn(CP nCT +R), (3.97)

setting the derivative equal to zero gives,

(CP n)T = Kn(CP nCT +R), (3.98)

solving for Kn gives,

Kn = P nCT (CP nCT +R)−1. (3.99)

Above equation is the optimal estimator gain equation, which minimizes the mean

81

square state estimation error. Substituting above equation into equation (3.93) yields,

P n = P n − P nCT (CP nCT +R)−1CP n,

= P n −KnCP n,

= (I −KnC)P n, (3.100)

which is the update equation for state error covariance matrix with optimal gain

Kn. Equations (3.89), (3.99) and (3.100) develop an estimate of variable xn. State

prediction is achieved using state equation,

¯ξn+1 = A′ξn, (3.101)

along with Neumann boundary condition g on Γout and equation (3.79) on Γin. To

complete the recursion it is important to find an equation which projects state error

covariance matrix into next θ-step, θ + 1. This is achieved by forming an expression

for the prior error (prediction error).

en+1 = ξn+1 − ¯ξn+1,

= A′en + ωn. (3.102)

Now extending equation (3.85) to n+ 1 gives,

P n+1 = E[en+1(en+1)T

],

= E[(A′en + ωn)(A′en + ωn)T

], (3.103)

en and ωn are uncorrelated as en is the error accumulated in previous n steps and ωn

82

is the process error for n-th step. This implies,

P n+1 = E[A′en(A′en)T

]+ E

[ωn(ωn)T

],

= A′P nA′T

+Q. (3.104)

This has completed the optimal estimator recursive loop.

3.3.4 Numerical Results

For all numerical examples presented in this section an annular domain with r ∈

[0.5, 1] is considered. Number of states ξi are chosen such that Kalman rank condition

given in equation (3.82) is satisfied. Pair (C,A) is observable for Nr ≤ 8, where Nr

is the number of discretization points along r. The results presented in this section

were obtained in just 3 to 4 iterations over the domain Ω1. Running the algorithm

for more iterations has no effect on results. The algorithm was tested for various

cases with smooth, non-smooth and noisy Cauchy data. Numerical results on a

(Nr ×Nθ) = (8× 2000) grid are summarized in following two subsections, where Nθ

is the number of discretization points along θ.

3.3.4.1 For Smooth Data

Figure. 3.13 shows the solution obtained by solving well-posed problem (3.70) with

h = sin(θ) + sin(3θ) and g = 0. Now (f, g) |Γout obtained by solving problem (3.70) is

used as Cauchy data to find unknown h on the inner boundary Γin. Figure. 3.12 shows

the solution obtained by the robust iterative observer. Figure. 3.14 compares the true

h = sin(θ) + sin(3θ) with the one obtained by the algorithm on Γin. For smooth data

case process and measurement noise co-variance was taken σ1 = σ2 ≈ 10−7 ∼ 10−6,

that is, putting more confidence into the process and also assuming very small error

83

in data with η1 = η2 = 0.

Figure 3.11: True solution or solutionobtained by solving the problem (3.70)with h = sin(θ) + sin(3θ) and g = 0over Ω1

Figure 3.12: Solution obtained fromoptimal iterative algorithm over Ω1,after a number of iterations in the di-rection of time-like variable θ

Figure 3.13: Difference between thetrue and optimal observer algorithmsolutions over Ω1

0 1 2 3 4 5 6

−1.5

−1

−0.5

0

0.5

1

1.5

θ−axis

So

lutio

n ξ

1 o

n Γ

in

Analytical Soln. vs. Robust Iterative Obsv. Soln. on Γin

Analytical Solution

Robust Obsv. Soluton

Figure 3.14: On Γin, comparison oftrue boundary h = sin(θ) + sin(3θ) tothe one recovered by optimal iterativealgorithm using Cauchy data from Γout

3.3.4.2 For Non-smooth Data

Figure. 3.15 shows the numerical solution over the whole domain Ω1 with a non-

smooth pulse signal h applied on Γin and the Neumann zero boundary condition (g =

0) on Γout. Figure. 3.16 shows the recovered solution over Ω1 obtained by using the

Cauchy data on Γout. Again as smooth data case, the process and measurement noise

84

covariance was assumed to be very small σ1 = σ2 ≈ 10−7 ∼ 10−6 and η1 = η2 = 0.

Figure. 3.17 shows the error over the whole domain. Figure. 3.18 shows the signal

recovery on the unknown data boundary Γin. It can be seen that this algorithm is

well-suited for edge detection of such a pulse-shaped signal.

Figure 3.15: Numerical solution overΩ1 obtained by solving the problem(3.70) with pulse shaped h on Γin andg = 0 on Γout

Figure 3.16: Solution obtained fromoptimal iterative algorithm over Ω1,after a number of iterations in the di-rection of time-like variable θ

Figure 3.17: Difference between thetrue and the recovered solutions overΩ1

0 1 2 3 4 5 6

−3

−2

−1

0

1

2

3

θ−axis

So

lutio

n ξ

1 o

n Γ

in

Analytical Soln. vs. Robust Iterative Obsv. Soln. on Γin

Analytical Solution

Robust Obsv. Soluton

Figure 3.18: On Γin, comparison of thetrue pulse-shaped boundary signal tothe one recovered by optimal iterativealgorithm using only the Cauchy datafrom Γout

85

3.3.4.3 For Noisy Data

In this section, robust iterative observer, which is also an optimal iterative algorithm,

is studied for the case of noisy Cauchy data. First, problem (3.70) is solved using

h = sin(θ) on Γin and Neumann zero boundary condition (g = 0) on Γout. Solution of

problem (3.70) provided f on Γout. Next an additive white Gausian noise with η2 = 0

and variance σ2 = 10−3 was added to f . This noisy f along with Neumann zero

(g = 0) was used as Cauchy data to solve the inverse problem. Process covariance σ1

was assumed to be ≈ 10−4.

Figure. 3.19 shows the noisy Dirichlet data on Γout. Figure. 3.20 compares the

true h = sin(θ) with the one recovered by the algorithm on the Γin using the noisy

measurement Dirichlet data and the homogeneous Neumann boundary data from

Γout. Figure. 3.21 shows the comparison of percentage relative error in the measure-

ment Dirichlet data on the outer boundary Γout with the percentage relative error in

the recovered solution on the inner boundary Γin (recovered solution obtained using

noisy Dirichlet measurement data and homogeneous Neumann on Γout). During all

these measurements, process noise variance σ1 was taken as 10−4, whereas, measure-

ment noise variance σ2 was adjusted according to the relative error in the Dirichlet

measurement data and η1 = η2 = 0. Percentage relative error is given by,

% Relative Error =1

‖h‖2

‖h− hrecovered‖2 × 100, (3.105)

where ‖.‖ is the Eucledian 2-norm. It is obvious from Figure. 3.20 that error in

the recovered solution is reduced with smaller and smaller error in Cauchy data.

However, for arbitrarily small noise in Cauchy data still there’s around 2% relative

error in solution, this might come from numerical discretization error.

86

0 1 2 3 4 5 6

−1

−0.5

0

0.5

1

θ−axis

Am

plit

ud

e

Noisy data on Γout

Noisy data on Γout

Figure 3.19: Noisy data on Γout Figure 3.20: On Γin: Comparison oftrue h = sin(θ) to the one recovered byrobust iterative algorithm using noisyCauchy data with measurement noisevariance σ2 = 10−3

3.4 Conclusion

An iterative observer and its robust iterative optimal counterpart are presented to

solve boundary data estimation problems for the Laplace equation. The algorithms

are developed using one of the space variables as time-like to solve steady-state bound-

ary data estimation problems. Stable and efficient numerical results for multiple test

cases are also presented. The iterative observer algorithms reflect the possibility of

considering a steady-state problem from a dynamical theory perspective by using

one of the space variables as time. Successful implementation of the algorithm and

promising results show a step forward in the direction of using dynamical systems’ in-

spired algorithms to solve steady-state problems modeled by time independent PDEs

and without introducing a particular notion of time. In the following chapter, the

iterative algorithms are used to solve source localization and estimation problems for

the Poisson equation.

87

Figure 3.21: Percentage relative error in the Dirichlet measurement data on Γout vs.percentage relative error in the recovered solution on Γin

88

Chapter 4

Iterative Observer-based Approach for Source Localization

and Estimation for Poisson Equation

Imagination is more important than knowledge. For knowledge is limited

to all we now know and understand, while imagination embraces the entire

world, and all there ever will be to know and understand.

(—Albert Einstein)

4.1 Introduction

Source localization and identification problems for Poisson equation are frequently

used to model physical phenomena in various engineering disciplines. Just to name a

few, heat source localization and identification problem [63], recovery and determina-

tion of cracks [64], [65]. Source identification problems in electromagnetic theory [66].

Electroencephalography (EEG) and Magnetoencephalography (MEG) problems to

monitor electric and magnetic activity in human brain [67]. Some classical source

estimation and localization problems can also be found in [68] and [69].

Various methods and strategies have been presented in the literature to tackle

such problems. For example [70] presents a Green’s functions based method to iden-

tify the unknown point sources. The technique involves the study of the method of

fundamental solutions and requires analytical skills to deal with Green’s functions. A

method based on Poisson integrals for solving source seeking task is proposed in [30].

The method involves a sensor-equipped vehicle to provide pointwise measurements of

quantity emitted by the source. A numerical method is presented by [71]. The draw-

89

back of this numerical method is that it needs prior approximate positions of point

sources which is not the case in many of the physical applications. Another work

based on a weighted residual approach using harmonic functions is presented by [32].

Again the algorithm works for the cases where an estimated localized region is avail-

able for the point sources. Tikhonov regularization based methods to solve inverse

source problem for Poisson equation are presented in [23]. The method presented is

useful for distributed source estimation only.

In this chapter, iterative observer-based method is developed to localize and es-

timate sources in a system governed by Poisson equation. Both smooth and noisy

observation data cases are tackled. The strategy developed in this chapter highlights

the fact that dynamical systems’ inspired algorithms can be used for source localiza-

tion/estimation problems for Poisson equation.

4.2 Iterative Observer-based Strategy for Point Source Lo-

calization

In this section, a method based on iterative observer design is presented to solve

point source localization problem for Poisson equation with given boundary data.

The procedure involves solutions of multiple boundary estimation sub problems using

the available overdetermined Dirichlet and Neumann data from different parts of the

boundary. An optimal weighted sum of these solution profiles localizes point sources

inside the domain. A method to compute these weights is also provided. Numerical

results are presented using finite differences in a rectangular domain.

90

4.2.1 Problem Statement

Let Ω2 be a bounded rectangular domain in R2 as shown in Figure 4.1 and ∂Ω2 =

Γ1 ∪ Γ2 ∪ Γ3 ∪ Γ4 be the boundary of Ω2. Consider the Poisson equation,

4u = f in Ω2, (4.1)

with Laplacian operator 4 =∂2

∂x2+

∂2

∂y2, along with Neumann boundary condition,

∂u

∂n= h on ∂Ω2, (4.2)

where∂

∂nrepresents normal derivative to the boundary. Let us assume that,

u = g on ∂Ω2, (4.3)

is obtained by solving boundary value problem (4.1) and (4.2) with known f and

h. Existence and uniqueness for the solution of problem (4.1),(4.2) is well-known

for consistent f and h, that is the case when f ∈ L2(Ω2) and h ∈ L2(∂Ω2) such

that g ∈ H12 (∂Ω2). Further, for this particular choice of f , h and g, the solution

u ∈ H1(Ω2). However, because of the steady-state diffusive nature of system, problem

of finding the unknown source f from observed g and h, is not obvious. Let us assume

that the steady-state potential field u is generated by a number of distinct point

sources inside the domain:

f(x, y) =N∑k=1

Ckδ(x− xk, y − yk), (x, y) ∈ Ω2, (4.4)

where δ’s represent Dirac delta point sources and scalars Ck are the corresponding

magnitudes.

91

Ω2

Γ1

Γ2

Γ3 Γ4

y

x

Figure 4.1: Left: Rectangular domain Ω2 with ∂Ω2 = Γ1 ∪ Γ2 ∪ Γ3 ∪ Γ4, Right:Co-ordinate axis.

The objective here is to find locations (xk, yk) for all k inside Ω2 from available overde-

termined data g and h on ∂Ω2.

In the following section, the boundary estimation subproblems are presented,

which can be solve using the iterative observer presented in chapter 3. Solutions

to these sub problems will then help to solve the source localization problem for

Poisson equation in a later section.

