Weierstrass Institute for Applied Analysis and Stochastics Direct and Inverse Elastic Scattering Problems for Diffraction Gratings

Transcription

1 Weierstrass Institute for Applied Analysis and Stochastics Direct and Inverse Elastic Scattering Problems for Diffraction Gratings Johannes Elschner & Guanghui Hu Mohrenstrasse Berlin Germany Tel PICOF 12, April 4, 2012

2 Content 1 Direct and inverse elastic scattering problems in periodic structures 2 Direct scattering problem: uniqueness and existence 3 Inverse scattering problem: uniqueness for polygonal gratings 4 Inverse scattering problem: a two-step algorithm Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 2 (27)

3 Elastic scattering by diffraction gratings Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 3 (27)

4 Elastic scattering by periodic structures: application in geophysics Characterize fractures using elastic waves in search for gas and liquids Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 4 (27)

5 Applications Geophysics -search for oil, gas and ore bodies Seismology -investigate earthquakes Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 5 (27)

6 Applications Geophysics -search for oil, gas and ore bodies Seismology -investigate earthquakes Nondestructive Testings (NDT) - detect cracks and flaws in concrete structures, such as bridges, buildings, dams, highways, etc. Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 5 (27)

7 Applications Geophysics -search for oil, gas and ore bodies Seismology -investigate earthquakes Nondestructive Testings (NDT) - detect cracks and flaws in concrete structures, such as bridges, buildings, dams, highways, etc. Problems: Understand the reflection and transmission of elastic waves through an interface Design efficient inversion algorithms using elastic waves Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 5 (27)

8 Basic assumptions The periodic surface Λ is invariant in one direction and 2π-periodic in another direction, φ = 0. The elastic medium above Λ is homogeneous, isotropic with the Lamé constants λ,µ. The mass density ρ = 1. The elastic displacement-field is time-harmonic, U(t,x) = u(x)e iωt. Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 6 (27)

9 Direct problems Navier equation in 2D (case φ = 0): ( + ω 2 )u = 0 in Ω Λ, := µ + (λ + µ)grad div, u = u in + u sc in Ω Λ. Angular frequency: ω > 0 Lamé constants: µ > 0,λ + µ > 0 Compressional wave number: k p := ω/ 2µ + λ Shear wave number: k s := ω/ µ Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 7 (27)

10 Direct problems Navier equation in 2D (case φ = 0): ( + ω 2 )u = 0 in Ω Λ, := µ + (λ + µ)grad div, u = u in + u sc in Ω Λ. Angular frequency: ω > 0 Lamé constants: µ > 0,λ + µ > 0 Compressional wave number: k p := ω/ 2µ + λ Shear wave number: k s := ω/ µ Incident angle: θ ( π/2, π/2) Incident plane pressure wave: u in p (x) = ˆθ exp(ik p x ˆθ), ˆθ := (sinθ, cosθ), x = (x 1,x 2 ) Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 7 (27)

11 Direct problems Navier equation in 2D (case φ = 0): ( + ω 2 )u = 0 in Ω Λ, := µ + (λ + µ)grad div, u = u in + u sc in Ω Λ. Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 8 (27)

12 Direct problems Navier equation in 2D (case φ = 0): ( + ω 2 )u = 0 in Ω Λ, := µ + (λ + µ)grad div, u = u in + u sc in Ω Λ. Quasi-periodicity: u(x 1 + 2π,x 2 ) = exp(2iαπ)u(x 1,x 2 ), α := k p sinθ. Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 8 (27)

13 Direct problems Navier equation in 2D (case φ = 0): ( + ω 2 )u = 0 in Ω Λ, := µ + (λ + µ)grad div, u = u in + u sc in Ω Λ. Quasi-periodicity: u(x 1 + 2π,x 2 ) = exp(2iαπ)u(x 1,x 2 ), α := k p sinθ. Dirichlet boundary condition on the grating profile: u = 0 on Λ. Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 8 (27)

14 Direct problems Navier equation in 2D (case φ = 0): ( + ω 2 )u = 0 in Ω Λ, := µ + (λ + µ)grad div, u = u in + u sc in Ω Λ. Quasi-periodicity: u(x 1 + 2π,x 2 ) = exp(2iαπ)u(x 1,x 2 ), α := k p sinθ. Dirichlet boundary condition on the grating profile: u = 0 on Λ. An appropriate radiation condition imposed on u sc as x 2. Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 8 (27)

15 Rayleigh Expansion Radiation Condition (RERC) u sc = u p + curl u s, ( + k 2 p)u p = 0, ( + k 2 s )u s = 0 Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 9 (27)

