Inverse obstacle scattering problems using multifrequency measurements

Transcription

1 Inverse obstacle scattering problems using multifrequency measurements Nguyen Trung Thành Inverse Problems Group, RICAM Joint work with Mourad Sini *** Workshop 3 - RICAM special semester 2011 Nov 21-25

2 Contents 1 Problem statement & motivation 2 Mathematical analysis 3 Numerical results 4 Conclusions & Future work Nguyen Trung Thành Multifrequency scattering 2

3 Forward scattering model Scattering of time-harmonic acoustic waves by a 2D sound-soft obstacle D: u(x)+k 2 u(x) = 0, x R 2 \ D, u(x) = 0, x D, ( u s (x) x x lim x k: wavenumber ( frequency) ) iku s (x) = 0. u = u i +u s : total wave, u i : incident wave, u s : scattered wave Incident plane wave: u i (x) := e ikx θ, θ S 1 : incident direction. Nguyen Trung Thành Multifrequency scattering 3

4 Far field measurements & Inverse problem Far field asymptotic expansion: u s (x) = eik x x u (ˆx)+O( x 3/2 ), x,ˆx := x x. u (ˆx): the far field pattern of u s (measured data). Nguyen Trung Thành Multifrequency scattering 4

5 Far field measurements & Inverse problem Far field asymptotic expansion: u s (x) = eik x x u (ˆx)+O( x 3/2 ), x,ˆx := x x. u (ˆx): the far field pattern of u s (measured data). The inverse problem: reconstruct D from measured far field patterns u m(ˆx,k), ˆx S 1, for one direction of incidence θ S 1 and multiple wavenumbers k in [k l,k h ]. Nguyen Trung Thành Multifrequency scattering 4

6 Far field measurements & Inverse problem Far field asymptotic expansion: u s (x) = eik x x u (ˆx)+O( x 3/2 ), x,ˆx := x x. u (ˆx): the far field pattern of u s (measured data). The inverse problem: reconstruct D from measured far field patterns u m(ˆx,k), ˆx S 1, for one direction of incidence θ S 1 and multiple wavenumbers k in [k l,k h ]. On the uniqueness: For only one frequency: Colton-Sleeman For a band of frequencies: Ramm Nguyen Trung Thành Multifrequency scattering 4

7 Why multifrequency data? Nguyen Trung Thành Multifrequency scattering 5

8 Why multifrequency data? At low frequency: Uniquely solvable Set of local convexity is large (see later). Stability is poor (Isakov 1991, Sincich-Sini 2008) difficult to reconstruct small details! Nguyen Trung Thành Multifrequency scattering 5

9 Why multifrequency data? At low frequency: Uniquely solvable Set of local convexity is large (see later). Stability is poor (Isakov 1991, Sincich-Sini 2008) difficult to reconstruct small details! Can we distinguish a flower and a circle with 5% noise? x x 1 Nguyen Trung Thành Multifrequency scattering 5

10 Why multifrequency data? At low frequency: Uniquely solvable Set of local convexity is large (see later). Stability is poor (Isakov 1991, Sincich-Sini 2008) difficult to reconstruct small details! Can we distinguish a flower and a circle with 5% noise? L 2 norm difference 5% noise level x x wave number Nguyen Trung Thành Multifrequency scattering 5

11 At high frequency: Why multifrequency data? (cont.) More stable but not uniquely solvable. Set of local convexity is small requires a good initial guess! Nguyen Trung Thành Multifrequency scattering 6

12 At high frequency: Why multifrequency data? (cont.) More stable but not uniquely solvable. Set of local convexity is small requires a good initial guess! The least squares objective functional associated with a circle: 25 k = 1 k = 8 20 Obj. fun radius Nguyen Trung Thành Multifrequency scattering 6

13 How to use multifrequency data? Nguyen Trung Thành Multifrequency scattering 7

14 How to use multifrequency data? Recursive optimization methods: 1 Step 1: Reconstruct a rough approximation of the obstacle s shape at the lowest frequency. 2 Step 2: Refine the reconstruction by using recursive optimization methods at higher frequencies. Nguyen Trung Thành Multifrequency scattering 7