4.2.2 Preliminary Analysis and Results

Let us observe that the available non-homogeneous boundary data as given in equation

(4.2) and (4.3) has four components on various parts of ∂Ω2:

g = ∪4i=1 gi|Γi , (4.5)

h = ∪4i=1 hi|Γi , (4.6)

Let us propose the boundary estimation problem:

Find ξ1 on Γ2:

∂ξ

∂x= Aξ in Ω2,

ξ1 = Cξ = g1 on Γ1,

∂ξ1

∂n= h1 on Γ1,

∂ξ1

∂nξ1 = 0 on Γ3 ∪ Γ4,

(4.7)

where ξ = (ξ1 ξ2)T represents state vector, C ∈ L(X,R) is the boundary observation

92

operator and operator matrix A : D(A)→ X:

X = H1Γ1

(α)× L2 (α) , (4.8)

D(A) =[H2 (α) ∩H1

Γ1(α)]×H1

Γ1(α) , (4.9)

where α is in the interval (y |Γ1 , y |Γ2) and without loss of generality let α ∈ [0, π/4],

and

H1Γ1

(α) =

p1 ∈ H1 (α) | dp1

dy|Γ1= h1

, (4.10)

here X is a Hilbert space with scalar product given by,

⟨ q1

q2

,

p1

p2

⟩ =

∫α

dq1

dy(y)

dp1

dy(y)dy

+

∫α

q1(y)p1(y)dy +

∫α

q2(y)p2(y)dy. (4.11)

The first equation in (4.7) represents Laplace equation,

4S1 = 0 on Ω2, (4.12)

with two auxiliary variables ξ1 = S1 and ξ2 =∂S1

∂x:

ξ =

ξ1

ξ2

, A =

0 1

− ∂2

∂y20

. (4.13)

Let us introduce the idea of iterations using time-like variable over the domain Ω2 as

follows. Let x be a variable defined over the interval [c, d) for all c, d ∈ R and d > c

then x[m], for all m ∈ Z = 0 ∪ Z+, represents x over mth iteration over the interval

[c, d). The idea of iteration is also shown in Figure. 3.2.

93

4.2.2.1 Iterative Observer Algorithm:

Theorem 6. Let ξ1 be the estimate of ξ1, then solution ξ1 of boundary value problem

(4.14) asymptotically (m→∞) converges to the solution of boundary value problem

(4.7) over Ω2:

∂

∂xξ(x[m]) = Aξ(x[m])−KC(ξ(x[m])− ξ) in Ω2,

∂

∂yξ1(x[m]) = h1 on Γ1,

∂2

∂y2ξ1(x[m]) = − ∂2

∂x2ξ1(x[m])−KC(ξ(x[m])− ξ) on Γ2,

ξ(x[m]) |initial= ξ(x[m−1]) in Ω2,

(4.14)

where ‘ ˆ ’ represents estimated quantity and ξ(x[m]) |initial represents the estimate

over the whole domain Ω2 at the start of mth iteration. Observer starts at index

m = 1 which represents first iteration. ξ(x[m=0]) is initial guess at the start of the

first iteration over the whole domain Ω2.

The proof of Theorem 6 is along similar lines as given in section 3.2.5. The

third equation in (4.14) is the assumption that Laplace equation is valid on the top

boundary and this provides necessary boundary condition required on Γ2. C is the

observation operator such that Cξ = ξ1 |Γ1 . K is the correction operator chosen in

such a way that state estimation error on Γ1 given by (ξ− ξ) exponentially converges

to zero over Ω2. Conditions for the existence of observer gain K are given in chapter

3. In the following section, a strategy to localize point sources inside domain Ω2 is

presented to solve problem (4.1) with known boundary conditions (4.2), (4.3) and

unknown point sources given by (4.4) by exploiting the iterative observer design.

94

4.2.3 Point Source Localization Strategy

The main theoretical result of this section is presented in form of a theorem as given

below.

4.2.3.1 Main Result

Theorem 7. For all i in integer set 1, 2, 3, 4, let Si be the solution over Ω2 obtained

using iterative observer (4.14) with Si = gi|Γi and∂

∂nSi = hi|Γi . Let for all i, wi be

the solution obtained by solving boundary value problem for Laplace equation,

4wi = 0 in Ω2,

wi = 1 on Γi,

wi = 0 on ∂Ω2\Γi,

(4.15)

then,

u =1

2

[4∑i=1

Siwi

]in Ω2, (4.16)

f =4∑i=1

∇Si.∇wi in Ω2, (4.17)

where u and f solve the boundary value problem for Poisson equation given by equa-

tions (4.1), (4.2) and (4.3).

Proof. We have Si and wi satisfy Laplace’s equation for all i ∈ 1, 2, 3, 4. From

equation (4.16) we can write,

∇u =1

2

4∑i=1

(Si∇wi + wi∇Si) , (4.18)

∇.(∇u) = ∇.

(1

2

4∑i=1

(Si∇wi + wi∇Si)

), (4.19)

95

which gives,

f = 4u =4∑i=1

∇Si.∇wi. (4.20)

thus we have that u as given in equation (4.16) satisfies Poisson equation 4u = f up

to an additive constant. Now using the properties of wi, Si we have to show that u

and f as given by equations (4.16) and (4.17) fully satisfy boundary value problem

given by equations (4.1), (4.2) and (4.3). We can write,

∫Ω2

[4∑i1

wi4Si

]dΩ2 = 0, (4.21)

using Green’s first identity,

∫∂Ω2

[4∑i=1

wi∂Si∂n

]d∂Ω2 =

∫Ω2

[4∑i=1

∇wi.∇Si

]dΩ2, (4.22)

We have wi = 1 on Γi and zero elsewhere on boundary. This gives,

∫∂Ω2

[4∑i=1

wi∂Si∂n

]d∂Ω2 =

4∑i=1

[∫Γi

∂Si∂n

d∂Ω2

]=

∫∂Ω2

h dΩ2, (4.23)

applying divergence theorem we have,

∫∂Ω2

h dΩ2 =

∫Ω2

∇.∇u dΩ2 =

∫Ω2

f dΩ2. (4.24)

Combining equations (4.22), (4.23) and (4.24),

∫Ω2

f dΩ2 =

∫Ω2

[4∑i=1

∇wi.∇ui

]dΩ2, (4.25)

above relation is true for all sizes of rectangular domains Ω2, thus we have,

f =4∑i=1

∇Si.∇wi. Q.E.D. (4.26)

96

Remark 2. The mathematical result presented in Theorem 7 is valid for all sizes of

rectangular domains.

4.2.3.2 Point Source Localization as a Two-step Strategy

The main theoretical result presented in the previous section can be used to localize

unknown point sources inside the domain. The process is illustrated as a two-step

strategy.

Step 1 : For all i ∈ 1, 2, 3, 4, compute iterative observer solutions Si over Ω2,

using Si = gi and∂Si∂n

= hi on Γi and iterative observer.

Step 2 : For all Si, compute corresponding weight solutions wi over Ω2. Find the

estimate of u using weighted sum as given in equation (4.16). This estimate of u

approximates the true solution u over the whole domain Ω2. The approximation

is fairly accurate for f ∈ L2(Ω2). However, for Dirac delta point sources local

minima or maxima provide their locations inside Ω2.

In the following section numerical simulation results are presented to locate un-

known point sources.

4.2.4 Numerical Simulations

4.2.4.1 Graphical Illustration of the Process

In this subsection, the point source localization strategy using iterative observer is

illustrated graphically with the help of MatLab1 simulations for the simple case of

a single point source in the middle of square domain of dimension [0, 1] × [0, 1] as

shown in Figure 4.2. It is important to note that any size of rectangular domain can

be considered. More complicated scenarios are presented in the next subsection.

1MatLab is a trademark of The MathWorks, Inc.

97

Figure 4.2: Numerical simulation result for the Poisson equation over domain Ω2 witha point source in the middle using homogeneous Neumann boundary data.

Step 1:

Figure 4.2 represents a point source in the middle of a square domain with homogenous

Neumann boundary data using finite difference discretization schemes on a 400× 400

uniform grid. This particular case under consideration is symmetric from all sides

hence would be easier for the illustrative purposes. Dirichlet boundary data can be

extracted from this solution profile. The objective is to recover the location of point

source inside this domain. Figure 4.3 shows g1 on bottom boundary. Because of

symmetry, g2, g3 and g4 on remaining parts of the boundary would be the same. Next

this Dirichlet data g1 is used to estimate S1 = ξ1 on opposite boundary Γ2 as given

in problem (4.7). Figure 4.4 shows g1 and estimated S1 = ξ1 on Γ2. It is important

to note that maximum number of states while solving problem (4.7) after full dis-

cretization depends on discrete observability condition also known as Kalman rank

condition. In this particular case 10 states were considered. Similarly for g2, g3 and

g4, estimated S1 = ξ1 on the corresponding opposite boundaries would be the same

because of the symmetry. Top plot in Figure 4.5 represents the full solution plot S1

over the whole domain Ω2 obtained using finite difference discretization on a 400×400

uniform grid with g1 on Γ1, corresponding S1 = ξ1 on opposite boundary Γ2 and in-

98

sulated (homogeneous Neumman) side boundaries. Similar the remaining three plots

in Figure 4.5 (from 2nd to 4th) represent S2, S3, S4 obtained by solving problem (4.7)

with g2, g3, g4 on Γ2,Γ3,Γ4 respectively and corresponding estimated ξ1 on respective

opposite boundary and insulated (homogeneous Neumann) side boundaries.

0 0.2 0.4 0.6 0.8 1

−6

−4

−2

0Dirichlet data on the boundary

Γi

Am

plit

ude

Figure 4.3: Dirichlet data g1 on bot-tom boundary Γ1. Because of symme-try g2|Γ2, g3|Γ3 and g4|Γ4 would be sim-ilar.

0 0.2 0.4 0.6 0.8 1−60

−50

−40

−30

−20

−10

0

Am

plit

ude

Dirichlet data and recovered signal on opposite boudaries

Dirichelt data g1 on Γ

1

Estimated S1 on opposite boundary

Figure 4.4: Dirichlet data g1|Γ1 and es-timated S1 = ξ1 on opposite boundaryΓ2 using iterative observer. Becauseof symmetry, qualitatively similar pro-files for other three cases.

Step 2:

Figure 4.6 represents the weight profiles w1, w2, w3 and w4 corresponding to solution

profiles shown in Figure 4.5. It is important to note that these weights are normalized.

Next these weights are applied to corresponding solution profiles by point to point

multiplication to obtain resultant weighted profiles. Finally these weighted solution

profiles are added to obtain weighted sum as shown in Figure 4.7. The minimum

point where the marker is placed represents the location of the point source.

4.2.5 Further Simulation Results

In this section different scenarios are presented to reflect the accuracy of the scheme to

localize point sources. All the numerical results were obtained using finite difference

discretization schemes on a square domain. The mesh size was 400 × 400 for all

99

Figure 4.5: Top plot: Solution profile S1 over Ω2 obtained by solving problem (4.7)with g1 on Γ1 and corresponding S1 = ξ1 on opposite boundary Γ2 and insulated(homogeneous Neumann) side boundaries.From 2nd to 4th: Plots for S2, S3 and S4 obtained using similar procedure.

100

Figure 4.6: From top to bottom: Weight profiles w1, w2, w3 and w4 over Ω2 corre-sponding to solution profiles S1, S2, S3, S4 respectively, as shown in Figure 4.5.

Figure 4.7: Weighted sum as given in equation (4.16) over Ω2. Marker in the middlerepresents the minima, where the negative point source is located.

101

Figure 4.8: Top: Non-centered point source inside Ω2. Bottom: Weighted sum withmarker representing minimum point and the location of point source.

simulation plots except for the Iterative observer algorithm where because of discrete

observability condition only 10 states were considered. The issue of limited number

of observable states after numerical discretization for distributed parameter systems

is well-known, [6,72]. Figure 4.8 represents the solution of the forward problem with

a non-centered point sources and the weighted sum obtained by applying two step

strategy. It can be seen that point source is very well localized. Figure 4.9 represents

the case with two point sources. The algorithm presented here works well for point

sources that are well separated, however, in case where multiple point sources are in

close vicinity, it not so easy to distinguish as shown in Figure 4.10.

4.3 Robust Iterative Algorithm-based Strategy for Inverse

Source Localization Problem

Source localization problem for Poisson equation with available noisy boundary data

is well-known to be highly sensitive to noise. The problem is ill-posed and lacks

to fulfill Hadamards stability criteria for well-posedness as presented in Chapter 2.

In this section, robust iterative observer algorithm along with the available noisy

boundary data from the Poisson problem is used to localize point sources inside a

102

Figure 4.9: Top: Two opposite polarity well seperated point sources in Ω2. Bot-tom: Weighted sum with markers representing minimum and maximum points andlocations of two point sources.

Figure 4.10: Top: Three closely located point sources in Ω2. Bottom: Weighted sumwith minima locating approximate position of point sources.

103

rectangular domain. The algorithm is inspired from Kalman filter design and is

implemented on a rectangular domain using finite difference discretization schemes.

Numerical implementation along with simulation results is detailed.