16 Rayleigh Expansion Radiation Condition (RERC) u sc = u p + curl u s, ( + k 2 p)u p = 0, ( + k 2 s )u s = 0 Radiation condition ( u sc (x) = A p,n n Z ( + A s,n n Z for x 2 > Λ + := max (x1,x 2 ) Λ x 2. Here, α n := α + n, β n = β n (θ) := α n β n ) γ n α n exp(iα n x 1 + iβ n x 2 ) ) exp(iα n x 1 + iγ n x 2 ) k 2 p αn 2 if α n k p i αn 2 k 2 p if α n > k p, γ n = γ n (θ) := { k 2 s α 2 n if α n k s i α 2 n k 2 s if α n > k s. The constants A p,n, A s,n C are called the Rayleigh coefficients. Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 9 (27)

17 Direct and inverse scattering problems Direct Problem (DP) Given Λ R 2 and u in, find u = u in + u sc H 1 loc (Ω Λ) 2 under the boundary, quasi-periodicity and radiation conditions. Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 10 (27)

18 Direct and inverse scattering problems Direct Problem (DP) Given Λ R 2 and u in, find u = u in + u sc H 1 loc (Ω Λ) 2 under the boundary, quasi-periodicity and radiation conditions. Uniqueness and existence of solutions Numerical solutions Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 10 (27)

19 Direct and inverse scattering problems Direct Problem (DP) Given Λ R 2 and u in, find u = u in + u sc H 1 loc (Ω Λ) 2 under the boundary, quasi-periodicity and radiation conditions. Uniqueness and existence of solutions Numerical solutions Inverse Problem (IP) Given incident field u in (x;θ) and the near-field data u(x 1,b;θ), determine the unknown scattering surface Λ. Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 10 (27)

20 Direct and inverse scattering problems Direct Problem (DP) Given Λ R 2 and u in, find u = u in + u sc H 1 loc (Ω Λ) 2 under the boundary, quasi-periodicity and radiation conditions. Uniqueness and existence of solutions Numerical solutions Inverse Problem (IP) Given incident field u in (x;θ) and the near-field data u(x 1,b;θ), determine the unknown scattering surface Λ. Can we identify Λ uniquely? How to recover Λ numerically? Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 10 (27)

21 Solvability results on the direct scattering problem Theorem If the grating profile Λ is a Lipschitz curve, then there always exists a solution of (DP). Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 11 (27)

22 Solvability results on the direct scattering problem Theorem If the grating profile Λ is a Lipschitz curve, then there always exists a solution of (DP). Uniqueness holds for small frequencies, or for all frequencies excluding a discrete set with the only accumulation point at infinity. Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 11 (27)

23 Solvability results on the direct scattering problem Theorem If the grating profile Λ is a Lipschitz curve, then there always exists a solution of (DP). Uniqueness holds for small frequencies, or for all frequencies excluding a discrete set with the only accumulation point at infinity. If Λ is the graph of a Lipschitz function, then for any frequency ω > 0, there exists a unique solution of (DP). Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 11 (27)

24 Solvability results on the direct scattering problem Theorem If the grating profile Λ is a Lipschitz curve, then there always exists a solution of (DP). Uniqueness holds for small frequencies, or for all frequencies excluding a discrete set with the only accumulation point at infinity. If Λ is the graph of a Lipschitz function, then for any frequency ω > 0, there exists a unique solution of (DP). Elschner J. and Hu G Variational approach to scattering of plane elastic waves by diffraction gratings M2AS Elschner J. and Hu G Scattering of plane elastic waves by three-dimensional diffraction gratings M3AS Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 11 (27)

25 Reduce the problem to one periodic cell Introduce Γ b := {(x 1,b) : 0 x 1 2π}, Ω b := {x Ω Λ : 0 < x 1 < 2π, x 2 < b} V α = V α (Ω b ) := {u H 1 α(ω b ) 2 : u = 0 on Λ}. D-to-N map 0 Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 12 (27)

26 Uniqueness under the Dirichlet boundary condition If Λ is the graph of a smooth function f, then using a periodic Rellich identity: 0 = 2Re ( + ω)u 2 udx Ω ( b ) = + 2Re(Tu 2 u) E (u,u)n 2 + ω 2 u 2 n 2 ds Λ Γ b ( = µ n u 2 + (λ + µ) div u 2) n 2 ds Λ Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 13 (27)

