differentiation of data parameter identification in
TRANSCRIPT
Parameter Identification in Partial Differential Equations
Martin Burger http://www.indmath.uni-linz.ac.at/people/burger
Johannes Kepler Universität LinzParameter Identification,
Winter School Inverse Problems, Geilo 2
Differentiation of data
Not strictly a parameter identification problem,but good motivation. Appears often as a subproblem.
Given noisy observation of a function f,
Find its derivative
Parameter Identification,Winter School Inverse Problems, Geilo 3
Differentiation of data
Pointwise noise at measurement points identically normally distributed with mean 0 and variance δ.
Law of large numbers yields
Parameter Identification,Winter School Inverse Problems, Geilo 4
Differentiation of data
Example:
Noise satisfies
But error in derivative is large
Parameter Identification,Winter School Inverse Problems, Geilo 5
Differentiation of data
Error:
Arbitrarily large since k can be arbitrarily large
Parameter Identification,Winter School Inverse Problems, Geilo 6
Conclusion 1
Parameter Identification,Winter School Inverse Problems, Geilo 7
Differentiation of dataAdditional information:Solution is smooth (e.g. twice differentiable)
Regularization: e.g. smoothing by elliptic PDE
Variational principle for elliptic PDEs: minimize
Parameter Identification,Winter School Inverse Problems, Geilo 8
Differentiaton of dataDetailed estimate:
Lecture notes, p. 7
Parameter Identification,Winter School Inverse Problems, Geilo 9
Computerized Tomography(A nice link between Austria & Norway)
Problem: reconstruct a spatial density f in a domain D from measurements of X-rays traveling through the domain.
X-rays travel along rays, parametrized by distance s from origin and its unit normal vector ω
Parameter Identification,Winter School Inverse Problems, Geilo 10
Computerized TomographyX-ray beam has intensity I
Model: decay of itensity in a small distance ∆t proportional to I, f, and distance
Limit ∆t to zero
Parameter Identification,Winter School Inverse Problems, Geilo 11
Computerized TomographyMeasurement: intensity at emitter and detector
Parameter identification problem for system of ordinary differential equations (first-order)
Overdetermined for given f since initial and final value are known
Parameter Identification,Winter School Inverse Problems, Geilo 12
Computerized TomographyIntegrate ODE
Leads to inversion of the Radon transform
Parameter Identification,Winter School Inverse Problems, Geilo 13
Computerized TomographyRadon Transform
Exact inversion formula by Johann Radon 1917.SVD computed by McCormick et. al. in 1960s, Nobel Prize for Medicine in the 1970
Parameter Identification,Winter School Inverse Problems, Geilo 14
Computerized TomographyRadial symmetry,
Parameter Identification,Winter School Inverse Problems, Geilo 15
Computerized TomographyRadial symmetry,
Abel Integral equation !
SVs decay with half speed as for differentiation
Parameter Identification,Winter School Inverse Problems, Geilo 16
Groundwater Filtration
Identification of diffusivity of sediments from an observation of the piezometric head (describing the flow)
Knowledge of diffusivity allows to draw conclusions about the structure of the groundwater
Parameter Identification,Winter School Inverse Problems, Geilo 17
Groundwater FiltrationMathematical model
Diffusivity a, piezometric head u (measured), density of water sources and sinks f
+ Appropriate boundary conditions (e.g. u=0)
Parameter Identification,Winter School Inverse Problems, Geilo 18
Groundwater FiltrationInstead of exact data, we only know noisy measurement
Typically the noise n(x) comes from identically normally distributed random variablesFrom a law of large numbers this gives an estimate
Parameter Identification,Winter School Inverse Problems, Geilo 19
Groundwater FiltrationConsider 1D version of inverse problem, forsimplicity with
We can integrate the equation to obtain
Hence, if the derivative of u does not vanish, a is determined uniquely (Identifiability)
Parameter Identification,Winter School Inverse Problems, Geilo 20
Groundwater FiltrationDetailed estimate:
Lecture notes, p. 9
Parameter Identification,Winter School Inverse Problems, Geilo 21
Groundwater FiltrationSome results (from Hanke, 1995)
Exact phantom
Parameter Identification,Winter School Inverse Problems, Geilo 22
Groundwater FiltrationReconstructions for two different noise levels(1% and 0.1%)