4.3.1 Problem Formulation

Let Ω2 be a bounded rectangular domain in R2 as shown in Figure 4.1 and ∂Ω2 =

Γ1 ∪ Γ2 ∪ Γ3 ∪ Γ4 be the boundary of Ω2. Let us consider the Poisson equation,

4u = f in Ω2, (4.27)

with Laplacian operator 4, along with Neumann boundary condition,

∂u

∂n= h on ∂Ω2, (4.28)

where ∂n represents normal derivative to the boundary. Existence and uniqueness for

the solution of problem (4.27),(4.28) is well-known for consistent f and h, that is the

case when f ∈ L2(Ω2) and h ∈ L2(∂Ω2) and the solution u in this case belongs to

H1(Ω2). Let the solution of boundary value problem (4.27),(4.28) gives,

u = g on ∂Ω2, (4.29)

Now let us assume that the steady-state potential field u is generated by a number

of distinct point sources inside the domain Ω2:

f(x, y) =N∑k=1

Ckδ(x− xk, y − yk), (x, y) ∈ Ω2, (4.30)

where δ(x − x, y − y), k = 1, · · · , N represent Dirac delta point sources localized

at (xk, yk) and scalars Ck, k = 1, · · · , N are the corresponding magnitudes. In this

section we are interested in unknown source localization problem from noisy bound-

ary measurements. We suppose that the measurement g and Neumann boundary h

104

are corrupted with additive white Gaussian noise ε1 and ε2 respectively, such that

ε1(η1, σ1) and ε2(η2, σ2) with mean values η1, η2 and variances σ1, σ2.

The objective, in this section, is to find locations (xk, yk) for all k = 1, · · · , N inside

Ω2 from available noisy measurement data g and known boundary h on ∂Ω2.

As discussed for the smooth data case, in the first part of this chapter, the source

localization problem for Poisson equation presented above boils down to a weighted

sum of solutions of multiple boundary estimation problems for Laplace equation. This

mathematical result is also presented in section 4.2.3. However, in the following sec-

tion, a robust iterative observer design is presented to solve the boundary estimation

problem for steady-state Laplace equation.

4.3.2 Robust Iterative Observer Design

In this section, we present robust iterative observer design for boundary estimation

for Laplace equation which helps to solve source localization problem for Poisson

equation. Let us observe that noisy boundary measurement g and Neumann boundary

h has four components on four different parts of boundary ∂Ω2,

g = g1|Γ1 ∪ g2|Γ2 ∪ g3|Γ3 ∪ g4|Γ4 , (4.31)

h = h1|Γ1 ∪ h2|Γ2 ∪ h3|Γ3 ∪ h4|Γ4 . (4.32)

Let us propose following boundary estimation problem for Laplace equation,

Find steady-state potential field S1 on Γ2:

4S1 = 0 in Ω2,

S1 = g1 on Γ1,

∂S1

∂n= h1 on Γ1,

∂S1

∂n= 0 on Γ3 ∪ Γ4.

(4.33)

105

Boundary value problem (4.33) is also known as Cauchy problem for Laplace equation.

Let us propose to reformulate this problem in a control-familiar form as following,

Find ξ1 on Γ2:

∂ξ

∂x= Aξ in Ω2,

ξ1 = Cξ = g1 on Γ1,

∂ξ1

∂n= h1 on Γ1,

∂ξ1

∂n= 0 on Γ3 ∪ Γ4,

(4.34)

with observation or measurement operator C and two auxiliary variables ξ1 = S1 and

ξ2 =∂S1

∂x:

ξ =

ξ1

ξ2

, A =

0 1

− ∂2

∂y20

, (4.35)

where A : D(A) → X as given in (4.8), (4.9) and (4.11). In chapter 3, a Kalman

filter-like robust optimal algorithm is already presented to solve ill-posed boundary

estimation problem for Laplace equation on an annulus domain. Here this algorithm

design is adapted to the rectangular domain using rectangular coordinates and with

the idea of iterations over Ω2 in the direction of time-like variable.

4.3.2.1 Numerical Discretization

Let us introduce Forward Euler-like approximation to discretize first derivative along

x as,∂ξ

∂x≈ 1

4x[ξn+1 − ξn

], (4.36)

where n is discrete positive integer index for variable x and 4x is a small step size.

State equation, after discretization in variable x, can be written as,

ξn+1 = (I + (4x)A) ξn, (4.37)

here I is 2× 2 identity matrix. Second equation in (4.37) can be written as,

106

(ξ2)n+1 = (ξ2)n +4x(− ∂2

∂y2

)(ξ1)n , (4.38)

to discretize second-order derivative term in above equation we use following second-

order accurate centered finite difference discretization formula,

∂2

∂y2(ξ1)n ≈ 1

(4y)2

[(ξ1)ni+1 − 2 (ξ1)ni + (ξ1)ni−1

], (4.39)

here i is discrete index for variable y, i = 1 on Γ2 and (4y) is a reasonably small step

size along y. Now from (4.38),

(ξ2)n+1i = (ξ2)ni −

4x(4y)2

[(ξ1)ni+1 − 2(ξ1)ni + (ξ1)ni−1

], (4.40)

rewriting above equation on Γ2, that is, at i = 1 gives,

(ξ2)n+11 = (ξ2)n1 −

4x(4y)2 [(ξ1)n2 − 2(ξ1)n1 + (ξ1)n0 ] , (4.41)

index i = 1 represents points on boundary Γ2, and index i = 0 represent fictitious

points outside the domain Ω2 along the boundary Γ2. Fictitious points are shown in

Figure 4.11. The equation for the fictitious points can be written as,

(ξ1)n0 = −(4y)2

4x[(ξ2)n+1

1 − (ξ2)n1]

+ 2(ξ1)n1 − (ξ1)n2 . (4.42)

Further at a particular index n fictitious point (ξ1)n0 can be determined uniquely as

it depends on (ξ2)n+11 . In each iteration at a particular n the fictitious point value

guess is determined using equation (4.42). This guess is then used in as a boundary

data to improve estimation on Γ2. Theoretically, this is equivalent to the assumption

that Laplace equation is satisfied on Γ2,

∂2u

∂x2= −∂

2u

∂y2on Γ2, (4.43)

107

which is possible as we are solving 2D Laplace equation as a first order state equation.

i=0;n

i=1;n

i=2;n

i=0;n− 1

i=1;n− 1

i=2;n− 1

i=0;n + 1

i=1;n + 1

i=2;n + 1

n

i

Γ1

Γ2

Figure 4.11: Left: Domain Ω2 after discretization and fictitious points outside Γ2,index i = 0 represents fictitious points (in blue).

4.3.2.2 Robust Iterative Observer Derivation

After full discretization equation (4.37) can be written as,

ξn+1 = (I + (4x)A) ξn, (4.44)

where A is discrete operator matrix. Further the output equation as given in (4.34)

can be written as,

ξn1 = Cξn + εn1 = gn1 on Γ1, (4.45)

where C is the discrete observation operator matrix. An important notion for exis-

tence of solution of state-space like system represented by equations (4.44) and (4.45)

is the discrete observability condition also called Kalman rank condition [6, 72]. The

system is observable if observability matrix formed by pair (C,A′) is full rank, where

A′ = I+(4x)A. Observability condition puts a restriction on the size of operator ma-

trix A, in other words on the possible number of discretization points in y-direction.

Let ε1 and ε2 be the measurement and process noise respectively such that their

covariance matrices be stationary over x and given by,

R = E[εn1 (εn1 )T

], (4.46)

Q = E[εn2 (εn2 )T

], (4.47)

108

here of R is 1 × 1 and Q is the size of A′. Let ξn be the estimate of true state ξn

and en represents the difference of true and estimated states at n. The goal is to

develop an iterative observer that is capable to predict true states over a period of

time-like variable x. For this, let positive definite error functional (en)2 = (ξn − ξn)2

be a feasible metric given by

ε(x) = E[(en)2

]= E

[en(en)T

]= P n, (4.48)

where P n is the error covariance matrix at x = n ×4x. The idea of iteration over

the rectangular domain can be introduced as following, let x be a variable defined in

the interval [c, d) for all c, d ∈ R and d > c then xm for all m ∈ N represents x over

mth iteration over the interval [c, d) .

In a similar way, we introduce idea of iterations over the rectangular domain

Ω2 in the direction of time-like variable. The right-end boundary is assumed to be

connected with the left end boundary to have time-like notion of the space variable.

Robust Iterative Observer Equations:

Prediction-Step:

¯ξn,m = A′ξn−1,m in Ω2,

P n,m = A′P n−1,mA′T +Q,

¯ξn=0,m=0 = P n=0,m=0 = 0,

(4.49)

with Neumann boundary condition h on Γ1 and fictitious points

close to Γ2 given by equation (4.42), taking care of boundary

conditions.

109

Correction-Step:ξn,m =

¯ξn,m +Kn,m

(yn − C ¯

ξn,m),

Kn,m = P n,mCT (CP n,mCT +R)−1,

P n,m = (I −Kn,mC)P n,m,

(4.50)

where ξn,m is an estimate of ξn,m,¯ξn,m is the prior-estimate of ξn,m, P n,m is the prior-

estimate of state error covariance matrix P n,m given by equation (4.48). Q and R as

given in equations (4.47) and (4.46) and Kn,m is the gain matrix.

At the start of iterative observer algorithm, (ξ1)n1 , (ξ1)n2 , (ξ2)n1 and (ξ2)n+11 given

in equation (4.42) come from the initial guess over the domain Ω2. The fictitious

points from equation (4.42) are then used in robust observer equations to tackle

boundary condition on Γ2, whereas Neumann boundary condition h is used to tackle

the boundary condition on Γ1.

Proposition 2. Let pair (C,A′) be observable and m be a discrete non-negative inte-

ger index of iteration for variable x. As m→∞ (that is for arbitrarily large number

of iterations over rectangular domain Ω2), the two-step algorithm given by equations

(4.49) and (4.50) minimizes the mean square error functional given in equation (4.48).

Proof. (Above proposition and the following proof are also detailed chapter 3 for

boundary estimation problem for Laplace equation on the annulus domain Ω1)

Let us re-write equation (4.48) as,

P n,m = E[(ξn,m − ξn,m)(ξn,m − ξn,m)T

]. (4.51)

Let us assume the prior estimate of ξn,m is called¯ξn,m and was obtained by the

knowledge of the system (prediction). The correction equation can be written using

the prior estimate with measurement data as follows,

ξn,m =¯ξn,m +Kn,m

(yn,m − C ¯

ξn,m). (4.52)

110

Kn,m is the gain matrix which will be derived in a moment. The term(yn − C ¯

ξn,m)

is the measurement residual. Substituting equation (4.45) into (4.52),

ξn,m =¯ξn,m +Kn,m

(Cξn,m + εn1 − C

¯ξn,m

). (4.53)

ε1, ε2 remain the same for particular n over all iterations m ∈ N. Now substituting

equation (4.53) into (4.51) gives,

P n,m = E[(

(I −Kn,mC)(ξn,m − ¯ξn,m)−Kn,mεn1

)(

(I −Kn,mC)(ξn,m − ¯ξn,m)−Kn,mεn1

)T], (4.54)

taking the error of prior estimate(ξn,m − ¯

ξn,m)

as uncorrelated to measurement noise

gives,

P n,m = (I −Kn,mC) E[(ξn,m − ¯

ξn,m)(ξn,m − ¯ξn,m)T

](I −Kn,mC)T +Kn,m E

[εn1 (εn1 )T

](Kn,m)T , (4.55)

now substituting equations (4.46) and (4.51) into above equation yields,

P n,m = (I −Kn,mC)P n,m(I −Kn,mC)T +Kn,mR(Kn,m)T . (4.56)

Above equation is the error covariance update equation and the diagonal of P n,m

contains mean squared error. This mean squared error has to be minimized with

respect to Kn,m. Re-writing equation (4.56) gives,

P n,m = P n,m −Kn,mCP n,m − P n,mCT (Kn,m)T

+Kn,m(CP n,mCT +R)(Kn,m)T , (4.57)

using the fact that trace of a matrix is equal to trace of its transpose, it can be seen

111

that,

T [P n,m] = T [P n,m]− 2T [Kn,mCP n,m]

+ T [Kn,m(CP n,mCT +R)(Kn,m)T ], (4.58)

where T [P n,m] is the trace of covariance matrix P n,m. Differentiating with respect to

Kn,m,

d

dKn,mT [P n,m] = −2CP n,m + 2Kn,m(CP n,mCT +R), (4.59)

setting derivative equal to zero gives,

(CP n,m)T = Kn,m(CP n,mCT +R), (4.60)

solving for Kn,m gives,

Kn,m = P n,mCT (CP n,mCT +R)−1. (4.61)

The above equation is the optimal estimator gain equation which minimizes the mean

square state estimation error. Substituting above equation into equation (4.56) yields,

P n,m = P n,m − P n,mCT (CP n,mCT +R)−1CP n,m,

= P n,m −Kn,mCP n,m,

= (I −Kn,mC)P n,m, (4.62)

which is the update equation for state error covariance matrix with optimal gain

Kn,m. Equations (4.52), (4.61) and (4.62) develop an estimate of variable xn,m. State

prediction is achieved using state equation,

¯ξn+1,m = A′ξn,m, (4.63)

112

along with Neumann boundary condition g on Γout and equation (4.43) on Γin. To

complete the recursion it is important to find an equation which projects state error

covariance matrix into next x-step, x+ 1. This is achieved by forming an expression

for the prior error (prediction error).

en+1,m = ξn+1,m − ¯ξn+1,m,

= A′en,m + εn2 . (4.64)

Now extending equation (4.48) to n+ 1 gives,

P n+1,m = E[en+1,m(en+1,m)T

],

= E[(A′en,m + εn2 )(A′en,m + εn2 )T

], (4.65)

en,m and εn2 are uncorrelated as en,m is the error accumulated in previous n steps and

εn2 is the process error for n-th step. This implies,

P n+1 = E[A′en,m(A′en,m)T

]+ E

[εn2 (εn2 )T

],

= A′P n,mA′T

+Q. (4.66)

This has completed the iterative observer loop.

