A Friendly Introduction to Inverse Scattering Theory Sam Cogar - - PowerPoint PPT Presentation

A Friendly Introduction to Inverse Scattering Theory

Sam Cogar Advisors: Peter Monk and David Colton

Summer Pizza Seminar

July 5, 2016

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 1 / 16

Outline

1

Scattering in an Inhomogeneous Medium

2

Solving the Inverse Problem

3

A Glimpse at Qualitative Methods

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 2 / 16

Scattering in an Inhomogeneous Medium

The Equations

Wave Equation ∂2U ∂t2 = c2(x)∆U

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 3 / 16

Scattering in an Inhomogeneous Medium

The Equations

Wave Equation ∂2U ∂t2 = c2(x)∆U Assuming time-harmonic wave propogation U(x, t) = Re{u(x)e−iωt}:

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 3 / 16

Scattering in an Inhomogeneous Medium

The Equations

Wave Equation ∂2U ∂t2 = c2(x)∆U Assuming time-harmonic wave propogation U(x, t) = Re{u(x)e−iωt}: Helmholtz Equation (Sort of) ∆u + k2n(x)u = 0 k = ω c0 (wavenumber), n(x) = c2 c2(x) (index of refraction)

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 3 / 16

Scattering in an Inhomogeneous Medium

The Direct Problem

Given: wavenumber k > 0, index of refraction n(x) with m := 1 − n compactly supported in R3, incident field ui satisfying ∆ui + k2ui = 0 in R3 such as ui(x) = eikx·d for some propogation direction d ∈ S2

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 4 / 16

Scattering in an Inhomogeneous Medium

The Direct Problem

Given: wavenumber k > 0, index of refraction n(x) with m := 1 − n compactly supported in R3, incident field ui satisfying ∆ui + k2ui = 0 in R3 such as ui(x) = eikx·d for some propogation direction d ∈ S2 Seek: total field u satisfying ∆u + k2n(x)u = 0 in R3 u = ui + us lim

r→∞ r

∂us ∂r − ikus

• = 0

(r = |x|)

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 4 / 16

Scattering in an Inhomogeneous Medium

The Direct Problem

Given: wavenumber k > 0, index of refraction n(x) with m := 1 − n compactly supported in R3, incident field ui satisfying ∆ui + k2ui = 0 in R3 such as ui(x) = eikx·d for some propogation direction d ∈ S2 Seek: total field u satisfying ∆u + k2n(x)u = 0 in R3 u = ui + us lim

r→∞ r

∂us ∂r − ikus

• = 0

(r = |x|) Note: us is called the scattered field

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 4 / 16

Scattering in an Inhomogeneous Medium

The Far Field

The scattered field us in the far field:

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 5 / 16

Scattering in an Inhomogeneous Medium

The Far Field

The scattered field us in the far field: us(x) = eik|x| |x| u∞(ˆ x, d) + O 1 |x|2

• , |x| → ∞, ˆ

x = x |x|

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 5 / 16

Scattering in an Inhomogeneous Medium

The Far Field

The scattered field us in the far field: us(x) = eik|x| |x| u∞(ˆ x, d) + O 1 |x|2

• , |x| → ∞, ˆ

x = x |x| The far field pattern is often written as u∞(ˆ x, d) to emphasize the propagation direction d.

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 5 / 16

Scattering in an Inhomogeneous Medium

The Far Field

The scattered field us in the far field: us(x) = eik|x| |x| u∞(ˆ x, d) + O 1 |x|2

• , |x| → ∞, ˆ

x = x |x| The far field pattern is often written as u∞(ˆ x, d) to emphasize the propagation direction d. The far field operator F : L2(S2) → L2(S2) is defined by (Fg)(ˆ x) =

• S2 u∞(ˆ

x, d)g(d)ds(d), ˆ x ∈ S2.

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 5 / 16

Scattering in an Inhomogeneous Medium

Some Amusing Facts

Unique Continuation If G is a domain in R3, n is piecewise continuous on G, and u ∈ H2(G) is a solution of ∆u + k2n(x)u = 0 in G which vanishes in a neighborhood of some x0 ∈ G, then u is identically zero on G.

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 6 / 16

Scattering in an Inhomogeneous Medium

Some Amusing Facts

Unique Continuation If G is a domain in R3, n is piecewise continuous on G, and u ∈ H2(G) is a solution of ∆u + k2n(x)u = 0 in G which vanishes in a neighborhood of some x0 ∈ G, then u is identically zero on G. Reciprocity Relation For all ˆ x, d ∈ S2, u∞(ˆ x, d) = u∞(−d, −ˆ x).