Parameter Identification,Winter School Inverse Problems, Geilo 23
Groundwater FiltrationReconstruction along the diagonal
Parameter Identification,Winter School Inverse Problems, Geilo 24
Groundwater FiltrationReconstructions of piecewise constant parameter, no noise (nonuniqueness curve), fine grid
Parameter Identification,Winter School Inverse Problems, Geilo 25
Groundwater FiltrationReconstructions of piecewise constant parameter, no noise (nonuniqueness curve), hierarchical grids
Parameter Identification,Winter School Inverse Problems, Geilo 26
Electrical Impedance Tomography
Application in Medical ImagingSet of electrodes placed around human chest, Response to various electrical impulses measured
Picture from RPI-EIT Project
Parameter Identification,Winter School Inverse Problems, Geilo 27
Electrical Impedance Tomography
Picture from RPI-EIT Project
Parameter Identification,Winter School Inverse Problems, Geilo 28
Electrical Impedance Tomography
Pictures from RPI-EIT Project
Parameter Identification,Winter School Inverse Problems, Geilo 29
Electrical Impedance Tomography
Mathematical model: Maxwell equations, reduced to potential equation
u is electrical potential, f applied voltage patterna denotes the (unknown) conductivity
Parameter Identification,Winter School Inverse Problems, Geilo 30
Electrical Impedance Tomography
Measurement: current density on the boundary for different voltage patterns
Idealized mathematical model: Dirichlet-Neumann map
Parameter Identification,Winter School Inverse Problems, Geilo 31
Electrical Impedance Tomography INVERSE CONDUCTIVITY PROBLEM
In practice finite number of measurements
N typically very large
Parameter Identification,Winter School Inverse Problems, Geilo 32
Electrical Impedance Tomography
To compute output, we have to solve solutions of N partial differential equations
Problem of extremely large scale
Parameter Identification,Winter School Inverse Problems, Geilo 33
Electrical Impedance Tomography
Interesting special case: piecewise constant conductivities
Real interest is the subset where a takes avalue different from a1 (e.g. a tumour)
Parameter Identification,Winter School Inverse Problems, Geilo 34
Semiconductor Devices
Related problem to impedance tomography appears for semiconductor devices: inverse dopant profiling
Parameter Identification,Winter School Inverse Problems, Geilo 35
Inverse Dopant Profiling
Identify the device doping profile from measurements Current-Voltage map
Analogous definitions of voltage and current, but more complicated mathematical model
Parameter Identification,Winter School Inverse Problems, Geilo 36
Mathematical ModelStationary Drift Diffusion Model: PDE system for potential V, electron density nand hole density p
in Ω (subset of R2)Doping Profile C(x) enters as source term
Parameter Identification,Winter School Inverse Problems, Geilo 37
Boundary ConditionsBoundary of Ω : homogeneous Neumann boundary conditions on ΓN and
on Dirichlet boundary ΓD (Ohmic Contacts)Parameter Identification,
Winter School Inverse Problems, Geilo 38
Device CharacteristicsMeasured on a contact Γ0 on ΓD :Outflow current density
Parameter Identification,Winter School Inverse Problems, Geilo 39
Numerical TestsTest for a P-N Diode
Real Doping Profile Initial Guess
Parameter Identification,Winter School Inverse Problems, Geilo 40
Numerical TestsDifferent Voltage Sources
Parameter Identification,Winter School Inverse Problems, Geilo 41
Numerical TestsReconstructions with first source
Parameter Identification,Winter School Inverse Problems, Geilo 42
Numerical TestsReconstructions with second source
Parameter Identification,Winter School Inverse Problems, Geilo 43
Polymer Crystallization
Parameter Identification,Winter School Inverse Problems, Geilo 44
Polymer CrystallizationMesoscale model for polymer crystallization
ξ = degree of crystallinityu,v = surface densities
T = temperature G = growth rate
N = nucleation
rate
Parameter Identification,Winter School Inverse Problems, Geilo 45
Polymer CrystallizationSource term
Parameter Identification,Winter School Inverse Problems, Geilo 46
Polymer CrystallizationTraditional way of determining nucleation rate as function of temperature:- Make separate experiment for each value of T- Count (by eyes !!!) the final number of crystals (typically >> 106).- Divide by volume
Extremely expensive, extremely time-consuming !