4.3.3 Two-step Process for Source Localization

The theoretical result presented previously can be used to devise a two-step source

localization method, similar to section 4.2.3.2, as follows,

Step 1 : For all i compute robust iterative observer solutions Si over Ω2, as given

in Proposition 1, using Si = gi and∂Si∂n

= hi on Γi.

Step 2 : For all i, compute weight profiles ωi by solving well-posed boundary value

problem (4.15). Combine Si’s and ωi’s using equation (4.16) to have an estimate

113

of Poisson problem solution u over Ω2. The position of Dirac delta sources inside

the domain is provided by the maxima or minima of estimated u.


For all numerical simulation results a square domain Ω2 of dimension [0, 1]× [0, 1] was

considered. For robust iterative observer implementation, number of state variables

were chosen such that pair (C,A′) fulfills discrete Kalman rank condition. Robust iter-

ative observer provided unknown boundary estimate for Cauchy problem for Laplace

equation using part of Cauchy data from Poisson problem. Cauchy data on Γi along

with this estimated boundary data on opposite boundary and Neumann zero side

boundaries were used to obtain Si for all i ∈ 1, 2, 3, 4 on a 400× 400 grid. Weight

profiles wi were also computed on the same grid size by numerically solving system

(4.15). All numerical results were obtained using MatLab2. Fig. 4.12 represents the

solution of problem (4.27),(4.28) over Ω2 with f = δ(0.5, 0.5) and h = 0 and using

centered finite difference discretization scheme on a 400 × 400 uniform grid. The

solution obtained on the boundary is then corrupted with additive white Gaussian

noise with η1 = 0 and σ1 = 5× 10−4. This noisy Dirichlet boundary data is used as a

measurement along with homogeneous Neumann boundary to recover the unknown

point source location. Fig. 4.13 represents noisy measurement data. Fig. 4.14 shows

noisy measurement on Γi and iterative observer boundary estimate on the opposite

boundary with η2 = 0 and σ2 = 0.1. From top to bottom Fig. 4.15 shows S1, S2, S3

and S4 respectively. Similarly from top to bottom Fig. 4.16 shows weight profiles

w1, w2, w3 and w4 respectively. Fig. 4.17 represents weighted sum obtained using

equation (4.16). This weighted sum gives estimate of solution u. The minimum point

represented with white marker locates the point source inside Ω2. Fig. 4.18 repre-

sents the localization of point sources for the case where two opposite polarity point

2MatLab is a trademark of The Mathworks Inc.

114

Figure 4.12: Square domain Ω2 with a point source in the middle using homogeneousNeumann boundary data.

sources are located inside domain Ω2. The white markers in the bottom figure rep-

resent approximate locations of point sources. The measurement data was corrupted

with additive white Gaussian noise with η1 = 0 and σ1 = 5× 10−4.

Figure 4.13: Noisy measurement datag with σ1 = 5× 10−4, because of sym-metry of the special case under consid-eration (one point source in the mid-dle of the domain), qualitatively simi-lar profiles on all parts of ∂Ω2.

Figure 4.14: Comparison of noisy mea-surement data g on Γi and robust it-erative observer solution on the oppo-site boundary. σ1 = 5 × 10−4 andσ2 = 1 × 10−1. Because of symme-try, qualitatively similar profiles on allparts of ∂Ω2.

115

Figure 4.15: Top figure: Solution profile S1 with measurement data g1 on Γ1, insulatedside boundaries and boundary estimate obtained using robust iterative observer onΓ2. From 2nd to 4th: Plots for S2, S3 and S4 respectively.

116

Figure 4.16: From top to bottom: Weight profiles w1, w2, w3 and w4 respectively,obtained by solving boundary value problem (4.15) for i ∈ 1, 2, 3, 4.

Figure 4.17: Weighted sum obtained using equation (4.16). Minimum point repre-sented with white marker in the middle provides the location of point source.

117

Figure 4.18: Top: Solution of problem (4.27),(4.28) with two point sources of oppositepolarity and homogeneous Neumann boundary on ∂Ω2. Below: Weighted sum ob-tained using equation (4.16). Minimum and maximum points represented with whitemarkers provide locations of point sources.

4.4 Distributed Potential Field Estimation for Poisson Equa-

tion

The iterative observer algorithms presented in chapter 3 can also be used for steady-

state distributed potential field estimation problem for the Poisson equation. The

distributed potential field estimation over the whole domain from boundary mea-

surements is a challenging problem. The numerical simulation results obtained using

the two-step strategy highlight the significance of observer-based algorithms. In the

following section the potential field estimation problem is formulated.


Let Ω2 be a rectangular domain with boundary ∂Ω2 = Γ1 ∪ Γ2 ∪ Γ3 ∪ Γ4 as shown

in Figure 3.1. Let us consider the steady-state diffusion model, with a source term,

known as Poisson equation,

4u = f in Ω2, (4.67)

118

where 4 is the Laplacian operator, u represents the steady-state potential field and

f is the source term.

The object is to estimate the unknown steady-state potential field u from the available

Cauchy data on the boundary ∂Ω2, given by,

u = g on ∂Ω2,

∂u

∂n= h on ∂Ω2.

(4.68)

The two-step solution strategy as presented in section 4.2.3 is used to estimation

steady-state distributed potential field u. In the following section, to avoid repetition,

only numerical simulation results are presented.


Various simulation results for steady-state distributed potential field estimation are

presented. All the simulations are done using finite difference discretization schemes

on a square domain of dimension 1×1. The two-step strategy as graphically illustrated

in the previous section is used for estimation. Figure 4.20 compares the weighted sum

obtained using two-step solution strategy to the exact numerical simulation result over

Ω2 (exact solution obtained with known f and Neumann boundary data) for various

test cases. As obvious from the results, the two-step solution strategy can well recover

the distributed potential field u using only boundary data.

4.5 Conclusion

An iterative observer-based technique is presented to solve source localization and

estimation problems for Poisson equation on rectangular domains. The idea of using

dynamical system like algorithm, using space variable as time-like, has not been

studied in the literature before. The two-step strategy using robust iterative observer

119

Figure 4.19: Steady-state potentialfield u over Ω, obtained by numeri-cally solving Poisson equation (4.67)for various f with homogeneous Neu-mann side boundaries.

Figure 4.20: Weighted sum obtainedfrom equation (4.16), which providesestimate to the steady-state potentialfield u over Ω2 for various test casesshown in Figure 4.19.

120

algorithm happens to be robust to noise for well separated point sources, further the

distributed potential field estimation is fairly good.

In the following chapter, a dimension decomposition approach is presented to

tackle 3D domains with two congruent parallel surfaces. Once the 3D problem is

decomposed into 2D sub problems, then the iterative observer-based methods are

used to solve the sub problems.

121

Chapter 5

Dimension Decomposition Approach for 3D Domains

Walls are tricky. Sometimes it’s not about breaking them down by force,

but about finding the weak spots and gently nudging.

(—My Kindred Spirit)

5.1 Introduction

In this chapter, a dimension decomposition approach is presented which helps to

solve boundary and source data estimation problems on three-dimensional domains

with two congruent parallel surfaces using iterative observer algorithms presented in

previous chapters. First of all, a boundary estimation problem for Laplace equation

on a 3D domain is presented in state-space-like representation and then the system

observability is studied. Based on the special observability result, the 3D problem is

subdivided into a number of 2D subproblems over rectangular cross-sections. These

subproblems can be independently solved using previously presented iterative ob-

servers. In the later half of this chapter, this dimension decomposition approach is

used to estimate unknown sources for Poisson problem in 3D.

5.2 Boundary Estimation Problem for Laplace Equation in

3D

In this section, boundary estimation problem for Laplace equation is formulated in a

control familiar form on a 3D domain. Using one of space variables, boundary value

problem is written as a forward evolving system in state-space-like representation.

122

Problem formulation and theoretical analysis is presented in the following.


Let Ω3 ∈ R3 with two congruent parallel surfaces ΓB,ΓT ∈ R2 as shown in Figure 5.1.

Cauchy data is given on one of the two surfaces and objective is to find the solution on

the opposite boundary surface. Boundary estimation problem for Laplace equation

can be given as,

Find u on ΓT :

4u = 0 in Ω3,

u = f on ΓB,

∂u

∂n= g on ΓB,

∂u

∂n= 0 on ∂Ω3\ ΓB ∪ ΓT ,

(5.1)

here∂

∂nrepresents normal derivative to the boundary surface. 4 is the Laplacian

operator in rectangular co-ordinates such that 4 =∂2

∂x21

+∂2

∂x22

+∂2

∂x23

.

5.2.1.1 Change of Variables

As proposed for a two-dimensional problem, we rewrite Laplace equation given in

(5.1), in rectangular coordinates, as a first-order state equation by introducing two

new auxiliary state variables ξ1 and ξ2 as follows,

ξ1(x1, x2, x3) = u(x1, x2, x3),

ξ2(x1, x2, x3) =∂u

∂x1

,

(5.2)

123

Ω3

ΓB

ΓT

x1 − plane

Figure 5.1: Left: Domain Ω3 with two congruent parallel surfaces ΓB and ΓT (ΓB andΓT Lipschitz continuous); Right: Plane containing time-like co-ordinate x1.

where x1, x2 and x3 represent rectangular coordinates with x1x2 plane parallel to the

two congruent parallel surfaces ΓB,ΓT , by introducing these auxiliary variables the

resulting Laplace equation can now be written as,

∂ξ

∂x1

= Aξ, (5.3)

where,

ξ =

ξ1

ξ2

; A =

0 1

− 4x2,x3

0

; − 4x2,x3

= − ∂2

∂x22− ∂2

∂x32. (5.4)

Note 3: The state operator matrix A has two positive definite operators on anti-

diagonal, this clearly shows that A has both positive and negative eigenvalues. As

given above, the state operator matrix does not generate a strongly continuous semi-

group. The existence of exponential of such an operator, under certain conditions, is

studied in the following subsections.

Let A : D(A) → X, as given in equation (5.4), be defined on a rectangular cross-

124

section ω of Ω3 (parallel to x2x3 plane, as shown in Figure 5.2) such that,

ω =

x ∈ ω : x =

x2

x3

∈ [a2, b2]

[a3, b3]

;

, (5.5)

where b2 > a2, b3 > a3, a2, a3, b2, b3 ∈ R+, ω = ω ∪ Γl∪r∪t∪b, Γb = ω ∩ ΓB and

Γt = ω ∩ ΓT .

X = H1Γb

(ω)× L2 (ω) , (5.6)

D(A) =

[f ∈ H2 (ω) ∩H1

Γb(ω) :

df

ds|Γb = c2

]×H1

Γb(ω) , (5.7)

where,

H1Γb

(ω) =f ∈ H1 (ω) : f |Γb = c1

, (5.8)

and c1, c2 are constants (coming from Cauchy data at a particular point on Γb) and


⟨ q1

q2

,

p1

p2

⟩ =

∫ω

∇q1(x).∇p1(x)dω +

∫ω

q1(x).p1(x)dω +

∫ω

q2(x).p2(x)dω.

(5.9)

here x as given in equation (5.5). It can be seen that D(A∞) is dense in X. ξ1 and ξ2

x2

x3

ω

Γb

Γt

Γl

(a3, a2)

(a3, b2)

(b3, a2)

(b3, a3)

Γr

Figure 5.2: Left: Cross-sectional plane ω of Ω3; Right: x2x3 plane orientation (ingray);

are called state variables and using these new variables, problem (5.1) can be written

125

in an equivalent form as,

Find ξ1 on ΓT :

∂ξ

∂x1

= Aξ in Ω3,

Cξ = ξ1 = f on ΓB,

∂ξ1

∂n= g on ΓB,

∂ξ1

∂n= 0 on ∂Ω3\ ΓB ∪ ΓT ,

(5.10)

where C : X → Y is the boundary observation operator and Y be the output space

given as,

Y =

(f1)|Γb :

f1

f2

∈ X, . (5.11)

Y forms a Hilbert space with respect to the norm,

〈q1(x3), p1(x3)〉 =

∫Γb

q1(x3)p1(x3)dx3. (5.12)

Boundary value problem (5.10) has a first order state equation in variable x1 and

overdetermined data is available on ΓB. The solution of this first order state equation

leads to the study of semigroup generated by unbounded differential operator matrix

A. Further, we study observability for the pair (C,A) in infinite-dimensional setting.

5.2.2 Theoretical Analysis and Results

5.2.2.1 Existence of Exponential of A

Theorem 8. Let m,n ∈ Z? be the non-zero set of integers, for A : D(A) → X (as

given in (5.4), (5.6) and (5.7)) there exists an infinite set of orthonormal eigenvectors

(Φmn) and corresponding eigenvalues (λmn). Furthermore, A generates a strongly

126

continuous semigroup for vectors

p1

p2

∈ X, if and only if, the decay rate of the

sequence

⟨ p1

p2

,Φmn

⟩is greater than the growth rate of the sequence eλmnx1 for

all m,n ∈ Z?.

