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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