Electrostatic Imaging via Conformal Mapping. R. Kress. joint work with I. Akduman, Istanbul and H. Haddar, Paris

Transcription

1 Electrostatic Imaging via Conformal Mapping R. Kress Göttingen joint work with I. Akduman, Istanbul and H. Haddar, Paris

2 Or: A new solution method for inverse boundary value problems for the Laplace equation Determine shape Γ 0 of a perfectly conducting or nonconducting inclusion or inclusion with different conductivity from overdetermined Cauchy data on Γ 1 Γ Γ 0 1 D Applications in the field of nondestructive testing via electrostatic imaging or thermal imaging, e.g., impedance tomography

3 Here: Perfectly conducting inclusion, i.e., inverse Dirichlet problem Extensions to other boundary conditions are in preparation 1. Brief survey on other methods (see 2. ed. of Linear Integral Equations) 2. Description of new method 3. Some numerical examples

4 The inverse problem u = 0 in D Γ Γ 0 1 D ν u = 0 on Γ 0 u = f on Γ 1 Inverse Problem: Given g = u ν on Γ 1 (and f), find boundary Γ 0 Uniqueness!!!

5 Uniqueness u = f u ν = g Γ 0 Γ0 Γ 1 Γ 0 u u = ũ Γ 0 ũ In shaded domain: u = 0 On boundary: u = 0 Schiffer 1960

6 For inverse boundary value problems, in general, wrong question to ask. Would need to characterize Cauchy data on Γ 1 for which the corresponding solution vanishes on a closed surface Γ 0 (or curve) within Γ 1. Existence??? u = f u ν = g Γ 1 Γ 0 u = 0 Main Task: Assuming correct data or perturbed correct data, design method for approximate and stable solution

7 Decomposition methods 1. Determine u from Cauchy data on Γ 1, for example via u(x) = u 0 (x)+ G(x, y)ϕ(y) ds(y) and integral equation of the first kind Kϕ = g u 0 / ν Γ u = f u ν = g Γ 1 Γ Γ 0 u = 0 2. Find Γ 0 as location of the zeros of u (in a least squares sense) Kirsch, K. (1987) Pros: Conceptionally simple No need for forward solver Contras: No high accuracy reconstructions Gap between theory and numerics

8 Newton type iterations 1. Interpret inverse problem as operator equation F (Γ 0 ) = g where F : Γ 0 u ν Γ1 Γ 0 Γ 1 u = 0 u = f u ν = g 2. Solve by regularized Newton iterations Pros: Conceptionally simple Accurate reconstructions Contras: Need forward solver Need good a priori information Convergence analysis difficult Hohage (1999), Potthast (2001)

9 Hybrid of decomposition and Newton methods 1. Determine u via u(x) = u 0 (x)+ G(x, y)ϕ(y) ds(y) and integral equation Kϕ = g 2. Update Γ Γ h = {x + h(x) : x Γ} via Newton step Γ u = f u ν = g Γ 1 Γ Γ 0 u = 0 u + grad u h = 0 Does not need a forward solver! Chapko, K. (2003) Inverse obstacle scattering: K. (2003), K., Rundell (2001), Potthast (2001)

10 Characterize unknown domain via spectral data of the Dirichletto-Neumann operator on Γ 1 A : u u ν Kirsch s factorization method Γ 0 Γ 1 u = 0 u = f u ν = g Hähner (1999), K. (1999), Kühn (2001) Brühl, Hanke (2000) Pros: Elegant mathematics Simple implementation No a priori information needed Contras: Need a lot of data No sharp boundaries ( =?) Very sensitive to noise

11 1. Solve nonlocal nonlinear ordinary differential equation for boundary values of a conformal mapping (by successive iterations) Our method Γ 1 u = 0 Γ 0 u = f u ν = g 2. Solve Cauchy problem for a holomorphic function in an annulus Pros: Conceptionally simple Satisfactory reconstructions Domain for Cauchy problem is known Contras: Restricted to two dimensions and to Laplace equation

12 Γ 1 D Γ 0 u = 0 u = f u ν = g = Ψ(1) B ρ C 0 C 1 u in D v = u Ψ in B D Ψ B Γ 1 = {γ(s) : s [0, L]} C 1 = {e it : t [0, 2π]} ϕ : arc length on C 1 arc length on Γ 1 Ψ(e it ) = γ(ϕ(t)) knowing ϕ equivalent to knowing Ψ C1

