A fast solver for the periodic Lippmann-Schwinger equation for - - PowerPoint PPT Presentation

▶
a fast solver for the periodic lippmann schwinger equation
SMART_READER_LITE
LIVE PREVIEW

A fast solver for the periodic Lippmann-Schwinger equation for - - PowerPoint PPT Presentation

Oct 06, 2023 368 likes •725 views

Universitt 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

slide-1
SLIDE 1

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

slide-2
SLIDE 2

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

slide-3
SLIDE 3

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

slide-4
SLIDE 4

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) = ui(x) + κ2

  • Ω

Φ(|x − y|) q(y) u(y) dy, x ∈ Ω LS

  • LS characterizes total field u for acoustic scattering of ui from a

bounded Ω with refractive index 1 + q ∈ L∞(Ω)

  • LS is uniquely solvable in C(Ω) ⇐

⇒ LS with ui ≡ 0 has only trivial solution (Fredholm alternative)

  • unique extension to R2\Ω by RHS of LS yields total field in R2
  • in R2\Ω, u is smooth (strict ellipticity), even analytic (analyticity

inherited from Φ)

1 / 22

slide-5
SLIDE 5

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) = ui(x) + κ2

  • Ω

Φ(|x − y|) q(y) u(y) dy, x ∈ Ω LS

  • LS characterizes total field u for acoustic scattering of ui from a

bounded Ω with refractive index 1 + q ∈ L∞(Ω)

  • LS is uniquely solvable in C(Ω) ⇐

⇒ LS with ui ≡ 0 has only trivial solution (Fredholm alternative)

  • unique extension to R2\Ω by RHS of LS yields total field in R2
  • in R2\Ω, u is smooth (strict ellipticity), even analytic (analyticity

inherited from Φ)

1 / 22

slide-6
SLIDE 6

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) = ui(x) + κ2

  • Ω

Φ(|x − y|) q(y) u(y) dy, x ∈ Ω LS

  • LS characterizes total field u for acoustic scattering of ui from a

bounded Ω with refractive index 1 + q ∈ L∞(Ω)

  • LS is uniquely solvable in C(Ω) ⇐

⇒ LS with ui ≡ 0 has only trivial solution (Fredholm alternative)

  • unique extension to R2\Ω by RHS of LS yields total field in R2
  • in R2\Ω, u is smooth (strict ellipticity), even analytic (analyticity

inherited from Φ)

1 / 22

slide-7
SLIDE 7

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) = ui(x) + κ2

  • Ω

Φ(|x − y|) q(y) u(y) dy, x ∈ Ω LS

  • LS characterizes total field u for acoustic scattering of ui from a

bounded Ω with refractive index 1 + q ∈ L∞(Ω)

  • LS is uniquely solvable in C(Ω) ⇐

⇒ LS with ui ≡ 0 has only trivial solution (Fredholm alternative)

  • unique extension to R2\Ω by RHS of LS yields total field in R2
  • in R2\Ω, u is smooth (strict ellipticity), even analytic (analyticity

inherited from Φ)

1 / 22

slide-8
SLIDE 8

Universität Karlsruhe (TH) Department of Mathematics

Research University · founded 1825

Motivation: Scattering from a periodic inhomogeneity

Π e Ω, q = 0 −π +π ∂e Ω R2\e Ω, q = 0 R+ R−

‘periodic’ ≡ ‘2π-periodic in x1’

  • Ω

periodic medium (Lipschitz) Π unit cell R± semi-infinite rectangles in Π above / below Ω q periodic contrast with q(x)|x1=−π = q(x)|x1=+π (q sufficiently regular)

2 / 22

slide-9
SLIDE 9

Universität Karlsruhe (TH) Department of Mathematics

Research University · founded 1825

Motivation: Scattering from a periodic inhomogeneity

Π e Ω, q = 0 −π +π ∂e Ω R2\e Ω, q = 0 R+ R−

Scattering problem: Find a periodic uper : Π → C such that ∆uper + κ2

0 (1 + q) uper = 0, uper = ui per + us per

in Π, us

per(x) = z∈Z u± z ei (z x1±βz x2)

in R±, βz =

  • κ2

0 − |z|2 = 0.

3 / 22

slide-10
SLIDE 10

Universität Karlsruhe (TH) Department of Mathematics

Research University · founded 1825

Motivation: Scattering from a periodic inhomogeneity

Π e Ω, q = 0 −π +π ∂e Ω R2\e Ω, q = 0 R+ R−

. . . equivalent to the periodic Lippmann-Schwinger equation uper(x) = ui

per(x) + κ2

  • Ω

Gper(x − y) q(y) uper(y) dy, x ∈ Π. pLS Ω = Ω ∩ Π, Gper periodic fundamental sol.n for ∆ + κ2

0 id

3 / 22

slide-11
SLIDE 11

Universität Karlsruhe (TH) Department of Mathematics

Research University · founded 1825

Outline

  • bject of interest: periodic Lippmann-Schwinger equation pLS

aim: its efficient numerical treatment tools: method by Gennadi Vainikko and my extension

4 / 22

slide-12
SLIDE 12

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

slide-13
SLIDE 13

Universität Karlsruhe (TH) Department of Mathematics

Research University · founded 1825

Periodization of the problem

2r

Π e Ω, q = 0 Ω −π +π ∂e Ω R2\e Ω, q = 0 Cr C2r r

Choose r > 0 so that Ω ⊂ Cr = {x ∈ Π : |x2| < r}. Consider restric- tions of Gper, ui

per, and q to C2r. Extend to R2 as (2π, 4r)-biperiodic

  • functions. Denote by Kext, ui

ext, and qext, respectively.

