Qualitative Methods for the Inverse Medium Problem Sam Cogar - - PowerPoint PPT Presentation

Qualitative Methods for the Inverse Medium Problem

Sam Cogar Advisors: David Colton and Peter Monk

Summer Research Symposium Department of Mathematical Sciences University of Delaware

August 12, 2016

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Outline

1

Inverse Medium Problem

2

Transmission Eigenvalues

3

Stekloff Eigenvalues

4

Future Work

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Inverse Medium Problem

The Direct Problem

Direct Scattering Problem for an Inhomogeneous Medium Seek total field u satisfying ∆u + k2n(x)u = 0 in R3, u = eikx·d + us, lim

r→∞ r

∂us ∂r − ikus

• = 0.

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Inverse Medium Problem

The Direct Problem

Direct Scattering Problem for an Inhomogeneous Medium Seek total field u satisfying ∆u + k2n(x)u = 0 in R3, u = eikx·d + us, lim

r→∞ r

∂us ∂r − ikus

• = 0.

k - wave number n(x) - refractive index (with 1 − n compactly supported) d - direct of propagation for incident field (|d| = 1) us - scattered field

Sam Cogar (University of Delaware) Qualitative Methods August 12, 2016 3 / 17

Inverse Medium Problem

The Direct Problem

Direct Scattering Problem for an Inhomogeneous Medium Seek total field u satisfying ∆u + k2n(x)u = 0 in R3, u = eikx·d + us, lim

r→∞ r

∂us ∂r − ikus

• = 0.

k - wave number n(x) - refractive index (with 1 − n compactly supported) d - direct of propagation for incident field (|d| = 1) us - scattered field Note: We let D = {x ∈ R3|n(x) = 1}.

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Inverse Medium Problem

The Direct Problem

u

s

D

i

u

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Inverse Medium Problem

The Direct Problem

u

s

D

i

u

Far Field Pattern Far from the inhomogeneity D, us(x) = eik|x| |x| u∞(ˆ x, d) + O 1 |x|2

• as |x| → ∞,

where ˆ x =

x |x| and u∞(ˆ

x, d) is the far field pattern.

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Inverse Medium Problem

The Inverse Problem

Inverse Medium Problem Given the far field pattern u∞(ˆ x, d) for ˆ x, d ∈ S2 and possibly multiple values of the wave number k, determine the refractive index n(x).

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Inverse Medium Problem

The Inverse Problem

Inverse Medium Problem Given the far field pattern u∞(ˆ x, d) for ˆ x, d ∈ S2 and possibly multiple values of the wave number k, determine the refractive index n(x). Theorem (The Good News) The refractive index n(x) is uniquely determined by a knowledge of the far field pattern u∞(ˆ x, d) for ˆ x, d ∈ S2 and a fixed wave number k.

Sam Cogar (University of Delaware) Qualitative Methods August 12, 2016 5 / 17

Inverse Medium Problem

The Inverse Problem

Inverse Medium Problem Given the far field pattern u∞(ˆ x, d) for ˆ x, d ∈ S2 and possibly multiple values of the wave number k, determine the refractive index n(x). Theorem (The Good News) The refractive index n(x) is uniquely determined by a knowledge of the far field pattern u∞(ˆ x, d) for ˆ x, d ∈ S2 and a fixed wave number k. The Bad News: This problem is ill-posed and nonlinear.

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Inverse Medium Problem

Solving the Inverse Medium Problem

(1) Iterative methods (expensive optimization, may require a priori information about D)

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Inverse Medium Problem

Solving the Inverse Medium Problem

(1) Iterative methods (expensive optimization, may require a priori information about D) (2) Decomposition methods (separation of ill-posedness and nonlinearity)

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Inverse Medium Problem

Solving the Inverse Medium Problem

(1) Iterative methods (expensive optimization, may require a priori information about D) (2) Decomposition methods (separation of ill-posedness and nonlinearity) (3) Sampling methods (determine D but not n)

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Inverse Medium Problem

Qualitative Methods

For applications such as non-destructive testing, knowing the refractive index is unnecessary.