Parameter Identification,Winter School Inverse Problems, Geilo 47
Polymer CrystallizationIdea: determine nucleation rate as function of temperature by single nonisothermal experiment
Measured data: temperature T at the boundary of the sample Degree of crystallinity at final time
Parameter Identification,Winter School Inverse Problems, Geilo 48
Polymer CrystallizationResults
Parameter Identification,Winter School Inverse Problems, Geilo 49
Polymer CrystallizationResults
Parameter Identification,Winter School Inverse Problems, Geilo 50
Polymer CrystallizationResults
Parameter Identification,Winter School Inverse Problems, Geilo 51
Polymer CrystallizationReconstruction error vs noise level
Parameter Identification,Winter School Inverse Problems, Geilo 52
Polymer CrystallizationReconstruction error vs noise level
Parameter Identification,Winter School Inverse Problems, Geilo 53
Polymer CrystallizationIteration numbers
Parameter Identification,Winter School Inverse Problems, Geilo 54
Polymer CrystallizationIteration numbers
Parameter Identification,Winter School Inverse Problems, Geilo 55
Iterative RegularizationFor nonlinear inverse problems, iterative regularization is of even higher importance as for linear ones We need to perform iteration to minimize nonlinear Tikhonov functional anyway
Start from least-squares functional
Parameter Identification,Winter School Inverse Problems, Geilo 56
Iterative RegularizationGradient of the least-squares functional
Perform simple gradient descent:
„Landweber iteration“
Parameter Identification,Winter School Inverse Problems, Geilo 57
Iterative RegularizationWhere is the regularization parameter ??!!
Parameter Identification,Winter School Inverse Problems, Geilo 58
Iterative RegularizationRegularization by appropriate early stopping, e.g. discrepancy principle
Each step of Landweber iteration is well-posed, so we obtain iterative regularization method
Parameter Identification,Winter School Inverse Problems, Geilo 59
Iterative RegularizationAnalysis of Landweber iteration: Hanke-Neubauer-Scherzer 1994, Scherzer 1995
Usual semi-convergence properties as for linear problems, but restriction on the nonlinearity of the problem (replacing regularity of )
Parameter Identification,Winter School Inverse Problems, Geilo 60
Iterative RegularizationLandweber iteration is forward Euler time discretization (with time step τ of the flow)
„Asymptotical regularization“ (Tautenhahn 1994)Iteration
Parameter Identification,Winter School Inverse Problems, Geilo 61
Iterative RegularizationWe can consider other time discretizations:Backward Euler
„Iterated Tikhonov Regularization“ (Hanke-Groetsch 1999, Groetsch-Scherzer 1999)
Parameter Identification,Winter School Inverse Problems, Geilo 62
Iterative RegularizationWe can consider other time discretizations:Semi-Implicit Euler
„Levenberg-Marquardt“ (Hanke 1995)
Arbitrary Runge-Kutta methods possibly, even inconsistent ones (Rieder, 2005)
Parameter Identification,Winter School Inverse Problems, Geilo 63
Iterative RegularizationLevenberg-Marquardt can be rewritten to
Tikhonov regularization of the linear Newton-equation
Parameter Identification,Winter School Inverse Problems, Geilo 64
Iterative RegularizationGeneral approach for Newton-type methods:Apply linear regularization to
„Iteratively regularized Gauss-Newton“ (Kaltenbacher-Neubauer-Scherzer 1996, Kaltenbacher 1997)„Newton-CG“ (Hanke, 1998)
Parameter Identification,Winter School Inverse Problems, Geilo 65
Iterative Regularization„Newton-Landweber“ (Kaltenbacher 1999)„Broyden“ (Kaltenbacher, 1997)„Multigrid / Discretization“ (Kaltenbacher, 2000-2003)
In particular for parameter identification:„Levenberg-Marquardt SQP“ (B-Mühlhuber 2002)„Newton-Kaczmarcz“ (B-Kaltenbacher 2004)
Parameter Identification,Winter School Inverse Problems, Geilo 66
Total Variation Denoising
Parameter Identification,Winter School Inverse Problems, Geilo 67
Total Variation Denoising
Parameter Identification,Winter School Inverse Problems, Geilo 68
TV-ImagesBasic problem in denoising:
given noisy version g of image f, find
„optimal“ approximation u of f
Basic problem in deblurring:
given noisy version g of Kf, find
„optimal“ approximation u of f
Parameter Identification,Winter School Inverse Problems, Geilo 69
TV-ImagesWhat is optimal ?
Satisfy other criterion, e.g., minimization
over a class of approximations
Parameter Identification,Winter School Inverse Problems, Geilo 70
ROF-Model
What is a suitable class of approximations ?