15 How to use multifrequency data? Recursive optimization methods: 1 Step 1: Reconstruct a rough approximation of the obstacle s shape at the lowest frequency. 2 Step 2: Refine the reconstruction by using recursive optimization methods at higher frequencies. Two types of recursive optimization methods: Nguyen Trung Thành Multifrequency scattering 7

16 How to use multifrequency data? Recursive optimization methods: 1 Step 1: Reconstruct a rough approximation of the obstacle s shape at the lowest frequency. 2 Step 2: Refine the reconstruction by using recursive optimization methods at higher frequencies. Two types of recursive optimization methods: Nonlinear frequency hopping algorithm: Chew-Lin 1995, Bucci-Isernia-Pascazio 2000, Dorn-Miller-Rappaport Nguyen Trung Thành Multifrequency scattering 7

17 How to use multifrequency data? Recursive optimization methods: 1 Step 1: Reconstruct a rough approximation of the obstacle s shape at the lowest frequency. 2 Step 2: Refine the reconstruction by using recursive optimization methods at higher frequencies. Two types of recursive optimization methods: Nonlinear frequency hopping algorithm: Chew-Lin 1995, Bucci-Isernia-Pascazio 2000, Dorn-Miller-Rappaport Recursive linearization algorithm (RLA): Chen 1997 Convergence for noiseless data: Bao-Triki Nguyen Trung Thành Multifrequency scattering 7

18 Mathematical analysis Convexity at low frequency Convergence of the RLA Stability at high frequency Nguyen Trung Thành Multifrequency scattering 8

19 Star-shaped obstacles We consider a smooth star-shaped obstacle D of class C 3 : D = {x 0 +x(t) R 2 : x(t) = r(t)(cost,sint),t [0,2π]} x 0 : a given internal point. Unknown: the radial function r(t). Parameter space: X := {ϕ C 3 [0,2π] : ϕ(0) = ϕ(2π)}. Denote by X + := {ϕ X : ϕ > 0}. Nguyen Trung Thành Multifrequency scattering 9

20 Far field operator Far field operator F : X [k l,k h ] Y := L 2 (S 1 ) is defined as F(r,k) = u (,k,r) Y. Domain derivative of F at r in the direction a X: F(r +ǫa,k) F(r,k) r F(r,k)a := lim. ǫց0 ǫ r F(r,k) is called the domain derivative of F at r. Nguyen Trung Thành Multifrequency scattering 10

21 Properties of the domain derivative Theorem (see Kirsch 1993, Potthast 1994) Let r X +, a X. Then r F(r,k)a exists and is given by r F(r,k)a = ũ, the far field patern of ũ(x)+k 2 ũ(x) = 0,x R 2 \ D, ũ(x) = a(t)(cost,sint) ν(x) u(x) ν,x = x(t) D,t [0,2π], [ ] lim x ũ(x) x ikũ(x) = 0. x Nguyen Trung Thành Multifrequency scattering 11

22 Properties of the domain derivative Theorem (see Kirsch 1993, Potthast 1994) Let r X +, a X. Then r F(r,k)a exists and is given by r F(r,k)a = ũ, the far field patern of ũ(x)+k 2 ũ(x) = 0,x R 2 \ D, ũ(x) = a(t)(cost,sint) ν(x) u(x) ν,x = x(t) D,t [0,2π], [ ] lim x ũ(x) x ikũ(x) = 0. x Corollary For each r X +, r F(r,k) is injective from X to Y. Nguyen Trung Thành Multifrequency scattering 11

23 Region of local convexity For noiseless measurements: G 0 (r,k) := 1 2 F(r,k) u m(,k) 2 Y. Nguyen Trung Thành Multifrequency scattering 12

24 Region of local convexity For noiseless measurements: G 0 (r,k) := 1 2 F(r,k) u m(,k) 2 Y. Theorem (Set of local convexity) Suppose that D B(x 0, π k ). Then G 0(,k) is convex in the set B( π k ) := {r X+ : r π k }, where r represents the maximum norm of r. Idea of proof: injectivity of r F(r,k) and uniqueness of the inverse problem. Nguyen Trung Thành Multifrequency scattering 12

