The Factorization Method for Inverse Scattering Problems Part I

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1 The Factorization Method for Inverse Scattering Problems Part I Andreas Kirsch Madrid 2011 Department of Mathematics KIT University of the State of Baden-Württemberg and National Large-scale Research Center of the Helmholtz Association

2 The Factorization Method for Inverse Scattering Problems Part I 2/21 Outline of the Course Part I: Introduction The Direct Scattering Problem, Dirichlet Boundary Conditions The Direct Scattering Problem, Inhomogeneous Medium The Inverse Scattering Problem (Dirichlet Boundary Conditions) Part II: A Factorization of the Far Field Operator Range Identities The Factorization Method Some Numerical Simulations Part III: The Inverse Scattering Problem for Inhomogeneous Media An Interior Transmission Eigenvalue Problem (see also Lassi!) An Electromagnetic Inverse Scattering Problem (perhaps)

3 The Factorization Method for Inverse Scattering Problems Part I 3/21 Introduction Propagation of (acoustic) waves is modelled by scalar wave equation in R d for d = 2 or 3: 2 U(x, t) t 2 = c 2 x U(x, t), x R d, t 0, where U(x, t) is potential at location x and time t. Speed of sound: c Velocity: x U(x, t) Pressure: U(x, t)/ t Special Case: Time harmonic (=periodic) waves, in complex form: U(x, t) = u(x) e iωt with circular frequency ω > 0; that is λ = 2π/ω is wavelength. Physical wave: Re U(x, t) = Re u(x) cos(ωt) + Im u(x) sin(ωt).

4 The Factorization Method for Inverse Scattering Problems Part I 4/21 Introduction u satisfies Helmholtz equation (reduced wave equation) u(x) + k 2 u(x) = 0 in (part of) R d, with wave number k = ω/c > 0. Examples: (a) Plane wave of direction ˆθ S d 1 (= unit sphere in R d ): u(x) = e ik ˆθ x, x R d. (b) Spherical wave with source point y R d : i 4 H(1) 0 (k x y ), d = 2, u(x) = Φ(x, y) := exp(ik x y ), 4π x y d = 3, x y.

5 The Factorization Method for Inverse Scattering Problems Part I 5/21 Introduction For a general scattering problem a given ( incident ) wave u inc is disturbed by a medium D and produces a scattered field u s Total field: u = u inc + u s u inc D u s

6 The Factorization Method for Inverse Scattering Problems Part I 6/21 Introduction The direct scattering problem is to determine the scattered and total field when the wave number k > 0, the incident field u inc, and the scattering medium D is given. In the inverse scattering problem the incident and the scattered fields are known ( measured ), and the medium D has to be determined. Literature on time harmonic (inverse) scattering theory: D. Colton, R. Kress: Inverse Acoustic and Electromagnetic Scattering Theory. 2nd edition, Springer, A. Kirsch: Introduction to the Mathematical Theory of Inverse Problems. Springer 1996, J.-C. Nédélec: Acoustic and Electromagnetic Equations. Springer, 2001.

7 The Factorization Method for Inverse Scattering Problems Part I 7/21 Direct Scattering Problem, Dirichlet BC Simple model for scattering problem: Helmholtz equation: u + k 2 u = 0 in R d \ D Boundary condition: u = 0 on D Sommerfeld s radiation condition (SRC): u s (x) r ik u s (x) = O ( r (d+1)/2), r = x, uniformly with respect to ˆx = x/ x S d 1. This is a classical boundary value problem in the (unbounded) exterior domain R d \ D for u s of the type: v + k 2 v = 0 in R d \ D, v = f on D, v satisfies SRC

8 Direct Scattering Problem, Dirichlet BC How to solve this exterior boundary value problem v + k 2 v = 0 in R d \ D, v = f on D, v satisfies SRC? First uniqueness: This is based on Lemma of Rellich: For k > 0 (real valued) and v + k 2 v = 0 for x > R 0 it holds that v 2 ds = 0 implies v = 0 for x > R 0 lim R x =R Proof of uniqueness: Assume f = 0. Green s theorem yields o(1) = v r ikv 2 ds = v 2 r + k 2 v 2 ds x =R + 2k Im x =R v v r ds }{{} x =R by Green s theorem in B R \ D and v = 0 on D. Thus v = 0 in R d \ D. = 0 The Factorization Method for Inverse Scattering Problems Part I 8/21

9 The Factorization Method for Inverse Scattering Problems Part I 9/21 Direct Scattering Problem, Dirichlet BC Existence: At least two approaches. Let f H 1/2 ( D). (A) Variational approach: Let B be ball with radius R such that D B. Solution of ext. bvp for B is given by series (d = 2): v(r, φ) = n Z f n H (1) n (kr) H(1) n (kr) e inφ, with f n = 1 2π f (Rφ) e inφ dφ. 2π 0 This defines Dirichlet-Neumann operator Λ : f v/ r B. Green s formula in B \ D: B\D [ v ψ k 2 vψ ] dx = B ψ v ds for all ψ X, ν where X = { ψ H 1 (B \ D) : ψ = 0 on D }.