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 6 / 16

Scattering in an Inhomogeneous Medium

Some Amusing Facts

Unique Continuation If G is a domain in R3, n is piecewise continuous on G, and u ∈ H2(G) is a solution of ∆u + k2n(x)u = 0 in G which vanishes in a neighborhood of some x0 ∈ G, then u is identically zero on G. Reciprocity Relation For all ˆ x, d ∈ S2, u∞(ˆ x, d) = u∞(−d, −ˆ x). Completeness of Far Field Patterns Given a countable dense set {dn} in S2, the set {u∞(·, dn)|n ∈ N} is complete in S2 if and only if F is injective.

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 6 / 16

Scattering in an Inhomogeneous Medium

The Inverse Medium Problem

Given: far field pattern u∞(ˆ x, d) for all ˆ x, d ∈ S2 (and possibly different values of k > 0)

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 7 / 16

Scattering in an Inhomogeneous Medium

The Inverse Medium Problem

Given: far field pattern u∞(ˆ x, d) for all ˆ x, d ∈ S2 (and possibly different values of k > 0) Seek: refractive index n(x) for x ∈ R3

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 7 / 16

Scattering in an Inhomogeneous Medium

The Inverse Medium Problem

Given: far field pattern u∞(ˆ x, d) for all ˆ x, d ∈ S2 (and possibly different values of k > 0) Seek: refractive index n(x) for x ∈ R3 Theorem (The Good News) The refractive index n is uniquely determined by a knowledge of the far field pattern u∞(ˆ x, d) for ˆ x, d ∈ S2 and a fixed wavenumber k.

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 7 / 16

Scattering in an Inhomogeneous Medium

The Inverse Medium Problem

Given: far field pattern u∞(ˆ x, d) for all ˆ x, d ∈ S2 (and possibly different values of k > 0) Seek: refractive index n(x) for x ∈ R3 Theorem (The Good News) The refractive index n is uniquely determined by a knowledge of the far field pattern u∞(ˆ x, d) for ˆ x, d ∈ S2 and a fixed wavenumber k. The Bad News:

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 7 / 16

Scattering in an Inhomogeneous Medium

The Inverse Medium Problem

Given: far field pattern u∞(ˆ x, d) for all ˆ x, d ∈ S2 (and possibly different values of k > 0) Seek: refractive index n(x) for x ∈ R3 Theorem (The Good News) The refractive index n is uniquely determined by a knowledge of the far field pattern u∞(ˆ x, d) for ˆ x, d ∈ S2 and a fixed wavenumber k. The Bad News: This problem is ill-posed.

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 7 / 16

Solving the Inverse Problem

Ill-Posed Problems

Hadamard’s Criteria for Well-Posedness of a Problem

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 8 / 16

Solving the Inverse Problem

Ill-Posed Problems

Hadamard’s Criteria for Well-Posedness of a Problem Existence of a solution ✓

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 8 / 16

Solving the Inverse Problem

Ill-Posed Problems

Hadamard’s Criteria for Well-Posedness of a Problem Existence of a solution ✓ Uniqueness of a solution ✓

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 8 / 16

Solving the Inverse Problem

Ill-Posed Problems

Hadamard’s Criteria for Well-Posedness of a Problem Existence of a solution ✓ Uniqueness of a solution ✓ Continuous dependence of the solution on the data ✗

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 8 / 16

Solving the Inverse Problem

Ill-Posed Problems

Hadamard’s Criteria for Well-Posedness of a Problem Existence of a solution ✓ Uniqueness of a solution ✓ Continuous dependence of the solution on the data ✗ If a problem fails any of these criteria, then it is called ill-posed.

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 8 / 16

Solving the Inverse Problem

Ill-Posed Problems

Hadamard’s Criteria for Well-Posedness of a Problem Existence of a solution ✓ Uniqueness of a solution ✓ Continuous dependence of the solution on the data ✗ If a problem fails any of these criteria, then it is called ill-posed. In Operator Terms Let A : X → Y be an operator from a normed space X to a normed space Y , and let f ∈ Y . The equation Aϕ = f is called well-posed if A is bijective and has continuous inverse. Otherwise, the equation is called ill-posed.

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 8 / 16

Solving the Inverse Problem

Regularization Techniques

To solve an ill-posed problem approximately in a stable manner, regularization techniques are used.

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 9 / 16

Solving the Inverse Problem

Regularization Techniques

To solve an ill-posed problem approximately in a stable manner, regularization techniques are used. For Hilbert spaces X and Y and a compact operator A : X → Y , Tikhonov regularization is common: Tikhonov Regularization Find the unique ϕα ∈ X such that Aϕα − f 2 + α f 2 = inf

ϕ∈X{Aϕ − f 2 + α f 2}

for some small α > 0. Equivalently, find the unique solution of αϕα + A∗Aϕα = A∗f .