Proof. Let m,n ∈ Z? set of all integers such that,

Φmn(x2, x3) = ρmn

αmnϕmn(x2, x3)

βmnϕmn(x2, x3)

, (5.13)

be the infinite set of orthonormal eigenvectors of operator A and λmn be the eigen-

values such that,

AΦmn = λmnΦmn, (5.14) βmnϕmn

− 4x2,x3

(αmnϕmn)

= λmn

αmnϕmn

βmnϕmn

.

Assuming that αmn, βmn do not depend on x2, x3. First equation in (5.14) gives,

λmn =βmnαmn

. (5.15)

Second equation above suggests that we are interested in finding the eigen-pairs of

the Laplacian operator − 4x2,x3

over the domain ω. Here ω is a cross-sectional view

of Ω3 parallel to x2x3 plane. Without loss of generality, let us assume that ω =

(0, a1)× (0, a2) for a1, a2 ∈ R+ as shown in Figure 5.3.

127

x2

x3

ω = (0, a1)× (0, a2)

Γb

Γt

Γl Γr

Figure 5.3: Left: Cross-sectional plane ω of Ω3 parallel to x2x3 plane; Right: x2x3

plane orientation (in gray);

The eigenvalue problem can be given as,

− 4x2,x3

ϕmn = λ2mnϕmn in ω,

ϕmn = f |Γb on Γb,

∂

∂nϕmn = g|Γb on Γb,

∂

∂nϕmn = 0 on Γl ∪ Γr,

(5.16)

Sign of λ2mn is positive from the well-known fact that − 4

x2,x3

is a positive definite

operator, further this can also be verified using Rayleigh quotient test. There can be

multiple set of eigenbasis functions and corresponding eigenvalues for the eigenvalue

problem in above form. Again, without loss of generality, let us assume that g|Γb = 0

and a1 = a2 = π/4 and compute an infinite set of eigenpairs as follows,

ϕmn(x2, x3) = −π2

64cos((6− 8m)x2) cos(4nx3), (5.17)

βmn = λmn = ±√

(6− 8m)2 + (4n)2, (5.18)

ϕmn for m,n ∈ Z form an orthonormal basis in L2(ω).

Finally, an orthonormal set of eigenvectors can be formed in X with respect to

128

norm defined by (5.9) as,

Φmn(x2, x3) = ρmnφmn(x2, x3) = ρmn

αmnϕmn(x2, x3)

βmnϕmn(x2, x3)

, (5.19)

where ρmn is a non-zero normalization factor, given as,

ρ2mn =

4096

π

(16384(6− 8m)2n2

1 + 16(6− 8m)2n2 (1 + (6− 8m)2 + 16n2)

). (5.20)

Also ρmn is observed to be a fast decaying sequence and fundamental modes (ρ21,1 =

ρ21,−1) ≥ ρ2

mn for all m,n ∈ Z?. Now let us try to write semigroup generated by

operator matrix A as an infinite series,

∑m,n∈Z?

eλmnx1

⟨ p1(x2, x3)

p2(x2, x3)

,Φmn(x2, x3)

⟩Φmn(x2, x3), ∀

p1

p2

∈ X.(5.21)

For x1 = 0 the above infinite series is clearly convergent, whereas for x1 → 0+ the

limit does not exist. Further we note that above series expression (5.21) satisfies

identity and semigroup properties as given in Definition 11, however, it lacks strong

continuity, except if we assume that the projection terms in angle brackets above are

decaying at a rate faster than the growth rate of eλmnx1 . Now with the introduction of

this assumption the limit x1 → 0+ exists. There is a large class of smooth analytical

functions that satisfy this condition. Once again, this reveals a historically known

fact about solving Cauchy problems for steady-state heat equation that unique and

stable solutions do not exist for non-smooth data [39, 54]. Let the Hilbert space X,

as given in equations (3.7) and (3.10), be composed of two mutually exclusive parts

as

X = X1 ⊕X2, (5.22)

129

where X1 satisfy conditions as stated above such that A forms a strongly continuous

semigroup and X1 and X2 both make the full space X. Thus with this additional

smoothness assumption, equation (5.23) represents the strongly continuous semigroup

generated by operator matrix A.

Wx1

p1

p2

=∑

m,n∈Z?eλmnx1

⟨ p1

p2

,Φmn

⟩Φmn, ∀

p1

p2

∈ X1, (5.23)

where X = X1 ⊕X2 such that equation (5.23) generates a strongly continuous semi-

group and the inner product in (5.23) is defined by equation (5.9).

5.2.2.2 System Observability

Proposition 2. Let W be the strongly continuous semigroup generated by operator

matrix A under the assumptions as given in theorem (8). For any arbitrarily small

ε > 0 such that if |x1 − x1| < ε, the pair (C,A) is final state observable (hence exactly

observable using Note 1 in time-like interval |x1 − x1| > 0 at a particular x1, where

C ∈ L(X, Y ).

Proof. Let ξ(0) be the initial guess at x1 = 0, given by,

ξ(0) =

ξ1(0)

ξ2(0)

=

p1(x2, x3)

p2(x2, x3)

. (5.24)

Φmn(x2, x3) for m,n ∈ Z? be an orthonormal basis in X. Let us first prove the final

state observability condition for a general mode Φm′n′ with corresponding eigenvalue

130

λm′n′ as follows,

‖WxΦmn‖X =

∥∥∥∥∥ ∑m,n∈Z?

eλmnx1 〈Φm′n′ ,Φmn〉Φmn

∥∥∥∥∥X

, ∀ Φm′n′ ∈ X, m′, n′ ∈ Z?,

= eλmnx1 ‖Φmn‖X ,

= eλmnx1 , (5.25)

also,

‖CWx1Φmn‖Y =

∥∥∥∥∥ ∑m,n∈Z?

eλmnx1 〈Φm′n′ ,Φmn〉 CΦmn

∥∥∥∥∥Y

, ∀ Φm′n′ ∈ X, m′, n′ ∈ Z?,

= eλmnx1 ‖CΦmn‖Y ,

= eλmnx1|ρmn| ‖cos(4nx3)‖Y ,

=π

8eλmnx1|ρmn|. (5.26)

Comparing equations (5.25) and (5.26) implies,

‖CWx1Φmn‖Y ≥ k1 ‖Wx1Φmn‖X , (5.27)

if and only if,

k1 ≤π

8|ρmn|, (5.28)

for a particular choice of Φmn there always exists k1 such that final state observability

condition (5.27) is satisfied.

C ∈ L(X, Y ) is a linear boundary observation operator. Now let ξ(0) =∑

m,n∈Z? γmnΦmn

where γmn are projection terms whose decay rate is greater than the growth rate of

eλmnx1 with λmn as eigenvalues of A corresponding to eigenvectors Φmn. Clearly,∑m,n∈Z? γmn and

∑m,n∈Z? ρmn are bounded from above, hence,

‖CWx1ξ(0)‖Y ≥ k2 ‖Wx1ξ(0)‖X , ∀ ξ(0) ∈ X1, (5.29)

131

where k1, k2 both are independent of x1. Further using Note 1 for arbitrarily small ε

pair (C,A) is exactly observable.

This special exact and final state observability result for arbitrarily small ε suggests

that to recover solution, on a line (ω ∩ Γ2) at a particular x1, we do not require any

other measurement from adjacent lines except the line (ω ∩ Γ1). This implies we can

decompose the solution along time-like variable x1. Further for each x1 solution can

be computed in parallel.

5.2.3 Dimension Decomposition

Observability result given in Proposition 2 suggests that problem (5.10) can be solved

on a particular cross-section ω independent of any help from the adjacent cross-

sections. This, in other words, suggests that, in this particular setting (domain Ω3

with two congruent parallel surfaces as given in Figure. 5.1), diffusion along time-like

variable x1 is zero and, on a particular ω (as shown in Figure. 5.3), problem (5.10)

boils down to,

For all ω ∈ Ω3, find u(x3) on Γt:

4x2,x3

u =∂2u

∂x22

+∂2u

∂x23

= 0 in ω,

u = fsub

on Γb,

∂u

∂n= g

subon Γb,

∂u

∂n= 0 on Γl ∩ Γr,

(5.30)

132

where Γt = ω ∩ Γ2, Γb = ω ∩ Γ1, fsub

= f |Γb , gsub

= g |Γb . Let us further introduce two

auxiliary variables ζ1, ζ2 as follows,

ζ1(x2, x3) = u(x2, x3),

ζ2(x2, x3) =∂u

∂x3

,

(5.31)

x2 and x3 are rectangular co-ordinates, by introducing these auxiliary variables the

resulting Laplace equation can now be written as,

∂ζ

∂x3

= Asubζ, (5.32)

where,

ζ =

ζ1

ζ2

; Asub =

0 1

−4x2

0

; −4x2

= − ∂2

∂x22. (5.33)

ζ1 and ζ2 are called new state variables and using these variables, problem (5.66) can

be written in equivalent form as,

For all ω ∈ Ω3, find ζ1(x2, x3) on Γt:

∂ζ

∂x3

= Asubξ in ω,

Csubζ(x3) = ζ1(x3) = f

subon Γb,

∂ζ1

∂x2

= gsub

on Γb,

∂ζ1

∂x3

= 0 on Γl∪r,

(5.34)

where Csub is the observation operator such that Csubζ = ζ1 |Γb . A boundary value

problem in this form has a first-order state equation in variable x3 and overdetermined

data is available on Γb. Before the introduction of iterative observer equations, let us

133

assume that left-hand boundary Γl is connected to right-hand boundary Γr to have

the notion of infinite time-like variable x3 over ω. The reason for having such an

assumption is that we are trying to develop an observer using space as time-like and

hoping that this observer will converge asymptotically in time-like variable x3. Let

m be a non-negative integer index of iteration over the domain ω in x3-direction. Let

x[m]3 , as given in Definition (15), represents x3 ∈ [a3, b3] for the m-th iteration over

the interval [a3, b3].

5.2.4 Observer Design for the Subproblem

Theorem 9. For consistent Cauchy data boundary value problem (5.35) asymptoti-

cally (m→∞) converges to the true solution of boundary value problem (5.34).

For all ω ∈ Ω3,

∂

∂x3

ζ(x2, x[m]3 ) = A

subζ(x2, x

[m]3 )−K C

sub(ζ(x2, x

[m]3 )− ζ) in ω,

∂

∂x2

ζ1(x2, x[m]3 ) = g

subon Γb,

∂2

∂x22

ζ1(x2, x[m]3 ) = − ∂2

∂x23

ζ1(x2, x[m]3 )−K C

sub(ζ(x2, x

[m]3 )− ζ) on Γt,

ζ(x2, x[m]3 ) |initial= ζ(x2, x

[m−1]3 ) in ω,

(5.35)

where ”ˆ” represents estimated quantity and ζ(x2, x[m]3 ) |initial represents the estimate

over the whole domain ω at the start of m-th iteration. Observer starts at index m = 1

which represents first iteration. ζ(x2, x[m=0]3 ) is initial guess at the start of the first

iteration over the whole domain Ω3. Any value of initial guess ξ(x2, x[m=0]3 ) can be

chosen at the start of first iteration. For each subsequent iteration, result of the

previous iteration is used as initial estimate as given in the last equation in (5.35).

Third equation in (5.35) is the assumption that Laplace equation is valid on the top

boundary and this provides necessary boundary condition required on Γt. Csub

is the

134

observation operator such that Csubζ = ζ1 |Γb . K is the correction operator chosen in

such a way that state estimation error on Γb given by ( Csubζ − C

subζ)) converges to zero

exponentially.

Proof. Semigroup generated by Asub

, observability study for pair ( Csub, Asub

) and proof of

theorem (9) is provided in chapter 3.

5.2.5 Numerical Implementation and Simulation Results

For the numerical simulations, a rectangular prism Ω3 is considered such that,

Ω3 =

x ∈ Ω3 : x =

x1

x2

x3

∈

[0, 2π][0, π

4

][0, 2π]

;

. (5.36)

A well-posed boundary value problem is solved using the method of separation of

variables to find analytical solution. The surface potential u = cos(x1) cos(x3) is

applied at Γ2 (x1-x3 plane at x2 = π/4). Homogeneous Neumann boundary∂u

∂n= 0

is considered on all other boundary surfaces. This gives analytical solution over the

whole domain as,

u(x1, x2, x3) =cosh(

√2x2)

cosh(√

2π4)

cos(x1) cos(x3). (5.37)

Analytical solution on Cauchy surface Γ1 (x1-x3 plane at x2 = 0) along with homoge-

neous Neumann boundary condition∂u

∂x2

= 0 on Γ1 is used as Cauchy data. Observer

algorithm is run for a number of iterations over all ω using parallel processing. The

analytical solution on Γ1 and Γ2 are shown in Figure 5.4 and 5.5 respectively. The

solution recovered using the iterative observer and divide and conquer approach is

presented in Figure 5.6. The difference between the analytical and recovered solution

on Γ2 is shown in Figure 5.7.