27 Uniqueness under the Dirichlet boundary condition If Λ is the graph of a smooth function f, then using a periodic Rellich identity: Since 0 = 2Re ( + ω)u 2 udx Ω ( b ) = + 2Re(Tu 2 u) E (u,u)n 2 + ω 2 u 2 n 2 ds Λ Γ b ( = µ n u 2 + (λ + µ) div u 2) n 2 ds Λ n 2 = 1/ 1 + f 2 < 0 on Λ, we have n u = u = 0 on Λ. Applying Holmgren s theorem leads to uniqueness at arbitrary frequency. Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 13 (27)

28 Uniqueness under the Dirichlet boundary condition If Λ is the graph of a smooth function f, then using a periodic Rellich identity: Since 0 = 2Re ( + ω)u 2 udx Ω ( b ) = + 2Re(Tu 2 u) E (u,u)n 2 + ω 2 u 2 n 2 ds Λ Γ b ( = µ n u 2 + (λ + µ) div u 2) n 2 ds Λ n 2 = 1/ 1 + f 2 < 0 on Λ, we have n u = u = 0 on Λ. Applying Holmgren s theorem leads to uniqueness at arbitrary frequency. Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 13 (27)

29 Elastic scattering by rough surfaces ( + ω 2 )u = g in Ω Λ, := µ + (λ + µ)grad div, u = 0 on Λ, radiation condition as x 2 + O Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 14 (27)

30 Elastic scattering by rough surfaces ( + ω 2 )u = g in Ω Λ, := µ + (λ + µ)grad div, u = 0 on Λ, radiation condition as x 2 + O If g L 2 (Ω Λ ) 2, then there admits a unique solution u H 1 (Ω Λ ) 2. Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 14 (27)

31 Uniqueness of (IP) for polygonal grating profiles A = { Λ f : f (x 1 )is a continuous piecewise linear function of period 2π, and is not a straight line parallel to ox 1 } Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 15 (27)

32 Uniqueness of (IP) for polygonal grating profiles A = { Λ f : f (x 1 )is a continuous piecewise linear function of period 2π, and is not a straight line parallel to ox 1 } Assume 1. Λ 1,Λ 2 A, and one of them has a corner point at the origin. 2. u 1 resp. u 2 satisfies the fourth kind boundary conditions on Λ 1 resp. Λ 2. (τ u = 0, n Tu = 0) 3. u 1 (x 1,b;θ) = u 2 (x 1,b;θ) holds for one incident plane pressure wave with the incident angle θ. Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 15 (27)

33 Uniqueness of (IP) for polygonal grating profiles A = { Λ f : f (x 1 )is a continuous piecewise linear function of period 2π, and is not a straight line parallel to ox 1 } Assume 1. Λ 1,Λ 2 A, and one of them has a corner point at the origin. 2. u 1 resp. u 2 satisfies the fourth kind boundary conditions on Λ 1 resp. Λ 2. (τ u = 0, n Tu = 0) 3. u 1 (x 1,b;θ) = u 2 (x 1,b;θ) holds for one incident plane pressure wave with the incident angle θ. Questions: 1. Can we obtain Λ 1 = Λ 2? (uniqueness) Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 15 (27)

34 Uniqueness of (IP) for polygonal grating profiles A = { Λ f : f (x 1 )is a continuous piecewise linear function of period 2π, and is not a straight line parallel to ox 1 } Assume 1. Λ 1,Λ 2 A, and one of them has a corner point at the origin. 2. u 1 resp. u 2 satisfies the fourth kind boundary conditions on Λ 1 resp. Λ 2. (τ u = 0, n Tu = 0) 3. u 1 (x 1,b;θ) = u 2 (x 1,b;θ) holds for one incident plane pressure wave with the incident angle θ. Questions: 1. Can we obtain Λ 1 = Λ 2? (uniqueness) 2. If not, can we describe the exceptional classes of grating profiles that generate the same near field on x 2 = b? Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 15 (27)

35 Uniqueness of (IP) for polygonal grating profiles A = { Λ f : f (x 1 )is a continuous piecewise linear function of period 2π, and is not a straight line parallel to ox 1 } Assume 1. Λ 1,Λ 2 A, and one of them has a corner point at the origin. 2. u 1 resp. u 2 satisfies the fourth kind boundary conditions on Λ 1 resp. Λ 2. (τ u = 0, n Tu = 0) 3. u 1 (x 1,b;θ) = u 2 (x 1,b;θ) holds for one incident plane pressure wave with the incident angle θ. Questions: 1. Can we obtain Λ 1 = Λ 2? (uniqueness) 2. If not, can we describe the exceptional classes of grating profiles that generate the same near field on x 2 = b? 3. How many incident elastic waves are sufficient to uniquely determine an arbitrary grating profile Λ A? Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 15 (27)