25 Region of local convexity (cont.) For noisy measurements: G γ (r,k) := 1 2 F(r,k) u,δ m (,k) 2 Y γ r 2 X. Nguyen Trung Thành Multifrequency scattering 13

26 Region of local convexity (cont.) For noisy measurements: Theorem G γ (r,k) := 1 2 F(r,k) u,δ m (,k) 2 Y γ r 2 X. Suppose that D B(x 0, π k ). Then for γ = dδ, d max rrf(r,k) L(X X,Y), G γ (,k) is convex in the set B( π r B( π k ) 2 k ). Nguyen Trung Thành Multifrequency scattering 13

27 Recursive linearization algorithm Consider a set of frequencies: k j = k l +j k,j = 0,1,...,N. Nguyen Trung Thành Multifrequency scattering 14

28 Recursive linearization algorithm Consider a set of frequencies: k j = k l +j k,j = 0,1,...,N. Step 1: Find an approximation r 0 at k 0. Nguyen Trung Thành Multifrequency scattering 14

29 Recursive linearization algorithm Consider a set of frequencies: k j = k l +j k,j = 0,1,...,N. Step 1: Find an approximation r 0 at k 0. Step 2: Given an approximation r j at k j, linearize: F(r,k j+1 ) F(r j,k j+1 )+A j (r r j ) with A j := r F(r j,k j+1 ) Nguyen Trung Thành Multifrequency scattering 14

30 Recursive linearization algorithm Consider a set of frequencies: k j = k l +j k,j = 0,1,...,N. Step 1: Find an approximation r 0 at k 0. Step 2: Given an approximation r j at k j, linearize: F(r,k j+1 ) F(r j,k j+1 )+A j (r r j ) with A j := r F(r j,k j+1 ) and update the shape r j+1 = r j + r j with } r j := argmin r { 1 2 A j r + F δ (r j,k j+1 ) 2 Y α r 2 X where F δ (r,k j+1 ) = F(r,k j+1 ) u,δ m (,k j+1 ). Nguyen Trung Thành Multifrequency scattering 14

31 Finite dimensional observable shapes Truncated Fourier series: r M (t) = β 0 + M (β m cosmt +γ m sinmt). m=1 Denote by X M = span{1,cost,sint,...,cosmt,sinmt}. Nguyen Trung Thành Multifrequency scattering 15

32 Finite dimensional observable shapes Truncated Fourier series: r M (t) = β 0 + M (β m cosmt +γ m sinmt). m=1 Denote by X M = span{1,cost,sint,...,cosmt,sinmt}. Definition For each k and a given δ > δ, a FD observable shape (or, in short, observable shape) D( r(k)) is defined as a domain whose radial function r(k) belongs to X + M for some M N and. F( r(k),k) u,δ m (,k) Y δ Remark: The FD observable shape may not exist if δ = δ! Nguyen Trung Thành Multifrequency scattering 15

33 Convergence of RLA Assumption 1: The radial functions of the observable shapes satisfy r(k j+1 ) r(k j ) X d 0 k j+1 k j, j = 0,...,N 1. Nguyen Trung Thành Multifrequency scattering 16

34 Convergence of RLA Assumption 1: The radial functions of the observable shapes satisfy r(k j+1 ) r(k j ) X d 0 k j+1 k j, j = 0,...,N 1. Noiseless data: rewrite F δ (r,k) = F(r,k)+f δ ( r(k),k) with F(r,k) := F(r,k) F( r(k),k) f δ ( r(k),k) := F( r(k),k) u,δ m (,k) and consider the residual F(r,k) instead of F δ (r,k). Nguyen Trung Thành Multifrequency scattering 16

35 Convergence rate: noiseless data Lemma (see also Bao-Triki 2010) Under Assumption 1 and with F δ being replaced by F, there exist constants α, c 0 and N 0 such that if r(k l ) r 0 X c 0 α, then the following error estimate holds r(k h ) r N X C N α, N N 0, where C is a constant independent of α and N. Remark: Bao-Triki assumed that r(k) is twice diff. w.r.t. k. Nguyen Trung Thành Multifrequency scattering 17

