Inverse Problems: Seeing the

unseen

Karthik Iyer

May 25, 2018

The Chess Mysteries of Sherlock Holmes

What did Black just play?

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Inverse Square Gravity Law

Only the inverse square law can account for the elliptical

focus-directed motion and hence only an inverse square

gravity law can explain Kepler’s orbits. (I. Newton)

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Dolphin Sonar

Can we build equipment to do the same?

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Imaging with Acoustic Waves

Occupies a large part of the brains of these mammals

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Seismic Imaging

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Oil Exploration

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Global Seismology

Inverse Problem: Determine inner structure of Earth by measuring

travel time of seismic waves.

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OCEAN ACOUSTICS

Ocean Acoustic Tomography is a tool with which we can study

average temperatures over large regions of the ocean. By measur-

ing the time it takes sound to travel between known source and

receiver locations, we can determine the sound speed. Changes in

soundspeed can then be related to changes in temperature.

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Medical Imaging

X-ray Tomography (CT Scan)

Problem: Can we recover the density

from attenuation of X-rays?

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X-ray Tomography

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Discrete Example

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Radon (1917) n = 2

f(x) = Unknown function

Idetector = e−∫L fIsource

Rf(s, θ) = g(s, θ) =∫⟨x,θ⟩=s

f(x)dH =∫Lf

f(x) =1

4π2p.v.

∫S1

dθ∫ d

dsg(s, θ)ds

⟨x, θ⟩ − s

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Magnetic Resonance Imaging

Nobel Prize in Medicine: P. Lauterbur and P. Mansfield (2003)

Magnetic resonance imaging involves using strong magnetic fields and radio

frequency energy to produce images based on the hydrogen content (primarily

water) of body tissues.

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Ultrasound

Ultrasound uses ultrasonic waves and measures the echo response

from the tissues.

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CALDERON’S PROBLEM

Ω ⊂ Rn

(n = 2,3)

Can one determine the electrical conductivity of Ω, γ(x),

by making voltage and current measurements at the

boundary?

(Calderon; Geophysical prospection)

Early breast cancer detection

Normal breast tissue 0.3 mhoCancerous breast tumor 2.0 mho

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Discrete Model

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Other applications:

- Non-destructive testing (corrosion, cracks)

- Seepage of groundwater pollutants

- Medical Imaging (EIT)

Tissue Conductivity (mho)Blood 6.7Liver 2.8

Cardiac muscle 6.3 (longitudinal)2.3 (transversal)

Grey matter 3.5White matter 1.5

Lung 1.0 (expiration)0.4 (inspiration)

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Photoacoustic Tomography

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Probes, Physical Properties and Imaging Systems

Probe Parameter System

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Different aspects of Inverse problems

- Uniqueness

- Stability

- Reconstruction techniques

- Numerical implementation

- Partial data issues

- Corrupted data

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Dirichlet-to-Neumann map

Consider a body Ω. An electrical potential u(x) causes

the current

I(x) = γ(x)∇u(x)

The conductivity γ(x) can be isotropic, that is, scalar,

or anisotropic, that is, a matrix valued function. If the

current has no sources or sinks, we have

div(γ(x)∇u) = 0 in Ω

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div(γ(x)∇u(x)) = 0

u∣∣∣∂Ω

= f

γ(x) = conductivity,

f = voltage potential at ∂Ω

Current flux at ∂Ω = (ν · γ∇u)∣∣∣∂Ω

were ν is the unit

outer normal.

Information is encoded in map Λγ(f) = ν · γ∇u∣∣∣∂Ω

Inverse problem

Does Λγ determine γ?

Λγ = Dirichlet-to-Neumann map

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Uniqueness in an Inverse problem

- Associated to every partial differential equation Lwith coefficients A, there is a Dirichlet-to-Neumann

(DN) operator ΛA.

- Uniqueness ≡ Invertibility of DN map

- My research dealt with uniqueness for two different

families of partial differential equations

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PDE 1: Polyharmonic operator

(−∆)m +A ·Du+ qu = 0 in Ω,m ≥ 2

- This operator arises in modeling plate equations in

theory of elasticity and in conformal geometry.

- Does ΛA,q determine A and q?

- Yes! [Assylbekov-I, 2017]

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Key ideas in the proof

- Don’t restrict to only real values! Consider complex

valued function v = ex·ζ such that −∆v = 0.

- Since the operator is essentially a perturbation of

(−∆m), choose ansatz of the form ex·ζ+ remainder .

- And keep track of the remainder.

- This approach essentially works because the differ-

ential operator is linear and complex numbers are

linear combinations of real numbers.

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PDE 2: Quasilinear conductivity operator

−div(A(x, u)∇u(x)) = 0 in Ω

- The above equation arises in modeling of thermal

conductivity of the Earth’s crust and heat conduc-

tion in composite materials.

- Does ΛA determine A?

- Yes, if A is scalar-valued and no if A is matrix valued.

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Properties

- There is a conformal equivalence which prevents

uniqueness in the matrix valued case.

- However, uniqueness goes through for the scalar

valued case!

- Can’t use ansatz since operator is non-linear.

- Linearize!

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- Conformal equivalence breaks down under degener-

acy of A.

- Transformation optics can be used to obtain param-

eters whose boundary measurements give no infor-

mation about certain parts of the domain.

- Theoretical basis of electrostatic cloaking.

- [Ghosh-I, 2018]

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Co-authors

Yernat Assylbekov Tuhin Ghosh

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