5 / 22

slide-14
SLIDE 14

Universität Karlsruhe (TH) Department of Mathematics

Research University · founded 1825

Periodization of the problem

2r

Π e Ω, q = 0 Ω −π +π ∂e Ω R2\e Ω, q = 0 Cr C2r r

For vext = qext uext and x ∈ C2r, we get (2π, 4r)-biperiodic L.-S. eqn. vext(x) = (qext ui

ext)(x) + κ2 0 qext(x)

  • C2r

Kext(x − y) vext(y) dy. bpLS

5 / 22

slide-15
SLIDE 15

Universität Karlsruhe (TH) Department of Mathematics

Research University · founded 1825

  • Fourier expansion of Kext w.r.t. trigon. basis {ϕj}j∈Z2 of L2(C2r)
  • It holds (∆ + κ2

0) ϕj = λj ϕj. Assume λj = 0 for all j ∈ Z2.

  • By Green’s representation theorem, we obtain
  • Kext(j) = −

c λj

  • 1 − (−1)j2ei βj1 2r

⇒

  • Kext(j) = O(|j|−2),
  • c normalization constant.

exploited: problem-specific periodicity in x1!

6 / 22

slide-16
SLIDE 16

Universität Karlsruhe (TH) Department of Mathematics

Research University · founded 1825

Trigonometric collocation for smooth contrast Define Z2

N =

  • j ∈ Z2 : − N

2 < jk ≤ N 2 , k = 1, 2

  • ,

TN = span

  • ϕj, j ∈ Z2

N

  • .

Define interpolation projection QN : H2

per(C2r) → TN by

(QN vper)(j ⊙ hN) = vper(j ⊙ hN), j ∈ Z2

N,

where hN = (2π, 4r)/N and ⊙ componentwise multiplication.

7 / 22

slide-17
SLIDE 17

Universität Karlsruhe (TH) Department of Mathematics

Research University · founded 1825

Trigonometric collocation for smooth contrast For q ∈ H2

per(C2r), solve bpLS by collocation

vN = QN(qext ui

ext) + κ2 0 QN(qext K vN),

bpLS-C where K : L2(C2r) → H2

per(C2r) given by

(K vper)(x) =

  • C2r

Kext(x − y) vper(y) dy. Note: (K ϕj)(x) = c−1 Kext(j) ϕj(x) by convolution theorem

  • vN(j), j ∈ Z2

N, are computed by fast Fourier transform (FFT)

8 / 22

slide-18
SLIDE 18

Universität Karlsruhe (TH) Department of Mathematics

Research University · founded 1825

Trigonometric collocation for smooth contrast For q ∈ H2

per(C2r), solve bpLS by collocation

vN = QN(qext ui

ext) + κ2 0 QN(qext K vN),

bpLS-C where K : L2(C2r) → H2

per(C2r) given by

(K vper)(x) =

  • C2r

Kext(x − y) vper(y) dy. = ⇒ numerical integration avoided, cheap expressions instead !

8 / 22

slide-19
SLIDE 19

Universität Karlsruhe (TH) Department of Mathematics

Research University · founded 1825

Theorem 1: Assume q ∈ H2

per(Π) and ui per ∈ H2 per, loc(Π). Let

pLS with ui

per ≡ 0 have only the trivial sol.n.

Then, bpLS has a unique sol.n vext ∈ H2

per(C2r), and the col-

location eqn. bpLS-C has a unique sol.n vN ∈ TN for N ≥ N0, and vN − vextλ ≤ c′ Nλ−2 vext2, 0 ≤ λ ≤ 2. Here, ·µ denotes the norm of Hµ

per(C2r).

9 / 22

slide-20
SLIDE 20

Universität Karlsruhe (TH) Department of Mathematics

Research University · founded 1825

Extension for discontinuous contrasts Assume q ∈ L2(Π). Instead of vext(x) = (qext ui

ext)(x) + κ2 0 qext(x)

  • C2r

Kext(x − y) vext(y) dy bpLS for vext = qext uext in C2r, consider uper(x) = ui

per(x) + κ2

  • Ω

Gper(x − y) q(y) uper(y) dy pLS for uper in Ω (total field). Recall Kext = Gper on C2r ⊃ (2·Ω) ∩ Π.

