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 Scattering in an Inhomogeneous Medium 1 Solving the Inverse Problem 2 A Glimpse at Qualitative Methods 3 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 ∂ 2 U ∂ t 2 = c 2 ( 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 ∂ 2 U ∂ t 2 = c 2 ( x )∆ U Assuming time-harmonic wave propogation U ( x , t ) = Re { u ( x ) e − i ω t } : 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 ∂ 2 U ∂ t 2 = c 2 ( x )∆ U Assuming time-harmonic wave propogation U ( x , t ) = Re { u ( x ) e − i ω t } : Helmholtz Equation (Sort of) ∆ u + k 2 n ( x ) u = 0 c 2 k = ω 0 (wavenumber) , n ( x ) = c 2 ( x ) (index of refraction) c 0 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 R 3 , incident field u i satisfying ∆ u i + k 2 u i = 0 in R 3 such as u i ( x ) = e ikx · d for some propogation direction d ∈ S 2 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 R 3 , incident field u i satisfying ∆ u i + k 2 u i = 0 in R 3 such as u i ( x ) = e ikx · d for some propogation direction d ∈ S 2 Seek: total field u satisfying ∆ u + k 2 n ( x ) u = 0 in R 3 u = u i + u s � ∂ u s � ∂ r − iku s lim = 0 ( r = | x | ) r →∞ r 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 R 3 , incident field u i satisfying ∆ u i + k 2 u i = 0 in R 3 such as u i ( x ) = e ikx · d for some propogation direction d ∈ S 2 Seek: total field u satisfying ∆ u + k 2 n ( x ) u = 0 in R 3 u = u i + u s � ∂ u s � ∂ r − iku s lim = 0 ( r = | x | ) r →∞ r Note: u s 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 u s 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 u s in the far field: � 1 u s ( x ) = e ik | x | � x = x | x | u ∞ (ˆ x , d ) + O , | x | → ∞ , ˆ | x | 2 | 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 u s in the far field: � 1 u s ( x ) = e ik | x | � x = x | x | u ∞ (ˆ x , d ) + O , | x | → ∞ , ˆ | x | 2 | 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 u s in the far field: � 1 u s ( x ) = e ik | x | � x = x | x | u ∞ (ˆ x , d ) + O , | x | → ∞ , ˆ | x | 2 | x | The far field pattern is often written as u ∞ (ˆ x , d ) to emphasize the propagation direction d . The far field operator F : L 2 ( S 2 ) → L 2 ( S 2 ) is defined by � x ∈ S 2 . ( Fg )(ˆ x ) = S 2 u ∞ (ˆ x , d ) g ( d ) ds ( d ) , ˆ 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 R 3 , n is piecewise continuous on G , and u ∈ H 2 ( G ) is a solution of ∆ u + k 2 n ( x ) u = 0 in G which vanishes in a neighborhood of some x 0 ∈ 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 R 3 , n is piecewise continuous on G , and u ∈ H 2 ( G ) is a solution of ∆ u + k 2 n ( x ) u = 0 in G which vanishes in a neighborhood of some x 0 ∈ G , then u is identically zero on G . Reciprocity Relation x , d ∈ S 2 , u ∞ (ˆ For all 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 R 3 , n is piecewise continuous on G , and u ∈ H 2 ( G ) is a solution of ∆ u + k 2 n ( x ) u = 0 in G which vanishes in a neighborhood of some x 0 ∈ G , then u is identically zero on G . Reciprocity Relation x , d ∈ S 2 , u ∞ (ˆ For all ˆ x , d ) = u ∞ ( − d , − ˆ x ). Completeness of Far Field Patterns Given a countable dense set { d n } in S 2 , the set { u ∞ ( · , d n ) | n ∈ N } is complete in S 2 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 x , d ∈ S 2 (and possibly Given: far field pattern u ∞ (ˆ x , d ) for all ˆ 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 x , d ∈ S 2 (and possibly Given: far field pattern u ∞ (ˆ x , d ) for all ˆ different values of k > 0) Seek: refractive index n ( x ) for x ∈ R 3 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 x , d ∈ S 2 (and possibly Given: far field pattern u ∞ (ˆ x , d ) for all ˆ different values of k > 0) Seek: refractive index n ( x ) for x ∈ R 3 Theorem (The Good News) The refractive index n is uniquely determined by a knowledge of the far x , d ∈ S 2 and a fixed wavenumber k . field pattern u ∞ (ˆ x , d ) for ˆ 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 x , d ∈ S 2 (and possibly Given: far field pattern u ∞ (ˆ x , d ) for all ˆ different values of k > 0) Seek: refractive index n ( x ) for x ∈ R 3 Theorem (The Good News) The refractive index n is uniquely determined by a knowledge of the far x , d ∈ S 2 and a fixed wavenumber k . field pattern u ∞ (ˆ x , d ) for ˆ 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 x , d ∈ S 2 (and possibly Given: far field pattern u ∞ (ˆ x , d ) for all ˆ different values of k > 0) Seek: refractive index n ( x ) for x ∈ R 3 Theorem (The Good News) The refractive index n is uniquely determined by a knowledge of the far x , d ∈ S 2 and a fixed wavenumber k . field pattern u ∞ (ˆ x , d ) for ˆ 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