Discrepancy principle: allow images with
with σ being the noise level:
Parameter Identification,Winter School Inverse Problems, Geilo 71
ROF-Model
Rudin-Osher-Fatemi 1989 (ROF):
Minimize total variation
subject to
Parameter Identification,Winter School Inverse Problems, Geilo 72
ROF-Model
Deblurring:
Minimize total variation
subject to
Parameter Identification,Winter School Inverse Problems, Geilo 73
ROF-Model
Acar-Vogel 1994:
ROF is regularization method (well-
posedness + convergence as σ to zero)
Chambolle-Lions 1997:
Solution unique, there exists Lagrange
multiplier λ, problem equivalent to
Parameter Identification,Winter School Inverse Problems, Geilo 74
ROF-Model
Acar-Vogel 1994:
ROF is regularization method (well-
posedness + convergence as σ to zero)
Chambolle-Lions 1997:
Solution unique, there exists Lagrange
multiplier λ, problem equivalent to
Parameter Identification,Winter School Inverse Problems, Geilo 75
Total Variation
Rigorous definition of total variation
Parameter Identification,Winter School Inverse Problems, Geilo 76
Total Variation
BV includes discontinuous functions:
1D example
Parameter Identification,Winter School Inverse Problems, Geilo 77
Total Variation
1D example
Parameter Identification,Winter School Inverse Problems, Geilo 78
Total Variation
Formal optimality for TV-Denoising
Term on the right-hand side corresponds to mean curvature of level sets of u
Parameter Identification,Winter School Inverse Problems, Geilo 79
Duality
Consider TV-regularization
Parameter Identification,Winter School Inverse Problems, Geilo 80
DualityExchange inf and sup
Solve inner inf problem for u
Parameter Identification,Winter School Inverse Problems, Geilo 81
DualityRemains sup problem for g
Rewrite equivalently
Parameter Identification,Winter School Inverse Problems, Geilo 82
DualityIntroduce
Dual TV problem (Chambolle 2003)
Dual space of BV
Parameter Identification,Winter School Inverse Problems, Geilo 83
Stair-CasingLecture notes p.20
Parameter Identification,Winter School Inverse Problems, Geilo 84
ROF-Model
Meyer 2001 (and others before):
ROF has a systematic error, since
This means that jumps in the reconstructed image are smaller than jumps in the true image
Parameter Identification,Winter School Inverse Problems, Geilo 85
Noise Decomposition
How to resolve the systematic error ?
Take some λ, and minimize TV-functional
to obtain . Then take residual („noise“)
Parameter Identification,Winter School Inverse Problems, Geilo 86
Noise Decomposition
Minimize
to obtain next iterate
The second step may increase total
variation !
Parameter Identification,Winter School Inverse Problems, Geilo 87
Iterated Total Variation
Obtain new iterate by minimizing
then decompose noise
Need not change the code, just the data !Parameter Identification,
Winter School Inverse Problems, Geilo 88
Some Results
Parameter Identification,Winter School Inverse Problems, Geilo 89
Some Results
Parameter Identification,Winter School Inverse Problems, Geilo 90
Some Results
Parameter Identification,Winter School Inverse Problems, Geilo 91
Some Results
Parameter Identification,Winter School Inverse Problems, Geilo 92
Some Results
Parameter Identification,Winter School Inverse Problems, Geilo 93
Some Results
Parameter Identification,Winter School Inverse Problems, Geilo 94
Convergence Analysis
So why does that work ???
Expand fitting term, throw away constant
terms:
Parameter Identification,Winter School Inverse Problems, Geilo 95
Convergence Analysis
Optimality condition implies
Write equivalent optimization problem
Parameter Identification,Winter School Inverse Problems, Geilo 96
Convergence Analysis
Hence, iterated TV is „proximal point algorithm“
Parameter Identification,Winter School Inverse Problems, Geilo 97
Convergence Analysis
The second term is called Bregman distance
with
Parameter Identification,Winter School Inverse Problems, Geilo 98
Convergence Analysis
Example: quadratic case
yields
Iterated Tikhonov / Levenberg-Marquardt Method !
Parameter Identification,Winter School Inverse Problems, Geilo 99
Inverse TV FlowDoes the reconstruction depend on λ ?
To understand the dependence on the
parameter, consider limit λ → 0.