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 9 / 16

Solving the Inverse Problem

Quantitative Methods

1) Iterative methods to determine n (expensive optimization)

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 10 / 16

Solving the Inverse Problem

Quantitative Methods

1) Iterative methods to determine n (expensive optimization) 2) Decomposition methods to determine n (solve an ill-posed linear integral equation first to reduce computational expense, then

• ptimize)

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 10 / 16

Solving the Inverse Problem

Quantitative Methods

1) Iterative methods to determine n (expensive optimization) 2) Decomposition methods to determine n (solve an ill-posed linear integral equation first to reduce computational expense, then

• ptimize)

3) Sampling methods to determine the support of m = 1 − n (avoid

• ptimization entirely, solve many ill-posed linear integral equations)

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 10 / 16

Solving the Inverse Problem

Quantitative Methods

1) Iterative methods to determine n (expensive optimization) 2) Decomposition methods to determine n (solve an ill-posed linear integral equation first to reduce computational expense, then

• ptimize)

3) Sampling methods to determine the support of m = 1 − n (avoid

• ptimization entirely, solve many ill-posed linear integral equations)

Some of these methods may not work if the wavenumber k > 0 is a transmission eigenvalue.

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 10 / 16

Solving the Inverse Problem

Transmission Eigenvalues

A transmission eigenvalue is a number k > 0 for which the homogeneous interior transmission problem has a nontrivial solution.

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 11 / 16

Solving the Inverse Problem

Transmission Eigenvalues

A transmission eigenvalue is a number k > 0 for which the homogeneous interior transmission problem has a nontrivial solution. Homogeneous Interior Transmission Problem Given D, the support of 1 − n, find v, w ∈ H2(D) such that the pair v, w satisfies ∆w + k2n(x)w = 0, ∆v + k2v = 0 in D and w = v, ∂w ∂ν = ∂v ∂ν on ∂D.

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 11 / 16

Solving the Inverse Problem

Transmission Eigenvalues

Theorem If k > 0 is a transmission eigenvalue and v is of the form v(x) =

• S2 e−ikx·dg(d)ds(d), x ∈ R3,

for some g ∈ L2(S2), then the far field operator F is not injective.

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 12 / 16

Solving the Inverse Problem

Transmission Eigenvalues

Theorem If k > 0 is a transmission eigenvalue and v is of the form v(x) =

• S2 e−ikx·dg(d)ds(d), x ∈ R3,

for some g ∈ L2(S2), then the far field operator F is not injective. Some of the methods given before require injectivity of F.

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 12 / 16

Solving the Inverse Problem

Transmission Eigenvalues

Theorem If k > 0 is a transmission eigenvalue and v is of the form v(x) =

• S2 e−ikx·dg(d)ds(d), x ∈ R3,

for some g ∈ L2(S2), then the far field operator F is not injective. Some of the methods given before require injectivity of F. Fortunately, transmission eigenvalues form a discrete set.

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 12 / 16

Solving the Inverse Problem

Transmission Eigenvalues

Theorem If k > 0 is a transmission eigenvalue and v is of the form v(x) =

• S2 e−ikx·dg(d)ds(d), x ∈ R3,

for some g ∈ L2(S2), then the far field operator F is not injective. Some of the methods given before require injectivity of F. Fortunately, transmission eigenvalues form a discrete set. Sampling methods may compute and utilize transmission eigenvalues to gain information about n.

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 12 / 16

A Glimpse at Qualitative Methods

Qualitative Methods

These methods compute eigenvalues of PDE problems to identify qualities of a material, such as the presence of cracks or cavities.

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 13 / 16

A Glimpse at Qualitative Methods

Qualitative Methods

These methods compute eigenvalues of PDE problems to identify qualities of a material, such as the presence of cracks or cavities. Transmission eigenvalues have some undesirable properties.

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 13 / 16

A Glimpse at Qualitative Methods

Qualitative Methods

These methods compute eigenvalues of PDE problems to identify qualities of a material, such as the presence of cracks or cavities. Transmission eigenvalues have some undesirable properties. An alternative is Stekloff eigenvalues λ for which there exists a nontrivial solution to the Stekloff eigenvalue problem. Stekloff Eigenvalue Problem Given a ball B containing the support of m = 1 − n, find w such that ∆w + k2n(x)w = 0 in B and ∂w ∂ν + λw = 0 on ∂B.

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 13 / 16

A Glimpse at Qualitative Methods

Qualitative Methods

The goal is to explore the existence of such eigenvalues and how changes in the eigenvalues may be used to infer changes in the refractive index.

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 14 / 16

A Glimpse at Qualitative Methods

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.

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 15 / 16

A Glimpse at Qualitative Methods

Questions?

Sam Cogar (Summer Pizza Seminar) A Friendly Intro to Inverse Scattering Theory July 5, 2016 16 / 16