Universität Karlsruhe (TH) Department of Mathematics Research University · founded 1825 A fast solver for the periodic Lippmann-Schwinger equation for smooth and piecewise constant contrasts Kai Sandfort sandfort@math.uni-karlsruhe.de Conference on A.I.P . 2009/07/24 Research Training Group 1294 Universit¨ at Karlsruhe (TH) supported by Deutsche Department of Mathematics Forschungsgemeinschaft
Universität Karlsruhe (TH) Department of Mathematics Research University · founded 1825 Introduction Scattering from a bounded inhomogeneity Scattering from a periodic inhomogeneity Vainikko’s method Periodization of the problem Trigonometric collocation Extension for discontinuous contrasts Numerical results
Universität Karlsruhe (TH) Department of Mathematics Research University · founded 1825 Introduction Scattering from a bounded inhomogeneity Scattering from a periodic inhomogeneity Vainikko’s method Periodization of the problem Trigonometric collocation Extension for discontinuous contrasts Numerical results
Universität Karlsruhe (TH) Department of Mathematics Research University · founded 1825 Reminder: The acoustic Lippmann-Schwinger equation We restrict to the 2D case throughout the talk. � u ( x ) = u i ( x ) + κ 2 Φ( | x − y | ) q ( y ) u ( y ) dy , x ∈ Ω LS 0 Ω LS characterizes total field u for acoustic scattering of u i from a • bounded Ω with refractive index 1 + q ∈ L ∞ (Ω) ⇒ LS with u i ≡ 0 has only • LS is uniquely solvable in C (Ω) ⇐ trivial solution (Fredholm alternative) unique extension to R 2 \ Ω by RHS of LS yields total field in R 2 • in R 2 \ Ω , u is smooth (strict ellipticity), even analytic (analyticity • inherited from Φ ) 1 / 22
Universität Karlsruhe (TH) Department of Mathematics Research University · founded 1825 Reminder: The acoustic Lippmann-Schwinger equation We restrict to the 2D case throughout the talk. � u ( x ) = u i ( x ) + κ 2 Φ( | x − y | ) q ( y ) u ( y ) dy , x ∈ Ω LS 0 Ω LS characterizes total field u for acoustic scattering of u i from a • bounded Ω with refractive index 1 + q ∈ L ∞ (Ω) ⇒ LS with u i ≡ 0 has only • LS is uniquely solvable in C (Ω) ⇐ trivial solution (Fredholm alternative) unique extension to R 2 \ Ω by RHS of LS yields total field in R 2 • in R 2 \ Ω , u is smooth (strict ellipticity), even analytic (analyticity • inherited from Φ ) 1 / 22
Universität Karlsruhe (TH) Department of Mathematics Research University · founded 1825 Reminder: The acoustic Lippmann-Schwinger equation We restrict to the 2D case throughout the talk. � u ( x ) = u i ( x ) + κ 2 Φ( | x − y | ) q ( y ) u ( y ) dy , x ∈ Ω LS 0 Ω LS characterizes total field u for acoustic scattering of u i from a • bounded Ω with refractive index 1 + q ∈ L ∞ (Ω) ⇒ LS with u i ≡ 0 has only • LS is uniquely solvable in C (Ω) ⇐ trivial solution (Fredholm alternative) unique extension to R 2 \ Ω by RHS of LS yields total field in R 2 • in R 2 \ Ω , u is smooth (strict ellipticity), even analytic (analyticity • inherited from Φ ) 1 / 22
Universität Karlsruhe (TH) Department of Mathematics Research University · founded 1825 Reminder: The acoustic Lippmann-Schwinger equation We restrict to the 2D case throughout the talk. � u ( x ) = u i ( x ) + κ 2 Φ( | x − y | ) q ( y ) u ( y ) dy , x ∈ Ω LS 0 Ω LS characterizes total field u for acoustic scattering of u i from a • bounded Ω with refractive index 1 + q ∈ L ∞ (Ω) ⇒ LS with u i ≡ 0 has only • LS is uniquely solvable in C (Ω) ⇐ trivial solution (Fredholm alternative) unique extension to R 2 \ Ω by RHS of LS yields total field in R 2 • in R 2 \ Ω , u is smooth (strict ellipticity), even analytic (analyticity • inherited from Φ ) 1 / 22
Universität Karlsruhe (TH) Department of Mathematics Research University · founded 1825 Motivation: Scattering from a periodic inhomogeneity Π R 2 \ e Ω , q = 0 R + ‘periodic’ ≡ ‘2 π -periodic in x 1 ’ � ∂ e Ω Ω periodic medium (Lipschitz) Π unit cell R ± semi-infinite rectangles in Π − π + π e Ω , q � = 0 above / below � Ω q periodic contrast with q ( x ) | x 1 = − π = q ( x ) | x 1 =+ π R − ( q sufficiently regular) 2 / 22
Universität Karlsruhe (TH) Department of Mathematics Research University · founded 1825 Motivation: Scattering from a periodic inhomogeneity Π R 2 \ e Ω , q = 0 R + ∂ e Ω Ω , q � = 0 − π + π e R − Scattering problem: Find a periodic u per : Π → C such that ∆ u per + κ 2 0 ( 1 + q ) u per = 0 , u per = u i per + u s in Π , per � per ( x ) = � 0 − | z | 2 � = 0 . u s z ∈ Z u ± z e i ( z x 1 ± β z x 2 ) κ 2 in R ± , β z = 3 / 22