13 Γ 1 D Γ 0 u = 0 u = f u ν = g Ψ, ϕ B ρ C 0 C 1 u, ũ and v = u Ψ, ṽ = ũ Ψ conjugate harmonics ṽ(t) = ũ(ϕ(t)) ṽ t = ũ s dϕ dt Cauchy Riemann equations v ν = u ν dϕ dt = A(f ϕ) g ϕ dϕ dt A = Dirichlet-to-Neumann map for B

14 B u u = f, ν = g on Γ 1 v C 0 ν ds = v C 1 ν ds = g ds Γ 1 { ln x v C 0 ν v 1 } ds = x C 1 v ln ρ C 0 ν ds = v ds C 1 ρ = exp ( ) 0 f ϕ dt Γ 1 g ds 2π ρ { ln x v ν v 1 } ds x C 0 C 1

15 dϕ dt = A(f ϕ) g ϕ A = Dirichlet-to-Neumann map for B L = 2π 0 A(f ϕ) g ϕ dt dϕ dt = A(f ϕ) g ϕ + L 2π 1 2π 2π 0 A(f ϕ) g ϕ dt Makes sure that throughout the iteration ϕ(2π) = L

16 ρ = exp ( 2π 0 f ϕ dt Γ 1 g ds ) dϕ dt = A(f ϕ) g ϕ + L 2π 1 2π 2π 0 A(f ϕ) g ϕ dt

17 ρ n = exp ( 2π 0 f ϕ n dt Γ 1 g ds ) dϕ n+1 dt = A n(f ϕ n ) g ϕ n + L 2π 1 2π 2π 0 A n (f ϕ n ) g ϕ n dt ϕ 0 (t) = L 2π t, correct if D annulus Theorem Under appropriate assumptions on D and f the successive approximations converge in H 1 [0, 2π].

18 dϕ n+1 dt = A n(f ϕ n ) g ϕ n + L 2π 1 2π 2π 0 A n (f ϕ n ) g ϕ n dt Numerical implementation: ϕ n (t) L 2π t + α n,0 + N [α n,k cos kt + β n,k sin kt] k=1 L n := 1 2π 2π 0 A n (f ϕ n ) g ϕ n dt L 2π (g ϕ n )(t j ) {ϕ n+1(t j ) + L n } A n (f ϕ n )(t j ) = 0, j = 1,..., J Solve by least squares to update coefficients Use trigonometric interpolation for f ϕ and g ϕ

19 Now, we know the radius ρ and Ψ = γ ϕ on the outer circle C 1, i.e., Fourier series γ(ϕ(t)) = b k e ikt, 0 t 2π k= Solve Cauchy problem for Ψ in B via Laurent expansion Ψ(z) = b k z k, ρ z 1 k= and obtain unknown boundary by { M } Γ 0 = Ψ(C 0 ) ρ k b k e ikt, 0 t 2π k= M B ρ Need to truncate because of exponential ill-posedness. C 0 C 1

20 1. Solve nonlocal nonlinear ordinary differential equation for boundary values of a conformal mapping (by successive iterations) Our method Γ 1 u = 0 Γ 0 u = f u ν = g 2. Solve Cauchy problem for a holomorphic function in an annulus

21 For numerical examples Γ 1 unit circle f(t) = 6 + exp(cos t) + exp(sin t) degree of trigonometric polynomials: N = cutoff in Laurent expansion : M = number of collocation points: J = 32 Between 6 to 10 iterations

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29 Remarks: Idemen, Akduman (1988) f = f 0 = const dϕ dt = 1 ln ρ f 0 g ϕ Nonconstant f required for extensions to other boundary conditions. However, no flux Γ 0 g ds = 0 has to be observed Inverse Problems 18, (2002)

30 Cracks!!! u = f u ν = g B ρ C 0 C 1 Γ 1 D Γ 0 u = 0 D Ψ B

31 Cracks!!! u = f u ν = g B ρ C 0 C 1 Γ 1 D Γ 0 D Ψ B Try Γ 0 Ψ( C 0 ) with radius ρ(1 + λ) for C 0

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33 Open problems: Other boundary conditions Incomplete data Satisfactory analysis for cracks Other regularizations in second step