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Inverse Medium Problem

Qualitative Methods

For applications such as non-destructive testing, knowing the refractive index is unnecessary. Oftentimes, we need only determine if a material has a flaw when compared to some reference material.

Sam Cogar (University of Delaware) Qualitative Methods August 12, 2016 7 / 17

Inverse Medium Problem

Qualitative Methods

For applications such as non-destructive testing, knowing the refractive index is unnecessary. Oftentimes, we need only determine if a material has a flaw when compared to some reference material. Qualitative methods utilize target signatures to detect changes in n

• r D for a penetrable object.

Sam Cogar (University of Delaware) Qualitative Methods August 12, 2016 7 / 17

Inverse Medium Problem

Qualitative Methods

For applications such as non-destructive testing, knowing the refractive index is unnecessary. Oftentimes, we need only determine if a material has a flaw when compared to some reference material. Qualitative methods utilize target signatures to detect changes in n

• r D for a penetrable object.

Target signatures may often be approximated using sampling methods.

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Transmission Eigenvalues

Transmission Eigenvalues

Homogeneous Interior Transmission Problem Given D, find v, w ∈ L2(D) such that w − v ∈ H2

0(D) and the pair v, w

satisfies ∆w + k2n(x)w = 0, ∆v + k2v = 0 in D and w = v, ∂w ∂ν = ∂v ∂ν on ∂D.

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Transmission Eigenvalues

Transmission Eigenvalues

Homogeneous Interior Transmission Problem Given D, find v, w ∈ L2(D) such that w − v ∈ H2

0(D) and the pair v, w

satisfies ∆w + k2n(x)w = 0, ∆v + k2v = 0 in D and w = v, ∂w ∂ν = ∂v ∂ν on ∂D. Definition (Transmission Eigenvalue) We say that k > 0 is a transmission eigenvalue if the homogeneous interior transmission eigenvalue problem has a nontrivial solution.

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Transmission Eigenvalues

Transmission Eigenvalues

Far Field Operator The far field operator F : L2(S2) → L2(S2) is defined as (Fg)(ˆ x) =

• S2 u∞(ˆ

x, d)g(d)ds(d), ˆ x ∈ S2.

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Transmission Eigenvalues

Transmission Eigenvalues

Far Field Operator The far field operator F : L2(S2) → L2(S2) is defined as (Fg)(ˆ x) =

• S2 u∞(ˆ

x, d)g(d)ds(d), ˆ x ∈ S2. – The operator F is injective with dense range unless k > 0 is a transmission eigenvalue with v of the form v(x) =

• S2 eikx·dg(d)ds(d), x ∈ R3,

for some g ∈ L2(S2).

Sam Cogar (University of Delaware) Qualitative Methods August 12, 2016 9 / 17

Transmission Eigenvalues

Transmission Eigenvalues

Far Field Operator The far field operator F : L2(S2) → L2(S2) is defined as (Fg)(ˆ x) =

• S2 u∞(ˆ

x, d)g(d)ds(d), ˆ x ∈ S2. – The operator F is injective with dense range unless k > 0 is a transmission eigenvalue with v of the form v(x) =

• S2 eikx·dg(d)ds(d), x ∈ R3,

for some g ∈ L2(S2). – Transmission eigenvalues may be computed using the sampling method with F, and they carry information about n(x).