Universität Karlsruhe (TH) Department of Mathematics Research University · founded 1825 Motivation: Scattering from a periodic inhomogeneity Π R 2 \ e Ω , q = 0 R + ∂ e Ω Ω , q � = 0 − π + π e R − . . . equivalent to the periodic Lippmann-Schwinger equation � u per ( x ) = u i per ( x ) + κ 2 G per ( x − y ) q ( y ) u per ( y ) dy , x ∈ Π . pLS 0 Ω Ω = � G per periodic fundamental sol.n for ∆ + κ 2 Ω ∩ Π , 0 id 3 / 22
Universität Karlsruhe (TH) Department of Mathematics Research University · founded 1825 Outline object of interest: periodic Lippmann-Schwinger equation pLS aim: its efficient numerical treatment tools: method by Gennadi Vainikko and my extension 4 / 22
Universität Karlsruhe (TH) Department of Mathematics Research University · founded 1825 Introduction Scattering from a bounded inhomogeneity Scattering from a periodic inhomogeneity Vainikko’s method Periodization of the problem Trigonometric collocation Extension for discontinuous contrasts Numerical results
Universität Karlsruhe (TH) Department of Mathematics Research University · founded 1825 Periodization of the problem Π R 2 \ e Ω , q = 0 C 2 r ∂ e Ω Ω − π + π e Ω , q � = 0 r C r 2r Choose r > 0 so that Ω ⊂ C r = { x ∈ Π : | x 2 | < r } . Consider restric- per , and q to C 2 r . Extend to R 2 as ( 2 π, 4 r ) -biperiodic tions of G per , u i functions. Denote by K ext , u i ext , and q ext , respectively. 5 / 22
Universität Karlsruhe (TH) Department of Mathematics Research University · founded 1825 Periodization of the problem Π R 2 \ e Ω , q = 0 C 2 r ∂ e Ω Ω − π + π e Ω , q � = 0 r C r 2r For v ext = q ext u ext and x ∈ C 2 r , we get ( 2 π, 4 r ) -biperiodic L.-S. eqn. � v ext ( x ) = ( q ext u i ext )( x ) + κ 2 0 q ext ( x ) K ext ( x − y ) v ext ( y ) dy . bpLS C 2 r 5 / 22
Universität Karlsruhe (TH) Department of Mathematics Research University · founded 1825 Fourier expansion of K ext w.r.t. trigon. basis { ϕ j } j ∈ Z 2 of L 2 ( C 2 r ) • It holds (∆ + κ 2 0 ) ϕ j = λ j ϕ j . Assume λ j � = 0 for all j ∈ Z 2 . • • By Green’s representation theorem, we obtain � 1 − ( − 1 ) j 2 e i β j 1 2 r � K ext ( j ) = − � c � � K ext ( j ) = O ( | j | − 2 ) , ⇒ λ j � c normalization constant. exploited: problem-specific periodicity in x 1 ! 6 / 22
Universität Karlsruhe (TH) Department of Mathematics Research University · founded 1825 Trigonometric collocation for smooth contrast Define � � j ∈ Z 2 : − N Z 2 2 < j k ≤ N N = 2 , k = 1 , 2 , � � ϕ j , j ∈ Z 2 T N = span . N Define interpolation projection Q N : H 2 per ( C 2 r ) → T N by j ∈ Z 2 ( Q N v per )( j ⊙ h N ) = v per ( j ⊙ h N ) , N , where h N = ( 2 π, 4 r ) / N and ⊙ componentwise multiplication. 7 / 22
Universität Karlsruhe (TH) Department of Mathematics Research University · founded 1825 Trigonometric collocation for smooth contrast For q ∈ H 2 per ( C 2 r ) , solve bpLS by collocation v N = Q N ( q ext u i ext ) + κ 2 0 Q N ( q ext K v N ) , bpLS-C where K : L 2 ( C 2 r ) → H 2 per ( C 2 r ) given by � ( K v per )( x ) = K ext ( x − y ) v per ( y ) dy . C 2 r c − 1 � Note: ( K ϕ j )( x ) = � K ext ( j ) ϕ j ( x ) by convolution theorem v N ( j ) , j ∈ Z 2 � N , are computed by fast Fourier transform (FFT) 8 / 22
Universität Karlsruhe (TH) Department of Mathematics Research University · founded 1825 Trigonometric collocation for smooth contrast For q ∈ H 2 per ( C 2 r ) , solve bpLS by collocation v N = Q N ( q ext u i ext ) + κ 2 0 Q N ( q ext K v N ) , bpLS-C where K : L 2 ( C 2 r ) → H 2 per ( C 2 r ) given by � ( K v per )( x ) = K ext ( x − y ) v per ( y ) dy . C 2 r ⇒ numerical integration avoided, cheap expressions instead ! = 8 / 22
Universität Karlsruhe (TH) Department of Mathematics Research University · founded 1825 Theorem 1: Assume q ∈ H 2 per (Π) and u i per ∈ H 2 per, loc (Π) . Let pLS with u i per ≡ 0 have only the trivial sol.n. Then, bpLS has a unique sol.n v ext ∈ H 2 per ( C 2 r ) , and the col- location eqn. bpLS-C has a unique sol.n v N ∈ T N for N ≥ N 0 , and � v N − v ext � λ ≤ c ′ N λ − 2 � v ext � 2 , 0 ≤ λ ≤ 2 . Here, �·� µ denotes the norm of H µ per ( C 2 r ) . 9 / 22
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