Let λN = T/N, set tk = k λN. We compute the
solution of the iterated TV flow with this specific parameter and set
Linear interpolation for other times.Parameter Identification,
Winter School Inverse Problems, Geilo 100
Inverse TV Flow
We obtain a limiting flow,
This flow is uniquely defined
Formally,
Parameter Identification,Winter School Inverse Problems, Geilo 101
Inverse TV FlowThe „inverse TV flow“ can be considered as
an „inverse scale space method“ (Scherzer-Weickert 99)
Start with zero image, and gradually insert smaller and smaller scales. Noise should return in the end, this happened in experiments
Parameter Identification,Winter School Inverse Problems, Geilo 102
Inverse TV Flow
The parameter λ can be interpreted as a time step for an implicit discretization of the inverse TV flow
Should work well and be independent of the
parameter, as long as λ is small enough. This can be observed in experiments, too.
Parameter Identification,Winter School Inverse Problems, Geilo 103
Nonlinear Inverse Problems
The algorithm can immediately be generalized to arbitrary linear and even nonlinear inverse problems:
Find next iterate as minimizer of
Parameter Identification,Winter School Inverse Problems, Geilo 104
Nonlinear Inverse Problems
Example: estimate diffusion coefficient q in
from measurement of u
Operator F: q → u(q)
Application: find material layers in ground-water modeling (piecewise constant)
Parameter Identification,Winter School Inverse Problems, Geilo 105
Nonlinear Inverse Problems
Exact data
Iterate 1
Parameter Identification,Winter School Inverse Problems, Geilo 106
Nonlinear Inverse Problems
Exact data
Iterate 2
Parameter Identification,Winter School Inverse Problems, Geilo 107
Nonlinear Inverse Problems
Exact data
Iterate 3
Parameter Identification,Winter School Inverse Problems, Geilo 108
Nonlinear Inverse Problems
Exact data
Iterate 4
Parameter Identification,Winter School Inverse Problems, Geilo 109
Nonlinear Inverse Problems
Exact data
Iterate 5
Parameter Identification,Winter School Inverse Problems, Geilo 110
Nonlinear Inverse Problems
Exact data
Iterate 6
Parameter Identification,Winter School Inverse Problems, Geilo 111
Nonlinear Inverse Problems
Exact data
Iterate 10
Parameter Identification,Winter School Inverse Problems, Geilo 112
Nonlinear Inverse Problems
Exact data
Iterate 20
Parameter Identification,Winter School Inverse Problems, Geilo 113
Nonlinear Inverse Problems
Exact data
Iterate 50
Parameter Identification,Winter School Inverse Problems, Geilo 114
Nonlinear Inverse Problems
Exact data
Iterate 100
Parameter Identification,Winter School Inverse Problems, Geilo 115
Nonlinear Inverse Problems
Exact data
Iterate 200
Parameter Identification,Winter School Inverse Problems, Geilo 116
Nonlinear Inverse Problems
1% noise
Iterate 1
Parameter Identification,Winter School Inverse Problems, Geilo 117
Nonlinear Inverse Problems
1% noise
Iterate 2
Parameter Identification,Winter School Inverse Problems, Geilo 118
Nonlinear Inverse Problems
1% noise
Iterate 3
Parameter Identification,Winter School Inverse Problems, Geilo 119
Nonlinear Inverse Problems
1% noise
Iterate 4
Parameter Identification,Winter School Inverse Problems, Geilo 120
Nonlinear Inverse Problems
1% noise
Iterate 5
Parameter Identification,Winter School Inverse Problems, Geilo 121
Nonlinear Inverse Problems
1% noise
Iterate 10
Parameter Identification,Winter School Inverse Problems, Geilo 122
Nonlinear Inverse Problems
1% noise
Iterate 50
Parameter Identification,Winter School Inverse Problems, Geilo 123
Nonlinear Inverse Problems
10% noise
Iterate 1
Parameter Identification,Winter School Inverse Problems, Geilo 124
Nonlinear Inverse Problems
10% noise
Iterate 2
Parameter Identification,Winter School Inverse Problems, Geilo 125
Nonlinear Inverse Problems
10% noise
Iterate 3
Parameter Identification,Winter School Inverse Problems, Geilo 126
Nonlinear Inverse Problems
10% noise
Iterate 4
Parameter Identification,Winter School Inverse Problems, Geilo 127
Nonlinear Inverse Problems
10% noise
Iterate 5
Parameter Identification,Winter School Inverse Problems, Geilo 128
Nonlinear Inverse Problems
10% noise
Iterate 10
Parameter Identification,Winter School Inverse Problems, Geilo 129
Parameter Identification
Parameter Identification,Winter School Inverse Problems, Geilo 130
Implementation AspectsLandweber iteration: Discretize parameter Discretize state + state equationCompute gradient by adjoint methodTry to use same kind of discretization for adjoint as for state equation („Discretized Adjoint = Adjoint of Discretized Problem“)For multiple state equations compute gradients corresponding to each state equation immediately
Parameter Identification,Winter School Inverse Problems, Geilo 131
Implementation AspectsLandweber iteration with Black Box Solver: Discretize parameter Discretize state + state equation (determined by solver)If adjoint method not possible, try to compute gradients by finite differencingFor multiple state equations compute gradients corresponding to each state equation immediately
Parameter Identification,Winter School Inverse Problems, Geilo 132
Implementation AspectsQuasi-Newton Methods (BFGS): Discretize parameter Discretize state + state equation (determined by solver)Compute gradients by adjoint method, use adjoint to update quasi-Newton matrixSolve linear system for update
Parameter Identification,Winter School Inverse Problems, Geilo 133
Implementation AspectsNewton Methods (LM / IRGN / NCG): Discretize parameter Discretize state + state equation Never compute Newton-Matrix (NOT SPARSE)!!