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Transmission Eigenvalues

Some Facts

– For real n with n(x) > 1 for all x ∈ D or n(x) < 1 for all x ∈ D, infinitely many transmission eigenvalues exist and have no finite accumulation point. ✓

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Transmission Eigenvalues

Some Facts

– For real n with n(x) > 1 for all x ∈ D or n(x) < 1 for all x ∈ D, infinitely many transmission eigenvalues exist and have no finite accumulation point. ✓ – Transmission eigenvalues may be used to estimate the average value

• f n on D. ✓

Sam Cogar (University of Delaware) Qualitative Methods August 12, 2016 10 / 17

Transmission Eigenvalues

Some Facts

– For real n with n(x) > 1 for all x ∈ D or n(x) < 1 for all x ∈ D, infinitely many transmission eigenvalues exist and have no finite accumulation point. ✓ – Transmission eigenvalues may be used to estimate the average value

• f n on D. ✓

– Transmission eigenvalues may only be utilized for materials with little

• r no absorption (i.e. when Imn ≈ 0). ✗

Sam Cogar (University of Delaware) Qualitative Methods August 12, 2016 10 / 17

Transmission Eigenvalues

Some Facts

– For real n with n(x) > 1 for all x ∈ D or n(x) < 1 for all x ∈ D, infinitely many transmission eigenvalues exist and have no finite accumulation point. ✓ – Transmission eigenvalues may be used to estimate the average value

• f n on D. ✓

– Transmission eigenvalues may only be utilized for materials with little

• r no absorption (i.e. when Imn ≈ 0). ✗

– Computing transmission eigenvalues requires sweeping through multiple frequencies. ✗

Sam Cogar (University of Delaware) Qualitative Methods August 12, 2016 10 / 17

Stekloff Eigenvalues

Stekloff Eigenvalues

Stekloff Eigenvalue Problem Given a ball B containing D, find w ∈ H1(B) such that ∆w + k2n(x)w = 0 in B and ∂w ∂ν + λw = 0 on ∂B.

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Stekloff Eigenvalues

Stekloff Eigenvalues

Stekloff Eigenvalue Problem Given a ball B containing D, find w ∈ H1(B) such that ∆w + k2n(x)w = 0 in B and ∂w ∂ν + λw = 0 on ∂B. Definition (Stekloff Eigenvalue) For fixed k, we say that λ ∈ C is a Stekloff eigenvalue if there exists a nontrivial solution to the Stekloff eigenvalue problem.

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Stekloff Eigenvalues

Stekloff Eigenvalues

Modified Far Field Operator The modified far field operator F : L2(S2) → L2(S2) is defined as (Fg)(ˆ x) =

• S2[u∞(ˆ

x, d) − h∞(ˆ x, d)]g(d)ds(d), ˆ x ∈ S2, where h∞ is the far field pattern corresponding to an exterior impedance problem which depends on a parameter λ.

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Stekloff Eigenvalues

Stekloff Eigenvalues

Modified Far Field Operator The modified far field operator F : L2(S2) → L2(S2) is defined as (Fg)(ˆ x) =

• S2[u∞(ˆ

x, d) − h∞(ˆ x, d)]g(d)ds(d), ˆ x ∈ S2, where h∞ is the far field pattern corresponding to an exterior impedance problem which depends on a parameter λ. – The operator F is injective with dense range unless λ is a Stekloff eigenvalue with w of a certain form.

Sam Cogar (University of Delaware) Qualitative Methods August 12, 2016 12 / 17

Stekloff Eigenvalues

Stekloff Eigenvalues

Modified Far Field Operator The modified far field operator F : L2(S2) → L2(S2) is defined as (Fg)(ˆ x) =

• S2[u∞(ˆ

x, d) − h∞(ˆ x, d)]g(d)ds(d), ˆ x ∈ S2, where h∞ is the far field pattern corresponding to an exterior impedance problem which depends on a parameter λ. – The operator F is injective with dense range unless λ is a Stekloff eigenvalue with w of a certain form. – Stekloff eigenvalues may be computed using the sampling method with F, and they carry information about n(x).