36 Convergence rate: noisy data Denote by X Mj the subspace of X containing both r j and r(κ j ). Assumption 2: The smallest singular value σ of à j XMj,j {0,...,N}, where à j := rf( r(k j ),k j ), satisfies ( ) ǫ 3/2 σ 3 3 δ 3 ǫ 2ξ(1 ǫ)ǫ (a+bσ) 0 for some fixed values ǫ,ξ (0,1) and a, b independent of δ and σ. Meaning of Assumption 2? Nguyen Trung Thành Multifrequency scattering 18

37 Relation between Assumption 2 and resolution 1 Fix k j : σ decreases if number of Fourier modes increases 2 Fix M: σ increases w.r.t. frequency k = 0.5 k = 2 k = 5 k = Singular value Number of Fourier coefficients Figure: Singular values of a flower at different frequencies. Nguyen Trung Thành Multifrequency scattering 19

38 Convergence rate: noisy data (cont.) Theorem Assume that Assumptions 1 & 2 are satisfied and α satisfies Then if ( ) ǫ 3/2 σ 3 α 3/2 3 ǫ r(κ l ) r 0 X 3 δ 2ξ(1 ǫ)ǫ (a+bσ). ǫ 3(a+bσ) α, there exists a constant N 0 = N 0 (α) such that r(κ h ) r N X C N α + δ 2 α(1 ǫ), for all N N 0, where C is independent of α and N. Nguyen Trung Thành Multifrequency scattering 20

39 Stability at high frequency Assumption: the obstacle is convex. Green s theorem: u (ˆx,θ,k) = eiπ/4 8πk Kirchhoff approximation: D u(y) ν(y) = 2 ui (y) ν(y) for y D +, u(y) ν(y) e ikˆx y ds(y). D + : illuminated part, D : shadowed part u(y) ν(y) = 0 for y D. Nguyen Trung Thành Multifrequency scattering 21

40 Stability at high frequency Assumption: the obstacle is convex. Green s theorem: u (ˆx,θ,k) = eiπ/4 8πk Kirchhoff approximation: D u(y) ν(y) = 2 ui (y) ν(y) for y D +, u(y) ν(y) e ikˆx y ds(y). D + : illuminated part, D : shadowed part For star-shaped obstacle: t 2 u (ˆx,θ,k) = ikeiπ/4 2πk where ϕ(t) := θ ˆx θ ˆx (cost,sint)r(t). u(y) ν(y) = 0 for y D. t 1 θ ν(x(t)) x (t) e ik θ ˆx ϕ(t) dt, Nguyen Trung Thành Multifrequency scattering 21

41 Stability estimate (cont.) Specular point: for s S 1, the specular point x 0 (s) D + is defined as x 0 (s) s := min x D + x s. The support function d(s) (Gutman-Ramm 2002): d(s) := x 0 (s) s, Nguyen Trung Thành Multifrequency scattering 22

42 Stability estimate (cont.) For an observation direction ˆx θ, denote by x 0 (s) := (cost 0,sint 0 )r(t 0 ) the specular point associated with s := θ ˆx θ ˆx. Property: t 0 is the unique minimum point of ϕ(t) and ϕ(t 0 ) = d(s). Nguyen Trung Thành Multifrequency scattering 23

43 Stability estimate (cont.) For an observation direction ˆx θ, denote by x 0 (s) := (cost 0,sint 0 )r(t 0 ) the specular point associated with s := θ ˆx θ ˆx. Property: t 0 is the unique minimum point of ϕ(t) and ϕ(t 0 ) = d(s). Stationary phase method implies: u (ˆx,θ,k) = 1 [ ] θ ˆx eik θ ˆx d(s) 1 1+O( 2 κ(x 0 (s)) k θ ˆx ), κ(x 0 (s)): the curvature of the shape at x 0 (s). Remark: the support function is not uniquely determined from this equation! Nguyen Trung Thành Multifrequency scattering 23