10 The Factorization Method for Inverse Scattering Problems Part I 10/21 Direct Scattering Problem, Dirichlet BC Substituting v/ ν B = Λv yields: Determine v H 1 (B \ D) with v = f on D and [ v ψ k 2 vψ ] dx = ψ Λv ds for all ψ X. B\D B Transformation to homogeneous boundary data: Choose F H 1 (B \ D) with F = f on D and F = 0 on B and make ansatz v = F + w. Then w X has to solve [ w ψ k 2 wψ ] dx ψ Λw ds = l(ψ), ψ X where B\D l(ψ) = B [ F ψ k 2 Fψ ] dx, ψ X. B\D

11 Direct Scattering Problem, Dirichlet BC B\D [ w ψ k 2 wψ ] dx ψ Λw ds = l(ψ), ψ X. B Remark: Λf = Λ k f = k n Z (H (1) n ) (kr) f n H (1) n (kr) e inφ is bounded from H 1/2 ( B) into H 1/2 ( B). Integral over B is dual form Λ k w, ψ. Difference Λ k Λ i compact, sesqui-linear forms: a(w, ψ) = w ψ dx Λ i w, ψ is coercive, b(w, ψ) = k 2 B\D B\D wψ dx (Λ i Λ k )w, ψ is compact. Theorem of Lax-Milgram and Riesz-theory yield existence. The Factorization Method for Inverse Scattering Problems Part I 11/21

12 The Factorization Method for Inverse Scattering Problems Part I 12/21 Direct Scattering Problem, Dirichlet BC (B) Boundary integral equation approach: Recall fundamental solution i 4 H(1) 0 (k x y ), d = 2, Φ(x, y) := exp(ik x y ), 4π x y d = 3, x y. Make ansatz for v as combination of double and single layer: [ ] v(x) = Φ(x, y) + i Φ(x, y) ϕ(y) ds(y), x / D, ν(y) D with density ϕ H 1/2 ( D). Then v satisfies SRC and v + k 2 v = 0 outside of D. Continuity properties of single and double layer with H 1/2 ( D) density yields boundary integral equation

13 Direct Scattering Problem, Dirichlet BC where (Dϕ)(x) = (Sϕ)(x) = ϕ + Dϕ + i Sϕ = f in H 1/2 ( D), D ϕ(y) Φ(x, y) ds(y), x D, ν(y) ϕ(y) Φ(x, y) ds(y), x D, D are compact in H 1/2 ( D). Again, Riesz-theory yields existence. Literature on boundary integral equation approach: D. Colton, R. Kress: Inverse Acoustic and Electromagnetic Scattering Theory, 2nd edition, Springer, W. McLean: Strongly Elliptic Systems and Boundary Integral Operators, Cambridge University Press, 2000 The Factorization Method for Inverse Scattering Problems Part I 13/21

14 The Factorization Method for Inverse Scattering Problems Part I 14/21 Direct Scattering Problem, Dirichlet BC Fundamental solution has asymptotic form Φ(x, y) = γ d exp(ik x ) x (d 1)/2 [ e ik ˆx y + O(1/ x ) ], x, uniformly with respect to ˆx := x/ x S d 1 and y D, where γ 2 = (1 + i)/(4 kπ) and γ 3 = 1/(4π). Ansatz yields v(x) = γ d exp(ik x ) x (d 1)/2 [ v (ˆx) + O(1/ x ) ], x, uniformly with respect to ˆx := x/ x S d 1. For special case f (x) = u inc (x) = exp(ik ˆθ x) the function u = u (ˆx, ˆθ) is called far field pattern or scattering amplitude.

15 Direct Scattering Problem, Inhom. Medium Karlsruhe Medium penetrable with index of refraction 1 + q. Institute of Technology Direct scattering problem: Given bounded domain D and k > 0 and q L (D) with q ˆq > 0 on D and u inc (x) = exp(ik ˆθ x), determine total field u and scattered field u s = u u inc such that u + k 2 (1 + q)u = 0 in R d, u s satisfies SRC. Uniqueness again by Rellich s Lemma and unique continuation. Existence by, e.g., Lippmann-Schwinger integral equation u(x) = u inc (x) + k 2 q(y) u(y) Φ k (x, y) dy, x D. Again: D u s (x) = γ d exp(ik x ) x (d 1)/2 [ u (ˆx) + O(1/ x ) ], x, uniformly with respect to ˆx := x/ x S d 1 with far field pattern u = u (ˆx, ˆθ). The Factorization Method for Inverse Scattering Problems Part I 15/21