135

Figure 5.4: Analytical solution u on Γ1 Figure 5.5: Analytical solution u on Γ2

Figure 5.6: Recovered solution on Γ2

using observer algorithmFigure 5.7: Difference of the analyti-cal solution and the one recovered byobserver algorithm on Γ2

5.3 Point Source Localization Problem for Poisson Equation

in 3D

In this section, an iterative observer-based method is developed to solve point source

localization problem for Poisson equation in a 3D rectangular prism with available

boundary data. The technique requires a weighted sum of solutions of multiple bound-

ary data estimation problems for Laplace equation over 3D domain. The solution of

each of these boundary estimation problems involves writing down the mathematical

problem in state-space-like representation using one of the space variables as time-like.

First system observability result for 3D boundary estimation problem is recalled in

136

an infinite dimensional setting. Then, based on the observability result, the boundary

estimation problem is then decomposed into a set of independent 2D sub-problems.

These 2D problems are then solved using an iterative observer to obtain the solution.

Theoretical results are provided. The method is implemented numerically using fi-

nite difference discretization schemes. Numerical illustrations along with simulation

results are provided.


Let Ω4 be a rectangular prism in R3 as shown in Figure 5.8 and ∂Ω4 = ∪6i=1Γi be the

boundary of Ω4. Let us consider Poisson equation,

4u = f in Ω4, (5.38)

with Laplacian operator 4 =∂2

∂x2+

∂2

∂y2+

∂2

∂z2. Let the steady-state potential field

u is generated by a number of distinct point sources inside the rectangular prism Ω4:

f(x, y, z) =N∑k=1

Ckδ(x− xk, y − yk, z − zk) (x, y, z) ∈ Ω4, (5.39)

where δ(x− x, y− y, z− z), k = 1, · · · , N represent Dirac delta point sources localized

at (xk, yk, zk) and scalars Ck, k = 1, · · · , N are the corresponding magnitudes.

The objective here is to localize point sources δ’s for all k = 1, · · · , N from available

Cauchy data on the boundary ∂Ω4 given by,

u = g on ∂Ω4,

∂u

∂n= h on ∂Ω4.

(5.40)

In the following section a strategy to localize point sources is presented.

137

x

y

z

Γ2

Γ1

Figure 5.8: Rectangular prism Ω4 with six boundary surfaces, ∂Ω4 = ∪6i=1Γi. Surfaces

Γ1, . . . ,Γ6 represent bottom, top, and side surfaces respectively.

5.3.2 Point Source Localization as Boundary Estimation Prob-

lem

First of all, let us consider that available Cauchy data as given in equation (5.40) has

various components on different surfaces of the Ω4 such that,

g = ∪6i=1gi|Γi , (5.41)

h = ∪6i=1hi|Γi . (5.42)

In this section, a mathematical result is presented that boils down the source lo-

calization problem given in section 5.3.1 to a weighted sum of following boundary

estimation problems for Laplace equation in 3D.

For i ∈ 1, . . . , 6, find steady-state potential field Si on Γi,opp:

4Si = 0 in Ω4,

Si = gi on Γi,

∂Si∂n

= hi on Γi,

∂Si∂n

= 0 on Γi,adjs,

(5.43)

where Γi,opp is the surface opposite to Γi and Γi,adjs are the surfaces connected to Γi.

138

Theorem 10. For all i in integer set 1, . . . , 6, let Si be the steady-state potential

field over Ω4 obtained by solving boundary estimation problem (5.43). Let for all i,

vi be the solution obtained by solving boundary value problem for Laplace equation,

4vi = 0 in Ω4,

vi = 1 on Γi,

vi = 0 on ∂Ω4\Γi,

(5.44)

then,

u =1

2

[6∑i=1

Sivi

]in Ω4, (5.45)

f =6∑i=1

∇Si.∇vi in Ω4, (5.46)

where u and f solve the boundary value problem for Poisson equation given by equa-

tions (5.38) and (5.40).

Proof. We have Si and vi satisfying Laplace equation for all i ∈ 1, . . . , 6. From

equation (5.45) we can write,

∇u =1

2

6∑i=1

(Si∇vi + vi∇Si) , (5.47)

∇.(∇u) = ∇.

(1

2

6∑i=1

(Si∇vi + vi∇Si)

), (5.48)

which gives,f = 4u =

6∑i=1

∇Si.∇vi. (5.49)

thus we have that u as given in equation (5.45) satisfies Poisson equation 4u = f up

to an additive constant. Now using the properties of vi, Si we have to show that u

and f as given by equations (5.45) and (5.46) satisfy boundary value problem given

139

by equations (5.38) and (5.40). We can write,

∫Ω4

[6∑i=1

vi4Si

]dΩ4 = 0, (5.50)

using Green’s first identity,

∫∂Ω4

[6∑i=1

vi∂Si∂n

]d∂Ω4 =

∫Ω4

[6∑i=1

∇vi.∇Si

]dΩ4, (5.51)

We have vi = 1 on Γi and zero elsewhere on boundary. This gives,

∫∂Ω4

[6∑i=1

vi∂Si∂n

]d∂Ω4 =

6∑i=1

[∫Γi

∂Si∂n

d∂Ω4

]=

∫∂Ω4

h dΩ4, (5.52)

applying divergence theorem we have,

∫∂Ω4

h dΩ4 =

∫Ω4

∇.∇u dΩ4 =

∫Ω4

f dΩ4. (5.53)

Combining equations (5.51), (5.52) and (5.53),

∫Ω4

f dΩ4 =

∫Ω4

[6∑i=1

∇vi.∇Si

]dΩ4, (5.54)

above relation is true for all sizes of 3D rectangular prisms Ω4, thus we have,

f =6∑i=1

∇Si.∇vi. (5.55)

Remark 3. The mathematical result presented in Theorem 10 is valid for all sizes

of 3D rectangular prisms.

140

5.3.3 Two-step Process for Source Localization:

Based on above result, a two-step source localization technique can be developed,

Step 1 : For all i, solve boundary estimation problem (5.43) to find steady-state

potential field Si over Ω4.

Step 2 : For all i, compute weight profiles vi by solving well-posed boundary value

problem (5.44). Combine Si’s and vi’s using equation (5.45) to have an estimate

of Poisson problem solution u over Ω4. The position of Dirac delta sources inside

the domain is provided by the maxima or minima of estimated u.

In the following section, the divide and conquer approach and iterative observer al-

gorithm are used to solve boundary estimation problem (5.43).

5.3.4 Boundary Estimation Problem for Laplace Equation

In this section, the divide and conquer approach, based on iterative observer design,

is used to solve boundary estimation problem for Laplace equation. The technique

is presented to solve problem (5.43) for index i = 1, however, the theoretical results

and implementation technique remain the same for all i ∈ 1, . . . , 6.

5.3.5 Preliminary Theoretical Results

Let us introduce two auxiliary variables ξ1 = S1 and ξ2 =∂S1

∂xto rewrite Laplace

equation in problem (5.43) as,

∂ξ

∂x= Aξ, (5.56)

where,

ξ =

ξ1

ξ2

; A =

0 1

−4y,z

0

; −4y,z

= − ∂2

∂y2− ∂2

∂z2. (5.57)

141

Let A : D(A) → X be defined on a rectangular cross-section ω of Ω4, parallel to yz

plane, as shown in Figure 5.9 such that,

ω =

α ∈ ω : α =

y

z

∈ [a2, b2]

[a3, b3]

;

, (5.58)

where b2 > a2, b3 > a3, a2, a3, b2, b3 ∈ R+, ω = ω ∪ Γl∪r∪t∪b, Γb = ω ∩ Γ1 and

Γt = ω ∩ Γ2.

X = H1Γb

(ω)× L2 (ω) , (5.59)

D(A) =

[f ∈ H2 (ω) ∩H1

Γb(ω) :

df

ds|Γb = c2

]×H1

Γb(ω) , (5.60)

where,H1

Γb(ω) =

f ∈ H1 (ω) : f |Γb = c1

, (5.61)

and c1, c2 are constants (coming from Cauchy data at a particular point on Γb) and


⟨ q1

q2

,

p1

p2

⟩ =

∫ω

∇q1(α).∇p1(α)dω

+

∫ω

q1(α).p1(α)dω +

∫ω

q2(α).p2(α)dω. (5.62)

here α as given in equation (5.58). It can be seen that D(A∞) is dense in X. Problem

(5.43) for index i = 1 can be written in control familiar form using auxiliary variables

as,

142

x

y

z

ωω

Γt

Γb

Γl Γr

Figure 5.9: Rectangular cross-section ω at a particular value of x in yz-plane insideΩ4.

Find S1 on Γ1,opp (:= Γ2):

∂ξ

∂x= Aξ in Ω4,

Cξ = ξ1 = g1 on Γ1,

∂ξ1

∂n= h1 on Γ1,

∂ξ1

∂n= 0 on Γ1,adjs,

(5.63)

where C : X → Y is the boundary observation operator and Y be the output space

given as,

Y =

(f1)|Γb :

f1

f2

∈ X, . (5.64)

Y forms a Hilbert space with respect to the norm,

〈q1(y), p1(y)〉 =

∫Γb

q1(y)p1(y)dy. (5.65)

Boundary value problem (5.63) has a first order state equation in variable x and

overdetermined data is available on Γ1. The solution of this first-order state equation

leads to the study of semigroup generated by unbounded differential operator matrix

A. Further we study observability for the pair (C,A) in infinite dimensional setting.

143

Proposition 3. Let m,n ∈ Z? be the non-zero set of integers, for A : D(A) → X

(as given in (5.57), (5.59) and (5.60)) there exists an infinite set of orthonormal

eigenvectors (Φmn) and corresponding eigenvalues (λmn). Furthermore, A generates

a strongly continuous semigroup for vectors

p1

p2

∈ X, if and only if, the decay

rate of the sequence

⟨ p1

p2

,Φmn

⟩is greater than the growth rate of the sequence

eλmnx for all m,n ∈ Z?.

Proof. Proof of above proposition is similar to the result presented in Theorem 8.

5.3.6 Observability Result

Theorem 11. Let W be the strongly continuous semigroup generated by operator

matrix A under the assumptions as given in Proposition 1. For any arbitrarily small

ε > 0 such that if |x− x| < ε, the pair (C,A) is exact and final state observable in

time-like interval |x− x| > 0 at a particular x, where C ∈ L(X, Y ).

Proof. The proof of above theorem is similar to Proposition 2.

This special exact and final state observability result for arbitrarily small interval ε

suggests that to recover solution, on a line (ω∩Γ2) at a particular x, we do not require

any other measurement from adjacent lines except the line (ω ∩ Γ1). This implies we

can decompose the solution along time-like variable x. Further for each x solution

can be computed in parallel.

5.3.7 Dimension Decomposition

Observability result given in Theorem 2 suggests that problem (5.63) can be solved on

a particular cross-section ω independent of any help from the adjacent cross-sections.

This, in other words, suggests that diffusion along time-like variable x is zero and, on

a particular ω (as shown in Figure 5.9), problem (5.63) boils down to,

144

For all ω ∈ Ω4, find u(y) on Γt:

4y,zu =

∂2u

∂y2+∂2u

∂z2= 0 in ω,

u = g1sub

on Γb,

∂u

∂n= h1

subon Γb,

∂u

∂n= 0 on Γl ∩ Γr,

(5.66)

where g1sub

= g1 |Γb , h1sub

= h1 |Γb . Let us further introduce two auxiliary variables ζ1, ζ2

as follows, ζ1(y, z) = u,

ζ2(y, z) =∂u

∂y,

(5.67)

further by introducing these auxiliary variables the resulting Laplace equation can

now be written as,

∂ζ

∂y= A

subζ, (5.68)

where,

ζ =

ζ1

ζ2

; Asub

=

0 1

−4z

0

; −4z

= − ∂2

∂z2. (5.69)

ζ1 and ζ2 are called new state variables and using these variables, problem (5.66) can

be written in equivalent form as,

For all ω ∈ Ω4, find ζ1(y, z) on Γt:

∂ζ

∂y= A

subξ in ω,

Csubζ(y) = ζ1(y) = g1

subon Γb,

∂ζ1

∂z= h1

subon Γb,

∂ζ1

∂y= 0 on Γl∪r.

(5.70)

145

where Csub

is the observation operator such that Csubζ = ζ1 |Γb . Boundary value problem

in this form has a first order state equation in variable y and overdetermined data

is available on Γb. Before the introduction of iterative observer equations, let us

assume that left-hand boundary Γl is connected to right-hand boundary Γr to have

the notion of infinite time-like variable y over ω as shown in Figure 5.11. The reason

for having such an assumption is that we present an observer using space as time-like

with asymptotic convergence in time-like variable y. Let m be a non-negative integer

index of iteration over the domain ω in y-direction. Let y[m] represents y ∈ [a2, b2] for

the m-th iteration over the interval [a2, b2].