Try to use iterative method for computing the Newton step, only implement application of and its adjoint (two PDEs)
Parameter Identification,Winter School Inverse Problems, Geilo 134
Implementation AspectsOne shot methods: Discretize parameter + state + state equation + adjoint + adjoint equation Solve the full KKT system simultaneously, e.g.
Parameter Identification,Winter School Inverse Problems, Geilo 135
Implementation AspectsOne shot methods: KKT system is sparse and symmetricKKT system is not positive definite (no CG !) Use BiCGStab, MINRES, QMR or GMRESApply appropriate preconditioning, note that there is a small parameter (α), acting like singular perturbationPreconditioning strategies: Battermann-Sachs 2000, Battermann-Heinckenschloss 2001, Ascher-Haber 2001-2003,B-Mühlhuber 2002, Griewank 2005
Parameter Identification,Winter School Inverse Problems, Geilo 136
Kaczmarcz MethodsFor problems like impedance tomography, the operators are of the form
Idea: perform multiplicative splitting (like in Gauss-Seidel), cyclic iteration over the N subproblems
Parameter Identification,Winter School Inverse Problems, Geilo 137
Kaczmarcz MethodsSubproblem j:
Landweber-Kaczmarcz (Natterer 1996, Kowar-Scherzer 2003)
Parameter Identification,Winter School Inverse Problems, Geilo 138
Kaczmarcz MethodsLevenberg-Marquardt Kaczmarcz (B-Kaltenbacher 2004)State equation:Linear KKT:
Parameter Identification,Winter School Inverse Problems, Geilo 139
Kaczmarcz MethodsSimple model problem: identify q in
from measurements
for different sources
Parameter Identification,Winter School Inverse Problems, Geilo 140
Numerical SolutionNewton-Kaczmarz approach is equivalent to minimizing
for in each step, subject to
Parameter Identification,Winter School Inverse Problems, Geilo 141
Numerical SolutionSince we need another function to realize the H-1/2-norm, corresponding KKT system is 4 x 4
Parameter Identification,Winter School Inverse Problems, Geilo 142
Numerical SolutionWith boundary conditions
and
Parameter Identification,Winter School Inverse Problems, Geilo 143
Numerical ResultsNumerical experiment with N=20 localized sources gj
(5 on each side)
Exact solution
Constant initial value
Parameter Identification,Winter School Inverse Problems, Geilo 144
Numerical ResultsIterates 1, 2, 3, 4, 5, 6
Parameter Identification,Winter School Inverse Problems, Geilo 145
Numerical ResultsIterates 10, 15, 20, 40, 60, 80 (1/2,3/4,1,2,3,4)
Parameter Identification,Winter School Inverse Problems, Geilo 146
Numerical ResultsResiduals and Error
Parameter Identification,Winter School Inverse Problems, Geilo 147
Numerical ResultsNumerical experiment with N=20 localized sources gj
(5 on each side)
Exact solution
Constant initial value
Parameter Identification,Winter School Inverse Problems, Geilo 148
Numerical ResultsIterates 1, 6, 11, 16, 21, 41
Parameter Identification,Winter School Inverse Problems, Geilo 149
Numerical ResultsIterates 80, 120, 160, 240, 320, 400 (4,6,8,12,16,20)
Parameter Identification,Winter School Inverse Problems, Geilo 150
Numerical ResultsResiduals and Error