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 Sam Cogar (University of Delaware) Qualitative Methods August 12, 2016 1 / 17
Outline Inverse Medium Problem 1 Transmission Eigenvalues 2 Stekloff Eigenvalues 3 Future Work 4 Sam Cogar (University of Delaware) Qualitative Methods August 12, 2016 2 / 17
Inverse Medium Problem The Direct Problem Direct Scattering Problem for an Inhomogeneous Medium Seek total field u satisfying ∆ u + k 2 n ( x ) u = 0 in R 3 , u = e ikx · d + u s , � ∂ u s � ∂ r − iku s r →∞ r lim = 0 . 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 + k 2 n ( x ) u = 0 in R 3 , u = e ikx · d + u s , � ∂ u s � ∂ r − iku s r →∞ r lim = 0 . k - wave number n ( x ) - refractive index (with 1 − n compactly supported) d - direct of propagation for incident field ( | d | = 1) u s - 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 + k 2 n ( x ) u = 0 in R 3 , u = e ikx · d + u s , � ∂ u s � ∂ r − iku s r →∞ r lim = 0 . k - wave number n ( x ) - refractive index (with 1 − n compactly supported) d - direct of propagation for incident field ( | d | = 1) u s - scattered field Note: We let D = { x ∈ R 3 | n ( x ) � = 1 } . Sam Cogar (University of Delaware) Qualitative Methods August 12, 2016 3 / 17
Inverse Medium Problem The Direct Problem i u D u s Sam Cogar (University of Delaware) Qualitative Methods August 12, 2016 4 / 17
Inverse Medium Problem The Direct Problem i u D u s Far Field Pattern Far from the inhomogeneity D , � 1 u s ( x ) = e ik | x | � | x | u ∞ (ˆ x , d ) + O as | x | → ∞ , | x | 2 x where ˆ x = | x | and u ∞ (ˆ x , d ) is the far field pattern . Sam Cogar (University of Delaware) Qualitative Methods August 12, 2016 4 / 17
Inverse Medium Problem The Inverse Problem Inverse Medium Problem x , d ∈ S 2 and possibly multiple Given the far field pattern u ∞ (ˆ x , d ) for ˆ values of the wave number k , determine the refractive index n ( x ). Sam Cogar (University of Delaware) Qualitative Methods August 12, 2016 5 / 17
Inverse Medium Problem The Inverse Problem Inverse Medium Problem x , d ∈ S 2 and possibly multiple Given the far field pattern u ∞ (ˆ x , d ) for ˆ 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 x , d ∈ S 2 and a fixed wave number k . field pattern u ∞ (ˆ x , d ) for ˆ Sam Cogar (University of Delaware) Qualitative Methods August 12, 2016 5 / 17
Inverse Medium Problem The Inverse Problem Inverse Medium Problem x , d ∈ S 2 and possibly multiple Given the far field pattern u ∞ (ˆ x , d ) for ˆ 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 x , d ∈ S 2 and a fixed wave number k . field pattern u ∞ (ˆ x , d ) for ˆ The Bad News: This problem is ill-posed and nonlinear. Sam Cogar (University of Delaware) Qualitative Methods August 12, 2016 5 / 17
Inverse Medium Problem Solving the Inverse Medium Problem (1) Iterative methods (expensive optimization, may require a priori information about D ) Sam Cogar (University of Delaware) Qualitative Methods August 12, 2016 6 / 17
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) Sam Cogar (University of Delaware) Qualitative Methods August 12, 2016 6 / 17
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 ) Sam Cogar (University of Delaware) Qualitative Methods August 12, 2016 6 / 17
Inverse Medium Problem Qualitative Methods For applications such as non-destructive testing, knowing the refractive index is unnecessary. 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. 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 or 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 or D for a penetrable object. Target signatures may often be approximated using sampling methods. Sam Cogar (University of Delaware) Qualitative Methods August 12, 2016 7 / 17
Transmission Eigenvalues Transmission Eigenvalues Homogeneous Interior Transmission Problem Given D , find v , w ∈ L 2 ( D ) such that w − v ∈ H 2 0 ( D ) and the pair v , w satisfies ∆ w + k 2 n ( x ) w = 0 , ∆ v + k 2 v = 0 in D and w = v , ∂ w ∂ν = ∂ v ∂ν on ∂ D . Sam Cogar (University of Delaware) Qualitative Methods August 12, 2016 8 / 17
Transmission Eigenvalues Transmission Eigenvalues Homogeneous Interior Transmission Problem Given D , find v , w ∈ L 2 ( D ) such that w − v ∈ H 2 0 ( D ) and the pair v , w satisfies ∆ w + k 2 n ( x ) w = 0 , ∆ v + k 2 v = 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. Sam Cogar (University of Delaware) Qualitative Methods August 12, 2016 8 / 17
Transmission Eigenvalues Transmission Eigenvalues Far Field Operator The far field operator F : L 2 ( S 2 ) → L 2 ( S 2 ) is defined as � x ∈ S 2 . ( Fg )(ˆ x ) = S 2 u ∞ (ˆ x , d ) g ( d ) ds ( d ) , ˆ Sam Cogar (University of Delaware) Qualitative Methods August 12, 2016 9 / 17
Transmission Eigenvalues Transmission Eigenvalues Far Field Operator The far field operator F : L 2 ( S 2 ) → L 2 ( S 2 ) is defined as � x ∈ S 2 . ( Fg )(ˆ x ) = S 2 u ∞ (ˆ x , d ) g ( d ) ds ( d ) , ˆ – The operator F is injective with dense range unless k > 0 is a transmission eigenvalue with v of the form � S 2 e ikx · d g ( d ) ds ( d ) , x ∈ R 3 , v ( x ) = for some g ∈ L 2 ( S 2 ). Sam Cogar (University of Delaware) Qualitative Methods August 12, 2016 9 / 17
Transmission Eigenvalues Transmission Eigenvalues Far Field Operator The far field operator F : L 2 ( S 2 ) → L 2 ( S 2 ) is defined as � x ∈ S 2 . ( Fg )(ˆ x ) = S 2 u ∞ (ˆ x , d ) g ( d ) ds ( d ) , ˆ – The operator F is injective with dense range unless k > 0 is a transmission eigenvalue with v of the form � S 2 e ikx · d g ( d ) ds ( d ) , x ∈ R 3 , v ( x ) = for some g ∈ L 2 ( S 2 ). – Transmission eigenvalues may be computed using the sampling method with F , and they carry information about n ( x ). Sam Cogar (University of Delaware) Qualitative Methods August 12, 2016 9 / 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. ✓ 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 of 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 of n on D . ✓ – Transmission eigenvalues may only be utilized for materials with little or no absorption (i.e. when Im n ≈ 0). ✗ Sam Cogar (University of Delaware) Qualitative Methods August 12, 2016 10 / 17
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