5.3.8 Observer Design for the Subproblem

For all ω ∈ Ω4:

∂

∂yζ(y[m], z) = A

subζ(y[m], z)−K C

sub(ζ(y[m], z)− ζ) in ω,

∂

∂zζ1(y[m], z) = h1

subon Γb,

∂2

∂z2ζ1(y[m], z) = − ∂2

∂y2ζ1(y[m], z)

−K Csub

(ζ(y[m], z)− ζ) on Γt,

ζ(y[m], z) |initial= ζ(y[m−1], z) in ω,

(5.71)

where“ˆ”represents estimated quantity and ζ(y[m], z) |initial represents the estimate

over the whole domain ω at the start of m-th iteration.

Observer starts at index m = 1 which represents first iteration. ζ(y[m=0], z) is initial

guess at the start of the first iteration over the whole domain Ω4. Any value of initial

guess ξ(z, y[m=0]) can be chosen at the start of first iteration. For each subsequent

iteration, result of the previous iteration is used as initial estimate as given in the

146

θ1

θ2

θ3

θ4

θ1

θ2

θ3

θ4

ω

Figure 5.10: Idea of iterations over rectangular cross-section ω.

last equation in (5.71). The third equation in (5.71) is the assumption that Laplace

equation is valid on the top boundary and this provides necessary boundary condition

required on Γt. Csub

is the observation operator such that Csubζ = ζ1 |Γb= g1

sub|Γb . K is

the correction operator chosen in such a way that state estimation error on Γb given

by ( Csubζ − C

subζ) converges to zero asymptotically.

Theorem 12. For consistent Cauchy data boundary value problem (5.71) asymptot-

ically (m→∞) converges to the true solution of boundary value problem (5.70).

Proof. Study of existence of exponential of Asub

under certain conditions, observability

study for pair ( Csub, Asub

) and proof of Theorem 12 are similar to the results presented

in Chapter 3.

5.3.9 Numerical Implementation and Results

In this section, numerical implementation and simulation results are presented. For

illustrative purposes, a unit cube is considered and finite difference discretization

schemes are used to implement two-step strategy for source localization. Figure 5.11

represents three orthogonal cross-sectional planes inside Ω4 which are used for 3D

visualization of different simulation plots. Figure 5.12 represents the solution of Pois-

son equation in the unit cube with homogeneous Neumann boundary data on ∂Ω4

and one point source in the middle of the domain. The solution is presented on a

147

x

y

z

Figure 5.11: Domain Ω4 with three orthogonal cross-sectional planes.

coarse 20× 20× 20 uniform grid. Solution over a particular cross-section ω is shown

in Figure 5.13 on a uniform 200 × 200 uniform grid. Because of symmetry solution

over all the three orthogonal cross-sectional planes look alike.

Now to localize this particular point source from boundary data it is required

to solve boundary estimation problem (5.43) for all i ∈ 1, . . . , 6 using dimension

decomposition approach presented in section 5.3.4. For this, using the special ob-

servability result for dimension decomposition presented in section 5.3.4, a particular

cross-section ω is considered. Dirichlet data g1sub

on Γb is extracted from ω shown in

Figure 5.13. This Dirichlet boundary data on Γb along with homogeneous Neumann

boundary data is used in iterative observer algorithm (5.71) to estimate boundary

data on the opposite boundary Γt as shown in Figure 5.14. Similarly, the iterative

observer solution is computed on all cross-sections parallel to ω to make up the full

solution profile S1 over Ω4. Similarly S2, . . . , S6 can be computed. Figure 5.15 shows

solution plot for S2 over Ω4.

Well-posed boundary value problem (5.44) is solved numerically for all i ∈ 1, . . . , 6

to compute weight profiles v1, . . . , v6. Figure 5.16 shows a particular weight profile v2

over Ω4. Finally the weighted sum given in equation (5.45) is computed numerically.

Figure 5.17 shows the weighted sum over Ω4. The local maxima in the center locate

the point source.

148

Figure 5.12: A single point source inthe middle of a 1 × 1 × 1 cube on auniform 20× 20× 20 grid.

Figure 5.13: Cross-sectional plane ωfrom Figure 5.12 on a 200 × 200 uni-form grid.

5.4 Conclusion

The divide and conquer approach presented in this chapter helps to solve boundary

estimation problem for the Laplace equation, in a particular 3D domain, as a num-

ber of independent 2D subproblems. The subproblems can be solved simultaneously,

using parallel implementation, which makes this algorithm advantageous compared

to the techniques in literature. Most of the previously existing techniques are op-

timization based methods and require the solution of the mathematical problem in

3D for a number of times with some cost criteria to obtain the convergence. Further

this demonstration of control-based algorithm solving a steady-state estimation prob-

lem in 3D highlights the potential that dynamical systems inspired algorithms can

be potentially advantageous and inspiring to develop solution techniques for steady-

state PDE problems. Further, the dimension decomposition approach presented in

this chapter is extended to source localization and estimation problems for Poisson

equation. Numerical simulation results highlight the usefulness of the proposed algo-

rithms.

149

0 0.2 0.4 0.6 0.8 1

−50

−40

−30

−20

−10

0

Am

plit

ude

Iterative observer solution

Dirichelt data g1sub

on Γb


Figure 5.14: Iterative observer solution from equation (5.71) with h1,sub = 0 andg1,sub extracted from Figure 5.13 on Γb and recovered boundary data on oppositeboundary. Similarly iterative observer solution can be computed on cross-sectionalplanes parallel to ω.

Figure 5.15: Solution profile S2 on a20 × 20 × 20 uniform grid, obtainedby solving boundary estimation prob-lem (5.43) for i = 2 using iterative ob-server.

Figure 5.16: Weight profile v2 obtainedby solving boundary estimation prob-lem (5.44) for i = 2 on a uniform20 × 20 × 20 grid. (only two orthog-onal cross-sectional planes displayed)

150

Figure 5.17: Weighted sum computed using equation (5.45). Local maxima in thecenter locate the position of the point source.

151

Chapter 6

Concluding Remarks

Sometimes when I consider what tremendous consequences come from little

things... I am tempted to think... There are no little things.

(—Bruce Barton)

In this chapter, concluding discussions are provided about the iterative observer

algorithms, for source and boundary data estimation problems, presented in this

thesis. The algorithms tackle time independent steady-state PDE systems with dy-

namical system inspired algorithms using space variable as time-like. The resultant

algorithms are shown to be robust to noise and at the same time easier to implement.

6.1 Summary of the Thesis Work

An iterative observer algorithm is presented to solve boundary data estimation prob-

lem for Laplace equation in chapter 3. The observer algorithm works for smooth data

case using one of the space variables as time. The algorithm sweeps over the whole

domain for a number of iterations to obtain convergence. Numerically accurate and

stable results are presented. However, the partially available boundary data is usually

corrupted with noise in most of the practical scenarios. Further, this boundary data

estimation problem with available noisy boundary data is highly ill-posed in the sense

of stability. An optimal iterative algorithm is presented that tackles this particular

ill-posed scenario.

The iterative observer algorithm is then used to develop a strategy for source

localization and estimation for a system governed by the Poisson equation in chapter

152

4. This strategy is detailed along with simulation results for both noisy and noise-

free cases. Several test scenarios are considered. Finally an approach to tackle higher

dimensional problems is provided. The method provides a way to divide and conquer

a three dimensional problem (three dimensional domain with two congruent parallel

surfaces) to a set of two dimensional independent subproblems. These subproblems

are then used using iterative observer algorithm. Similarly, a source estimation and

localization strategy is developed for the Poisson equation problem.

The algorithms presented in this thesis are formulated as a state space like system,

that is, by writing a second-order steady-state elliptic PDE system as a first order

evolutionary ODE system. The numerical implementation is done using finite differ-

ences and is fairly simple compared to the existing optimization based techniques.

The main advantage is that the problem is formulated on a sub domain, that is,

rather than solving a three dimensional problem over three dimensional domain for

several times, a number of two dimensional problems are solved for a number of times

to obtain convergence.

6.2 Future Research Directions

The work presented in this thesis proposal can be extended in the following directions.

6.2.1 Extensions to Arbitrary Shaped Domains

The idea of using space variable as time-like put some restrictions on the shape of the

domain under consideration. The algorithms presented in this thesis work are applied

two dimensional regular shaped domains and three dimensional domains with two

congruent parallel surfaces. The extension of these algorithms to other arbitrarily

shaped domains need further investigation. Similarly, the source localization and

estimation strategy has been applied to similar kind of two and three dimensional

domains.

153

Ω

Γ∗

Γ∗

Ω

Ω Γ∗

Ω

Γ∗

Figure 6.1: Domains under considera-tion for the boundary estimation prob-lem for Laplace equation with Γ∗, theunknown data boundary.

Ω∗

∗

*Ω

Ω

*

*

Figure 6.2: Domains under consider-ation for the source localization andestimation problems for Poisson equa-tion.

6.2.2 Iterative Observer Applications to Other Steady-State

PDE Systems

The algorithms presented in this document tackle Laplace and Poisson types of linear

systems. The results presented in this thesis are first steps towards the development

of dynamical systems inspired algorithms for steady-state PDEs, thus extending the

existing concepts of control theory towards all kinds of PDE systems. The results

presented in this document are promising. However, further investigations need to be

done to tackle other steady-state linear and non linear PDE systems.

6.2.3 Steady-State Energy Field Imaging Technique

Well-known tomographic imaging techniques in radiology, geophysics, biology, arche-

ology and materials science use penetrating waves to construct the the object’s form,

especially inner visual representation. Almost all of these techniques are sensitive to

the mass density variations inside the object. For example, X-ray imaging in biomed-

ical works on the principle that high energy X-rays pass through different parts of a

patient’s body, and depending on the attenuations caused by mass density variations

154

(tissues and bones), a different intensity of X-ray will come out of the body from the

opposite end. The X-rays coming out are recorded on a film, which then provides the

information about mass density variations inside the body.

The source estimation strategy presented in Chapter 5 can be extended and used

as a steady-state energy field imaging technique in an n-D rectangular prism. The

method does not use wave penetration, rather it uses steady-state diffusive model to

recover distributed potential field inside a homogenous medium. The technique is

not sensitive to mass density variations, however, it can well recover the steady-state

forced fields inside the medium. Some preliminary numerical results are presented, in

a 3D rectangular prism domain, for the steady-state diffusion model with distributed

sources.

6.2.3.1 Initial Simulation Results

In this section, numerical implementation is detailed and initial simulation results

are presented. For illustrative purposes, a unit cube is considered and finite differ-

ence discretization schemes are used to implement two-step strategy for steady-state

potential field imaging. As shown in Chapter 5, Fig. 5.11 represents three orthogo-

nal cross-sectional planes inside Ω4 which are used for 3D visualization of different

simulation plots. Fig. 6.3 represents the solution of Poisson equation in the unit

cube with homogeneous Neumann boundary data on ∂Ω4 and a distributed sources

f = exp(−(2.5(x−0.5))2−(2.5(y−0.5))2−(2.5(z−0.5))2) centered inside the domain.

This particular choice for f is for illustrative purposes as it is centered inside the unit

cube domain. The solution is presented on a coarse 50 × 50 × 50 uniform grid. A

cross-sectional view of this numerical solution is also shown in Fig. 6.4 at x = 0.5.

Because of the qualitative similarity of this particular example, the cross-sectional

view will look alike at y = 0.5 and z = 0.5. The distributed potential field inside the

Ω4, as shown in Fig. 6.3, is to be recovered applying the imaging technique, presented

155

here, using only the boundary data with added noise.

Now, to apply the steady-state potential field imaging technique, the Dirichlet

boundary data g is extracted from the solution presented in Fig. 6.3 over ∂Ω4.

Based on the dimension decomposition approach, and the special observability result

presented in chapter 5, each particular cross-section ω is considered independent of

adjacent cross-sections. The Dirichlet data is extracted for each particular ω. This

Dirichlet data is then corrupted with additive white Gaussian noise with η1 = 0 and

σ1 = 5 × 10−3. This noisy Dirichlet data g1sub

at Γb = ω ∩ Γ1 is shown in blue in

Fig. 6.5. The data on the remaining boundaries of ω|x=0.5 will look alike because

of qualitative similarity. The solution of boundary value problem (5.70) is obtained

by applying optimal iterative algorithm (4.49), (4.50) and is shown in Fig. 6.5 in

black. The optimal iterative algorithm recovers the boundary estimate at the opposite

boundary Γt by solving the boundary estimation problem for the Laplace equation

(5.70). The corresponding solution profile S1 at ω|x=0.5 is shown in Fig. 6.6. Similarly

the solution profile S1 is computed on all cross-sections parallel to ω|x=0.5. The

full tomographic image profile S1 (solution of problem 5.43 for i = 1) over Ω4 is

shown in Fig. 6.7, this solution is obtained by stacking up all the S1 solutions over

all the cross-sections parallel to ω|x = 0.5. The weight profile v1, corresponding to

tomographic image S1 over Ω4, is obtained by numerically solving boundary value

problem (5.44), for i = 1, as shown in Fig. 6.8. In a similar way S2, . . . , S6 and

v2, . . . , v6 are computed numerically over Ω4. Finally, steady-state potential field

image is generated using equation (5.45) as shown in Fig. 6.9. A cross-sectional view

at ω|x=0.5 is also shown in Fig. 6.10. The algorithm recovers the distributed steady-

state potential field reasonably well, Fig.s 6.3 and 6.4 can be compared with Fig.s 6.9

and 6.10 respectively.