36 Uniqueness under the fourth kind boundary conditions Theorem If u 1 (x 1,b;θ) = u 2 (x 1,b;θ), x 1 (0,2π) then either Λ 1 = Λ 2 or Λ 1,Λ 2 D 2 (θ,k p ). In the latter case, the total field takes the form u = ˆθ exp(ik p x ˆθ) ˆθ exp( ik p x ˆθ) e 1 exp(ik p x 1 ) + e 1 exp( ik p x 1 ) in R 2, where e 1 = (1,0). Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 16 (27)

37 Uniqueness under the fourth kind boundary conditions Theorem If u 1 (x 1,b;θ) = u 2 (x 1,b;θ), x 1 (0,2π) then either Λ 1 = Λ 2 or Λ 1,Λ 2 D 2 (θ,k p ). In the latter case, the total field takes the form u = ˆθ exp(ik p x ˆθ) ˆθ exp( ik p x ˆθ) e 1 exp(ik p x 1 ) + e 1 exp( ik p x 1 ) in R 2, where e 1 = (1,0). Corollary If Λ A and Λ / D 2 (θ,k p ), then Λ can be uniquely determined by the near-field data u(x 1,b;θ), under the boundary conditions of the fourth kind. Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 16 (27)

38 Uniqueness with minimal number of incident elastic waves Theorem Under the fourth kind boundary conditions, two incident pressure waves are enough to uniquely determine a grating Λ A. If Rayleigh frequencies of the compressional part are excluded, the minimal number is one incident pressure wave. Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 17 (27)

39 Uniqueness with minimal number of incident elastic waves Theorem Under the fourth kind boundary conditions, two incident pressure waves are enough to uniquely determine a grating Λ A. If Rayleigh frequencies of the compressional part are excluded, the minimal number is one incident pressure wave. -based on the reflection principle for the Navier equation (Elschner & Yamamoto 2010). -can be extended to the third kind boundary value problem in both 2D and 3D. -the inverse Dirichlet boundary value problem is challenging. Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 17 (27)

40 Numerical algorithm for (IP): a two-step algorithm ( ) ( ) Π(x,y) = 1 G ks (x,y) x 2 1 x1 [ ] x2 µ 0 G ks (x,y) ω 2 x2 x1 x 2 G (x,y) G (x,y) ks kp 2 Step 1 Reconstruct the scattered field u sc from near-field measurement data. Making the ansatz for u sc in the form u sc = 1 2π Π(x 1,x 2 ;t,0)ϕ(t)dt, x 2 f (x 1 ), 2π 0 we only need to solve the first kind integral equation Jϕ(x 1 ) := 1 2π Π(x 1,b ;t,0)ϕ(t)dt = u sc (x 1,b). 2π 0 Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 18 (27)

41 Numerical algorithm for (IP): a two-step algorithm ( ) ( ) Π(x,y) = 1 G ks (x,y) x 2 1 x1 [ ] x2 µ 0 G ks (x,y) ω 2 x2 x1 x 2 G (x,y) G (x,y) ks kp 2 Step 1 Reconstruct the scattered field u sc from near-field measurement data. Making the ansatz for u sc in the form u sc = 1 2π Π(x 1,x 2 ;t,0)ϕ(t)dt, x 2 f (x 1 ), 2π 0 we only need to solve the first kind integral equation Jϕ(x 1 ) := 1 2π Π(x 1,b ;t,0)ϕ(t)dt = u sc (x 1,b). 2π 0 Step 2 Find f by minimizing the defect u in (x 1, f (x 1 )) + 1 2π Π(x 1, f (x 1 ) t,0)ϕ(t)dt 2 2π L 0 2 (0,2π) inf, f M where M M = { f (x 1 ) = a 0 + a m cos(mx 1 ) + a M+m sin(mx 1 ),a j R, j = 0,1,,2M}. m=1 Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 18 (27)

42 Numerical examples u in : incident pressure wave θ = 0, k p = 4.2, k s = 4.5, ω = 5 γ : Tikhonov regularization parameter δ : noise level of the measurement u(x 1,b) 2K + 1: the number of propagating modes involved in computation K < 4: part of far-field data K = 4: all far-field data K > 4: all far-field data + some evanescent modes Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 19 (27)