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Stekloff Eigenvalues

Some Facts

– If n is real-valued, then Stekloff eigenvalues exist, and they are real and discrete. ✓

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Stekloff Eigenvalues

Some Facts

– If n is real-valued, then Stekloff eigenvalues exist, and they are real and discrete. ✓ – If n = n1 + i n2

k with n1 > 0 and n2 > 0, then infinitely many Stekloff

eigenvalues exist in the complex plane, and they form a discrete set without finite accumulation points. ✓

Sam Cogar (University of Delaware) Qualitative Methods August 12, 2016 13 / 17

Stekloff Eigenvalues

Some Facts

– If n is real-valued, then Stekloff eigenvalues exist, and they are real and discrete. ✓ – If n = n1 + i n2

k with n1 > 0 and n2 > 0, then infinitely many Stekloff

eigenvalues exist in the complex plane, and they form a discrete set without finite accumulation points. ✓ – Stekloff eigenvalues may be utilized for absorbing materials. ✓

Sam Cogar (University of Delaware) Qualitative Methods August 12, 2016 13 / 17

Stekloff Eigenvalues

Some Facts

– If n is real-valued, then Stekloff eigenvalues exist, and they are real and discrete. ✓ – If n = n1 + i n2

k with n1 > 0 and n2 > 0, then infinitely many Stekloff

eigenvalues exist in the complex plane, and they form a discrete set without finite accumulation points. ✓ – Stekloff eigenvalues may be utilized for absorbing materials. ✓ – The testing frequency may be freely chosen since the target signature is no longer the wave number k. ✓

Sam Cogar (University of Delaware) Qualitative Methods August 12, 2016 13 / 17

Stekloff Eigenvalues

Some Facts

– If n is real-valued, then Stekloff eigenvalues exist, and they are real and discrete. ✓ – If n = n1 + i n2

k with n1 > 0 and n2 > 0, then infinitely many Stekloff

eigenvalues exist in the complex plane, and they form a discrete set without finite accumulation points. ✓ – Stekloff eigenvalues may be utilized for absorbing materials. ✓ – The testing frequency may be freely chosen since the target signature is no longer the wave number k. ✓ – Stekloff eigenvalues lack many of the nice properties of transmission eigenvalues and are not as well understood. ✗

Sam Cogar (University of Delaware) Qualitative Methods August 12, 2016 13 / 17

Stekloff Eigenvalues

Some Facts

– If n is real-valued, then Stekloff eigenvalues exist, and they are real and discrete. ✓ – If n = n1 + i n2

k with n1 > 0 and n2 > 0, then infinitely many Stekloff

eigenvalues exist in the complex plane, and they form a discrete set without finite accumulation points. ✓ – Stekloff eigenvalues may be utilized for absorbing materials. ✓ – The testing frequency may be freely chosen since the target signature is no longer the wave number k. ✓ – Stekloff eigenvalues lack many of the nice properties of transmission eigenvalues and are not as well understood. ✗ – The extension to Maxwell’s equations is problematic. ✗

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Future Work

A New Problem

The n0 Problem For a fixed k > 0 and a (generally complex) parameter n0, find v, w such that ∆w + k2n(x)w = 0, ∆v + k2n2

0v = 0 in D

and w = v, ∂w ∂ν = ∂v ∂ν on ∂D.

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Future Work

A New Problem

The n0 Problem For a fixed k > 0 and a (generally complex) parameter n0, find v, w such that ∆w + k2n(x)w = 0, ∆v + k2n2

0v = 0 in D

and w = v, ∂w ∂ν = ∂v ∂ν on ∂D. We use the same form of the modified far field operator F, but instead h∞ is the far field pattern corresponding to a transmission problem which depends on a parameter n0.

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Future Work

Future Work

– Study the relationship between the n0 problem and the refractive index n(x) – Prove that the sampling method may be used to detect n0 eigenvalues – Extend the results to Maxwell’s equations – Examine the usefulness of n0 eigenvalues as a target signature

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Future Work

References

Colton D, Kress R (2013) Inverse Acoustic and Electromagnetic Scattering Theory, 3rd edn. Springer, New York. Cakoni F, Colton D (2014) A Qualitative Approach to Inverse Scattering Theory. Springer, New York.

• F. Cakoni, D. Colton, S. Meng, and P. Monk, Stekloff Eigenvalues in

Inverse Scattering, to appear in SIAM J. Appl. Math.

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Future Work

Questions?

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