156

Figure 6.3: Numerical solution of thePoisson equation 4u = f over Ω4 withf = exp(−(2.5(x − 0.5))2 − (2.5(y −0.5))2 − (2.5(z − 0.5))2) and homoge-neous Neumann boundary data at ∂Ω4.The solution is computed over a 50 ×50 × 50 uniform grid using 2nd orderaccurate centered finite difference dis-cretization schemes.

Figure 6.4: A particular cross-sectionalview of the numerical solution pre-sented in Fig. 6.3, at x = 0.5.

0 0.2 0.4 0.6 0.8 1

−30

−20

−10

0

Am

plit

ude

Dirichlet data and recovered signal on opposite boundaries

Noisy Dirichlet data on Γb


Figure 6.5: (In blue): Dirichlet data, extracted from Fig. 6.4 at Γb = ω∩Γ1, corruptedwith added white Gaussian noise with η1 = 0 and σ1 = 5×10−3; (In black) Numericalsolution of boundary estimation problem (5.66), obtained by using optimal iterativealgorithm (4.49), (4.50) at cross-section ω|x=0.5 at Γt = ω ∩ Γ2.

157

Figure 6.6: Full solution profile overcross-section ω|x=0.5 obtained by usingfinite difference discretization schemesand the estimated boundary data fromFig. 6.5

Figure 6.7: Tomographic image S1 overΩ4 obtained by solving boundary esti-mation problem for Laplace equation(5.43) using dimension decompositionapproach and optimal iterative algo-rithm.

Figure 6.8: Weight profile v1 obtained by numerically solving boundary value problem(5.44) for index i = 1.

158

Figure 6.9: Weighted sum obtainedfrom equation (5.45) by combiningtomographic image profiles S1, . . . , S6

and corresponding weights v1, . . . , v6.The simulation result shows recovery ofthe distributed potential field presentedin Fig. 6.3, using only the noise cor-rupted boundary data.

Figure 6.10: A cross-sectional view ofweighted sum over ω|x=0.5. The resultabove to be compared with the cross-sectional view given in Fig. 6.4.

159

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APPENDICES

A Appendix :

Proof of Some Theorems

Theorem 1

Proof. If Ψτξ0 = 0, then the analytic function y(t) = CetAξ0 vanishes identically on

[0, τ ]. In particular, CAkξ0 = y(k)(0) = 0 for all k ≥ 0 which shows that ξ0 is also in

kernel of the matrix appearing in the right-hand side of (2.4).

Conversely, if CAkξ0 = 0 for all 0 ≤ k ≤ n − 1, then CAkξ0 = 0 for all k ≥ 0,

since by Cayley-Hamilton theorem, the powers Ak for k ≥ n are linear combinations

of Ak for 0 ≤ k ≤ n− 1. Consequently, y(t) = CetAξ0 =∑∞

k=0

tk

k!CAkξ0 = 0 for all t

and thus ξ0 ∈ KerΨτ . The fact that (2.4) provides characterization for observability

follows immediately from (2.5).

A.1 An Example of Exactly Observable System based on

String Equation [11]

In this section a semigroup, associated with the equations modelling the vibration of

an elastic string of length π fixed at both ends, is constructed. Initial boundary value

166

problem for one dimensional wave equation can be written as,

∂2w

∂t2(x, t) =

∂2w

∂x2(x, t) x ∈ (0, π), t ≥ 0,

w(0, t) = 0, w(π, t) = 0 t ∈ [0,∞),

w(x, 0) = f(x),∂w

∂t(x, 0) = g(x) x ∈ (0, π).

(A.1)

by setting,

ξ(t) =

w(., t)

∂w

∂t(., t)

, (A.2)

system of equations (A.1) can be written as,

ξ(t) = Aξ(t) ∀t ≥ 0, ξ(0) =

f

g

. (A.3)

A.1.1 Semigroup Generated by A

Let us denoteX= H10 (0, π)×L2(0, π), which is a Hilbert space with the scalar product,

⟨ f1

g1

,

f2

g2

⟩ =

∫ π

0

df1

dx(x)

df2

dx(x)dx+

∫ π

0

g1(x)g2(x)dx. (A.4)

Define, A: D(A)→ X by

D(A) = [H2(0, π) ∩H10 (0, π)]×H1

0 (0, π), (A.5)

A

f

g

=

g

d2f

dx2

∀

f

g

∈ D(A). (A.6)

Let us denote by Z∗ the set of non-zero integers. For n ∈ Z∗, denote φn(x) =√2

πsin(nx). Family of functions (φn)n∈N is an orthonormal basis in L2[0, π]. This

167

implies that the family, defined by

Φn =1√2

1

ιnφn

φn

∀n ∈ Z∗, (A.7)

is an orthonormal basis in X with respect to the norm (A.4). The vectors φn from

(A.5) are eigenvectors ofA and the corresponding eigenvalues are λ = ιn, with n ∈ Z∗.

Operator A generates a semigroup T on X. T is given by

Tt

f

g

=∑n∈Z∗

eιnt

⟨ f

g

, φn

⟩φn ∀

f

g

∈ X. (A.8)

From the above relation it follows that,

Tt

f

g

=1√2

∑n∈Z∗

eιnt

(ι

n

⟨df

dx,dφndx

⟩L2[0,π]

+ 〈g, φn〉L2[0,π]

)φn. (A.9)

A.1.2 Observability

Proposition 4. Let X= H10 (0, π) × L2(0, π), and let A be the operator defined by

(A.6). Denote Y= C and consider the observation operator C∈ L(X,Y ) defined by,

C

f

g

=df

dx(0) ∀

f

g

∈ D(A). (A.10)

Then the pair (A, C) is exactly observable in any time τ ≥ 2π.

Proof. By using formulas (A.7) and (A.9), we have that, for all

f

g

∈ D(A),

CTt

f

g

=1√2

∑n∈Z∗

eιnt

(⟨df

dx, ψn

⟩L2[0,π]

− ι 〈g, φn〉L2[0,π]

), (A.11)

168

where ψn(x) =

√2

πcos(nx) for all n ∈ Z. The above formula and orthogonality of

family (eιnt)n∈Z∗ in L2[0, 2π] imply that,

∫ 2π

0

∣∣∣∣∣∣∣CTt f

g

∣∣∣∣∣∣∣2

dt =∑n∈Z∗

∣∣∣∣∣⟨df

dx, ψn

⟩L2[0,π]

− ι 〈g, φn〉L2[0,π]

∣∣∣∣∣2

. (A.12)

Since φ−n = −φn and ψ−n = −ψn, from (A.12) it follows that,

∫ 2π

0

∣∣∣∣∣∣∣CTt f

g

∣∣∣∣∣∣∣2

dt = 2∑n∈Z∗

∣∣∣∣∣⟨df

dx, ψn

⟩L2[0,π]

∣∣∣∣∣2

+∣∣∣〈g, φn〉L2[0,π]

∣∣∣2 . (A.13)

The above relation together with the fact that (ψn)n≥0 and (φn)n≥0 are orthonormal

basis in L2[0, π],

⇒ ∫ 2π

0

∣∣∣∣∣∣∣CTt f

g

∣∣∣∣∣∣∣2

dt = 2

∥∥∥∥∥∥∥ f

g

∥∥∥∥∥∥∥

2

∀

f

g

∈ D(A). (A.14)

Hence C is an admissible operator for semigroup T and pair (A, C) is exactly observ-

able in any time τ ≥ 2π.

B List of Papers

B.1 Journal Papers

J1: M.U. Majeed and T.M. Laleg-Kirati, “A dimension decomposition approach

based on iterative observer design for an elliptic Cauchy problem”, 2016. (archive

pre-print , under-review)

J2: M.U. Majeed and T.M. Laleg-Kirati, “Iterative Observer for Boundary Estima-

http://repository.kaust.edu.sa/kaust/bitstream/10754/559953/1/MUMajeed_TMLalegKirati.pdf

http://repository.kaust.edu.sa/kaust/bitstream/10754/559953/1/MUMajeed_TMLalegKirati.pdf

169

tion for Elliptic Equations”, 2015. (archive pre-print , under-review)

J3: M.U. Majeed and T.M. Laleg-Kirati, “Iterative Observers for Distributed Source

Estimation for Poisson Equation”, 2016. (submitted, under-review)

J4: M.U. Majeed and T.M. Laleg-Kirati, “An Optimal Iterative Algorithm Based

Technique for Steady-State Potential Field Imaging in 3D”, 2017. (under-

preparation)

B.2 Reviewed Conference Papers & Proceedings

C1: M.U. Majeed and T.M. Laleg-Kirati, “Iterative Observer Based Method for

Source Localization Problem for Poisson Equation in 3D”, The 2017 American

Control Conference (ACC 2017), Seattle, WA, USA, 2017.

C2: M.U. Majeed∗ and T.M. Laleg-Kirati, “Robust Iterative Observer for Source

Localization for Poisson Equation”, 55th Conference on Decision and Control

(CDC 2016), Las Vegas, NV, USA, 2016.

C3: M.U. Majeed∗ and T.M. Laleg-Kirati, “Localization of Point Sources for Poisson

Equation using State Observers”, 2nd IFAC Workshop on Control of Systems

Governed by Partial Differential Equations (CPDE’16), Bertinoro Italy, 2016.

(online link)

C4: M.U. Majeed and T.M. Laleg-Kirati∗, “An optimal iterative algorithm to solve

Cauchy problem for Laplace equation”, 3rd International Conference on Control

Engineering and Information Technology (CEIT), Tlemcen Algeria, 2015.

(online link) (Best Paper Award)

C5: M.U. Majeed∗ and T.M. Laleg-Kirati, “Boundary Estimation Problem for An

Infinite Dimensional Elliptic Cauchy Problem”, SIAM Conference on Control

http://arxiv.org/abs/1404.6957

http://acc2017.a2c2.org/

http://cdc2016.ieeecss.org/

http://www.cpde2016.org/

http://www.sciencedirect.com/science/article/pii/S2405896316306218

http://lat.univ-tlemcen.dz/ceit2015/index.php

http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=7233079&tag=1

170

and Its Applications (CT15), Paris France, 2015.

C6: M.U. Majeed and T.M. Laleg-Kirati∗, “Two-step observer approach to solve

Cauchy problem for Laplace equation”, (PICOF’14) Inverse Problems, Control

and Shape Optimization, Hammamet Tunisia, 2014.

C7: M.U. Majeed∗ and T.M. Laleg-Kirati, “Cauchy Problem for Laplace Equation

on a Square Domain using Observers”, 8th International Conference on Inverse

Problems in Engineering (ICIPE), Krakow Poland, 2014.

C8: M.U. Majeed, C. Zayane-Aissa∗ and T.M. Laleg-Kirati, “Cauchy Problem for

the Laplace’s Equation: An Observer based Approach”, The 3rd International

Conference on Systems and Control (ICSC’13), Algiers Algeria, 2013.

(online link)

( “∗” : author who presented in the corresponding conference)

B.3 Talks & Presentations

P1: Cyberphysical Systems Laboratory (CPSLab), New York University, Abu Dhabi,

UAE, July 2017.

P2: The 2017 American Control Conference (ACC), Seattle, WA, USA, May 2017.

P3: Mobile Sensors Lab, University of California Berkeley (UC Berkeley), CA, USA,

January 2017.

P4: 55th Conference on Decision and Control (CDC), Las Vegas, NV, USA, Decem-

ber 2016.

P5: 2nd IFAC Workshop on Control of Systems Governed by Partial Differential

Equations (CPDE16), Bertinoro, Italy, June 2016.

http://www.siam.org/meetings/ct15/

http://www.lamsin.tn/picof14/

http://www.icipe2014.org/index.php/icipe2014/ICIPE2014

http://lias.labo.univ-poitiers.fr/icsc/icsc2013/

http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=6750929&tag=1

171

P6: Cymer Center for Control Systems and Dynamics, University of California, San

Diego (UCSD), CA, USA, January 2016.

P7: SIAM Conference on Control and Its Applications (CT15), Paris, France, July

2015.

P8: Inverse Problems - from Theory to Applications (IPTA2014), Bristol, U.K. Au-

gust 2014.

P9: 8th International Conference on Inverse Problems in Engineering (ICIPE), Krakow,

Poland, May 2014.

P10: Winter Enrichment Program (WEP), King Abdullah University of Science and

Technology (KAUST), KSA, January 2014.

P11: Franco-German Summer School, Inverse Problems and Partial Differential Equa-

tions, Bremen, Germany, October 2013.

P12: Applied Inverse Problems Conference, Korean Advanced Institute of Science

and Technology (KAIST) Daejeon, South Korea, July 2013.

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