43 Example 1: sensitivity w.r.t. parameter K (δ = 0,γ = ) Figure: K = 1,2,3,4,5,6 K=1 2.5 K=2 2.5 K= computed target initial computed target initial computed target initial K=4 2.5 K=5 2.5 K=6 2.4 computed target initial 2.4 computed target initial 2.4 computed target initial Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 20 (27)

44 Example 2: smooth gratings (K = 7,γ = 10 12,δ = 0) Suppose that f (t) = exp(sin(3t)) + 0.3exp(sin(3t)), f M (t) = a 0 + a m cos(mt) + a M+m sin(mt), M = 8 m=1 Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 21 (27)

45 Example 2: smooth gratings (K = 7,γ = 10 12,δ = 0) Suppose that f (t) = exp(sin(3t)) + 0.3exp(sin(3t)), f M (t) = a 0 + a m cos(mt) + a M+m sin(mt), M = 8 m= computed (δ=0) target initial Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 21 (27)

46 Example 2: smooth gratings (K = 7,γ = 10 12,δ 0) Suppose that f (t) = exp(sin(3t)) + 0.3exp(sin(3t)), f M (t) = a 0 + a m cos(mt) + a M+m sin(mt), M = 8 m=1 Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 22 (27)

47 Example 2: smooth gratings (K = 7,γ = 10 12,δ 0) Suppose that f (t) = exp(sin(3t)) + 0.3exp(sin(3t)), f M (t) = a 0 + a m cos(mt) + a M+m sin(mt), M = 8 m= computed (δ=0.05) computed (δ=0.1) target initial Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 22 (27)

48 Example 2: different initial guesses (K = 4,γ = 10 12,δ = 0) K=4,M=8 computed target initial K=4,M= computed target initial Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 23 (27)

49 Example 3: binary gratings (K = 4,δ = 0,γ = 10 4 ) A priori information: the unknown surface is a binary grating with a fixed number of corner points target initial computed target initial computed Reconstruct a binary grating profile from the far-field data corresponding to three incident angles θ = π/4,0,π/4 (left) or one incident angle θ = 0 (right) Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 24 (27)

50 Example 4: piecewise linear gratings (δ = 0,γ = 10 4 ) target initial computed with K=7 or K= target initial computed with K= K = 4: only far-field data ( n 4) (left) K = 7: far-field data and part of evanescent modes (4 < n 7) K = 3: part of far field data ( n 3) (right) (left) Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 25 (27)

51 Remarks In our reported numerical examples, the unknown grating profile is given by a finite number of parameters (e.g. Fourier coefficients or corner points); Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 26 (27)

52 Remarks In our reported numerical examples, the unknown grating profile is given by a finite number of parameters (e.g. Fourier coefficients or corner points); synthetic near-field data are generated by discrete trigonometric Galerkin method applied to integral equation formulation of direct problem; Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 26 (27)

53 Remarks In our reported numerical examples, the unknown grating profile is given by a finite number of parameters (e.g. Fourier coefficients or corner points); synthetic near-field data are generated by discrete trigonometric Galerkin method applied to integral equation formulation of direct problem; we can readily obtain the singular value decomposition of the first-kind integral operator and solve the nonlinear least-squares minimization problem; Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 26 (27)

54 Remarks In our reported numerical examples, the unknown grating profile is given by a finite number of parameters (e.g. Fourier coefficients or corner points); synthetic near-field data are generated by discrete trigonometric Galerkin method applied to integral equation formulation of direct problem; we can readily obtain the singular value decomposition of the first-kind integral operator and solve the nonlinear least-squares minimization problem; we need not solve the direct scattering problem at each iteration. Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 26 (27)

55 Remarks In our reported numerical examples, the unknown grating profile is given by a finite number of parameters (e.g. Fourier coefficients or corner points); synthetic near-field data are generated by discrete trigonometric Galerkin method applied to integral equation formulation of direct problem; we can readily obtain the singular value decomposition of the first-kind integral operator and solve the nonlinear least-squares minimization problem; we need not solve the direct scattering problem at each iteration. Compared to the Kirsch-Kress optimization method based on the combined cost functional, F(ϕ, f ) = T ϕ u b 2 L 2 (0,2π) + ρ T ϕ + uin 2 L 2 (Λ f ) + γ ϕ 2 L 2 (0,2π), the two-step algorithm can considerably reduce the computational effort. However, for the combined functional, a convergence result can be proved. Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 26 (27)

56 Thank you very much for your attention! Direct and Inverse Elastic Scattering Problems for Diffraction Gratings PICOF 12, April 4, 2012 Page 27 (27)