16 The Factorization Method for Inverse Scattering Problems Part I 16/21 Inverse Scattering Problem (Dirich. BC) Recall: Incident plane wave u inc (x) = exp(ik ˆθ x) is scattered by obstacle D R d and produces scattered field u s = u s (x, ˆθ) and total field u = u inc + u s satisfying u + k 2 u = 0 in R d \ D, u = 0 on D, and u s satisfies Sommerfeld s radiation condition (SRC). u s has asyptotic behaviour u s (x, ˆθ) = γ d exp(ik x ) x (d 1)/2 [ u (ˆx, ˆθ) + O(1/ x ) ], x, uniformly with respect to ˆx := x/ x S d 1 and ˆθ S d 1. The inverse scattering problem is to determine the shape of D from the knowledge of the far field pattern u (ˆx, ˆθ) for all ˆx, ˆθ S d 1.

17 The Factorization Method for Inverse Scattering Problems Part I 17/21 The Inverse Scattering Problem Which domain D R 2 corresponds to the following far fields u (φ, θ), φ, θ [0, 2π]? Re u Im u Re u Im u Left example simple: Theorem of Karp: If u (ˆx, ˆθ ) = ψ(ˆx ˆθ) for all ˆx, ˆθ S d 1, then D is a ball in R d 1.

18 The Inverse Scattering Problem First uniqueness: Theorem (Schiffer, before 1967) The far field patterns u (ˆx, ˆθ) determine D uniquely; that is, if D j u j (ˆx, ˆθ) for j = 1, 2, then: u 1 (ˆx, ˆθ) = u 2 (ˆx, ˆθ) for all ˆx, ˆθ S d 1 = D 1 = D 2. Second stability: Theorem (Isakov 1991, 1993) Let D 1, D 2 be star-shaped with respect to the origin; that is, D j = {r ˆx : r < d j (ˆx), ˆx S d 1 }. Assume 1 c 0 d j 2+τ c 0 and u 1 (ˆx, ˆθ) u2 (ˆx, ˆθ) ε for all ˆx S d 1 and some ˆθ S d 1. Then where c depends only on c 0. d 1 d 2 c ln( ln ε) 1/c The Factorization Method for Inverse Scattering Problems Part I 18/21

19 The Factorization Method for Inverse Scattering Problems Part I 19/21 The Inverse Scattering Problem Third reconstruction techniques: (A) Iterative methods. Define mapping (for fixed incident field) F : D u. Apply iterative method to solve F(D) = f for D where f is given (measured) far field pattern. After parametrization (for, e.g., star-shaped obstacles) this leads to F : C 2 (S d 1, R >0 ) C(S d 1, C). Possible members of this group: Newton-type methods, gradient-type methods, second order methods. Derivatives are computed via, e.g., domain derivatives. Advantages: Very general, accurate, incorporation of a priori information possible. Disadvantages: Expensive, only local convergence, a priori information necessary (number of components, type of boundary condition).

20 The Factorization Method for Inverse Scattering Problems Part I 20/21 The Inverse Scattering Problem (B) Methods based on analytic continuation. First step: Given (measured) far field pattern f on D d 1 determine (approximation of) scattered field u s with u = f. Second step: Determine surface Γ such thatu s + u inc = 0 on Γ. Usually, both steps are combined into one functional to be minimized. Members of this group: Dual space method by Colton/Monk, continuation method by Kirsch/Kress, point source method by Potthast, contrast source inversion method by Kleinman/van den Berg. Advantages: Avoids computation of direct problems, quite general, incorporation of a priori information possible. Disadvantages: Only local convergence since methods are based on minimization of non-quadratic functionals, a priori information necessary (number of components, type of boundary condition).

21 The Factorization Method for Inverse Scattering Problems Part I 21/21 The Inverse Scattering Problem (C) Sampling Methods. Choose set of sampling objects, e.g. points z R d, and construct binary criterium which uses only the data u to decide whether or not z belongs to D. Members of this group: Linear sampling method by Colton/Kirsch, Factorization method by Kirsch (both use points z R d as sampling objects), Probe method by Ikehata (curves), No-response-test by Luke/Potthast (domains), Singular sources method by Potthast (points, in combination with point source method) We discuss only Factorization method. Advantages: Fast, avoids computation of direct problems, no a priori information on type of boundary condition or number of components necessary, mathematically elegant and rigorous, gives characteristic function explicitely. Disadvantages: Needs u (ˆx, ˆθ) for many (in theory: all) ˆx, ˆθ, no incorporation of a-priory information possible, very sensitive to noise