44 Point-wise stability estimate Nguyen Trung Thành Multifrequency scattering 24

45 Theorem Point-wise stability estimate Assume that the support functions d(s) and d(s) of D and D satisfy d d(s), d(s) d with d d ǫ π k for a fixed value ǫ (0,1). Then, for δ small and k large enough, we have d(s) d(s) 4 κ u k θ ˆx 3/2O(δ)+O( 1 k 2 θ ˆx 2), where κ u is the upper bound of the curvature of the obstacles. Nguyen Trung Thành Multifrequency scattering 24

46 Theorem Point-wise stability estimate Assume that the support functions d(s) and d(s) of D and D satisfy d d(s), d(s) d with d d ǫ π k for a fixed value ǫ (0,1). Then, for δ small and k large enough, we have d(s) d(s) 4 κ u k θ ˆx 3/2O(δ)+O( 1 k 2 θ ˆx 2), where κ u is the upper bound of the curvature of the obstacles. Conclusion (Höder stability) The illuminated parts of D can be reconstructed with stability O( δ) in the high frequency regime. Nguyen Trung Thành Multifrequency scattering 24

47 Numerical results Incident angle: 30 o. k l = 0.3, k h = 8 with 20 wavenumbers. α = 10 2, 16 observation directions in the full aperture. Noise level = 5%. Nguyen Trung Thành Multifrequency scattering 25

48 Numerical results Incident angle: 30 o. k l = 0.3, k h = 8 with 20 wavenumbers. α = 10 2, 16 observation directions in the full aperture. Noise level = 5%. Reconstruction with a far away initial guess: 3 Estimate, κ =8 Estimate, κ =0.3 Initial guess True shape 3 Estimate, κ =8 Estimate, κ =0.3 Initial guess True shape x 2 x x x 1 Nguyen Trung Thành Multifrequency scattering 25

49 Effect of no. of frequencies & the highest frequency 10 frequencies [0.3, 8] 20 frequencies [0.3, 3] 3 2 Estimate, κ =8 Estimate, κ =0.3 Initial guess True shape 3 2 Estimate, κ =3 Estimate, κ =0.3 Initial guess True shape 1 1 x 2 0 x x x 1 Nguyen Trung Thành Multifrequency scattering 26

50 Limited aperture measurements Obs. angle: 30 o 210 o Obs. angle: 210 o 360 o Estimate, κ =8 Estimate, κ =0.3 Initial guess True shape Estimate, κ =8 Estimate, κ =0.3 Initial guess True shape x 2 0 x x x 1 Nguyen Trung Thành Multifrequency scattering 27

51 Improve the accuracy using multiple incident directions Two perpendicular incident directions: 30 o and 120 o. 3 2 Estimate, κ =8 Estimate, κ =0.3 Initial guess True shape 1 x x 1 Nguyen Trung Thành Multifrequency scattering 28

52 A glance at multiple scattering Incident direction: 120 o. When a test circle is known: Estimate, κ =8 Estimate, κ =0.3 Initial guess True shape x x Estimate, κ =8 Estimate, κ =0.3 Initial guess True shape x x 1 Nguyen Trung Thành Multifrequency scattering 29

53 A glance at multiple scattering When both obstacles are unkown: Estimate, κ =8 Estimate, κ =0.3 Initial guess True shape x x 1 Nguyen Trung Thành Multifrequency scattering 30

54 Conclusions: Conclusions & Future work Using multifrequency data, we can reconstruct the illuminated part quite well without a good initial guess. The more frequencies used, the better the convergence The smaller the lowest frequency, the more freely the initial guess we can choose. The higher the highest frequency, the more accurate the reconstruction. Nguyen Trung Thành Multifrequency scattering 31

55 Conclusions: Conclusions & Future work Using multifrequency data, we can reconstruct the illuminated part quite well without a good initial guess. The more frequencies used, the better the convergence The smaller the lowest frequency, the more freely the initial guess we can choose. The higher the highest frequency, the more accurate the reconstruction. Future work: Other types of boundary conditions Locally adaptive parametrization Higher order recursive optimization algorithms Waveguide models, stratified media Nguyen Trung Thành Multifrequency scattering 31