10/ 22

slide-21
SLIDE 21

Universität Karlsruhe (TH) Department of Mathematics

Research University · founded 1825

Extension for discontinuous contrasts uper(x) = ui

per(x) + κ2

  • Ω

Kext(x − y) q(y) uper(y) dy, x ∈ Ω pLS

  • Define P u =
  • u

in Ω in CR\Ω and R u = u|Ω, then new integral

  • perator

K : L2(Ω) → H2

per(Ω) by

K = R ◦ K ◦ P.

  • Use extension operator Eper : H2

per(Ω) → H2 per(C2r) to setup

collocation uN = QN Eper(ui

per) + κ2 0 QN Eper

K(q R uN). E-pLS-C Note: The extension performed by Eper is artificial ! Implicit purpose: Obtain good approximant for argument as restriction to Ω of a function in TN !

11/ 22

slide-22
SLIDE 22

Universität Karlsruhe (TH) Department of Mathematics

Research University · founded 1825

Illustration of the extension by Eper : H2

per(Ω) → H2 per(C2r)

(left) data points in Ω, (right) extension with data and guiding points

12/ 22

slide-23
SLIDE 23

Universität Karlsruhe (TH) Department of Mathematics

Research University · founded 1825

Theorem 2: Assume q ∈ L2(Π) and ui

per ∈ H2 per, loc(Π). Let pLS

with ui

per ≡ 0 have only the trivial sol.n.

Then, pLS has a unique sol.n uper ∈ H2

per(Ω), and the collo-

cation eqn. E-pLS-C has a unique sol.n uN ∈ TN for N ≥ N0, and uN − Eper uperλ ≤ c′ Nλ−2 Eper uper2, 0 ≤ λ ≤ 2. Cost: extremely efficient eval. of K applied to vN ∈ TN not applicable !

13/ 22

slide-24
SLIDE 24

Universität Karlsruhe (TH) Department of Mathematics

Research University · founded 1825

For piecewise constant q = L

i=1 qi idCi:

good performance maintained up to some N0 ≥ N by precalculation

  • f “generalized Fourier coefficients”
  • K (i)

ext(x, j) =

  • e

Ci

Gper(x − y) ϕj(y) dy, j ∈ Z2

N0,

  • n grid G = Z2

N0 ⊙ hN0 ∋ x, where for specific contrast Ci = k∈Ii

Ck.

14/ 22

slide-25
SLIDE 25

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

slide-26
SLIDE 26

Universität Karlsruhe (TH) Department of Mathematics

Research University · founded 1825

Projection errors for piecewise constant q

(left) piecewise constant q, (right) error in the orthogonal (PN) and in the interpolation (QN) projection of q

15/ 22

slide-27
SLIDE 27

Universität Karlsruhe (TH) Department of Mathematics

Research University · founded 1825

Projection errors for smooth q

(left) smooth q, (right) error in the orthogonal (PN) and in the interpolation (QN) projection of q

16/ 22

slide-28
SLIDE 28

Universität Karlsruhe (TH) Department of Mathematics

Research University · founded 1825

Error in the solution vN to the collocation bpLS-C

(left) for piecewise constant q, (right) for smooth q w.r.t. v128 (Theorem 1 applies!)

17/ 22

slide-29
SLIDE 29

Universität Karlsruhe (TH) Department of Mathematics

Research University · founded 1825

Projection errors for smoothly extended q

(left) smoothly extended q, (right) error in the orthogonal (PN) and in the interpolation (QN) projection of the extension

18/ 22

slide-30
SLIDE 30

Universität Karlsruhe (TH) Department of Mathematics

Research University · founded 1825

Error in the solution uN to the collocation E-pLS-C

error in uN w.r.t. u64 (Theorem 2 applies!)

19/ 22

slide-31
SLIDE 31

Universität Karlsruhe (TH) Department of Mathematics

Research University · founded 1825

Evolution of vN (for exemplary ui

per)

v16 v32 v64

20/ 22

slide-32
SLIDE 32

Universität Karlsruhe (TH) Department of Mathematics

Research University · founded 1825

Evolution of uN cropped to Ω (for same ui

per)

u16 u32 u64

21/ 22

slide-33
SLIDE 33

Universität Karlsruhe (TH) Department of Mathematics

Research University · founded 1825

Application of the Lippmann-Schwinger solvers: Field simulation and numerical validation of the Factorization Method for inverse scattering from periodic inhomogeneous media (topic of my PhD thesis) For field and reconstruction plots, see the gallery at

http://www.mathematik.uni-karlsruhe.de/grk1294/˜sandfort

Reference: Saranen, J. and Vainikko, G., Periodic Integral and Pseudodifferential Equa- tions with Numerical Approximation, Springer-Verlag, 2002

22/ 22

slide-34
SLIDE 34

Universität Karlsruhe (TH) Department of Mathematics

Research University · founded 1825

Thank you!

For field and reconstruction plots, see the gallery at

http://www.mathematik.uni-karlsruhe.de/grk1294/˜sandfort

Reference: Saranen, J. and Vainikko, G., Periodic Integral and Pseudodifferential Equa- tions with Numerical Approximation, Springer-Verlag, 2002

22/ 22