Reconstructing inclusions from Electrostatic Data

Transcription

1 Reconstructing inclusions from Electrostatic Data Isaac Harris Texas A&M University, Department of Mathematics College Station, Texas Joint work with: W. Rundell Purdue University - November 2017 Reconstructing inclusions from Electrostatic Data 1 / 35

2 Outline Introduction Formulation of the Problem Solution of the Inverse Problem Solution to the Inverse Shape Problem Solution to the Inverse Impedance Problem Current/Future Research Reconstructing inclusions from Electrostatic Data 2 / 35

3 What is an Inverse Problem? Input = Physical Model = Output Input: Boundary Data Physical Model: Partial Differential Equation Output: Values of the Solution Forward Problem: Input + Physical Model = Output Inverse Problem: Output + Input = Physical Model Reconstructing inclusions from Electrostatic Data 3 / 35

4 An Inverse Scattering Problem The Physical Model: Acoustic or Electromagnetic wave scattering. Inverse Problem Reconstruct the support of the defective region in a known material from the measured scattered wave. Reconstructing inclusions from Electrostatic Data 4 / 35

5 Some Important Questions Assume your inverse problem is given by A(f ) = g Well-Posedness: Solvability, Injectivity and Stability Does A 1 exists as a continuous mapping? (Usually Not!!!) Regularization Techniques Approximate (point-wise) A 1 by a continuous mapping Inversion Algoritms Develop a stable and computable method to solve for f Reconstructing inclusions from Electrostatic Data 5 / 35

6 Formulation of the Problem I. Harris and W. Rundell A direct method for reconstructing inclusions and boundary conditions from electrostatic data Submitted (arxiv: ) Reconstructing inclusions from Electrostatic Data 6 / 35

7 Electrical Impedance Tomography (EIT) EIT is a non-destructive imaging method in which the physical parameters are recovered from surface electrode measurements. Figure: Picture taken from Reconstructing inclusions from Electrostatic Data 7 / 35

8 The Inverse Electrostatic Problem Inverse Problem Recover the inclusion Ω and the boundary condition B from the knowledge of the Dirichlet to Neumann operators Λf = ν u D and Λ 0 f = ν u 0 D. Reconstructing inclusions from Electrostatic Data 8 / 35

9 In this talk we focus on B(u 0 ) = ν u 0 + γ(x)u 0 Assumptions: 1. D R d be a bounded open region with D is class C 2 2. Ω D such that dist( D, Ω) > 0 with Ω is class C 2 3. The function γ(x) L + ( Ω) is the impedance L + ( Ω) := { γ L ( Ω) : γ(x) 0 } for all x Ω The results in this talk still hold for the case when u is replaced by A(x) u + c(x)u with symmetric A(x) C 1 (D, C d d ), where the Re(A) is positive definite and Im(A) is non-positive. As well as c(x) L (D) such that Re(c) is non-negative. Reconstructing inclusions from Electrostatic Data 9 / 35

10 Some Previous Works Uniqueness for the inverse problem for 2 pairs of input/output. V. Bacchelli Uniqueness for the determination of unknown boundary and impedance with the homogeneous Robin condition, Inverse Problems 25, (2009) An iterative method for one inclusion and γ(x) small. F. Ben Hassen, Y. Boukari and H. Haddar Inverse impedance boundary problem via the conformal mapping method: the case of small impedances, Revue ARIMA, 13 (2010), pp Newton s method for recovering the inclusion and γ(x). W. Rundell Recovering an obstacle and its impedance from Cauchy data, Inverse Problems 24 (2008), Reconstructing inclusions from Electrostatic Data 10 / 35

11 Iterative Reconstruction Methods Iterative/Optimization Methods: Solve nonlinear model using iterative scheme to reconstruct the material parameters, which requires a priori information, can be computationally expensive. Now assume that we have f j on D. Λ 0 f j : ( Ω, γ) ν u (j) 0 for j = 1, 2 Then you would solve the system of non-linear (exponentially) ill-posed equation using a iterative scheme. Reconstructing inclusions from Electrostatic Data 11 / 35

12 Restrictions for Iterative Methods Good initial estimates are usually needed to insure convergence. This requires a priori information that may not be known: The number of inclusions. Knowledge of the type of Boundary Condition. For the impedance condition a initial estimate is needed. QUESTION: Can one develop an inversion algorithm that requires little to no a priori information? A lack of information cannot be remedied by any mathematical trickery -C. Lanczos Reconstructing inclusions from Electrostatic Data 12 / 35

13 Alternative Reconstruction Method Direct (Qualitative) Methods: Using nonlinear model to reconstruct limited information with no a priori information, Usually by solving many linear integral equations. Our Reconstruction Method We split our inverse problem into 2 parts: 1. Reconstruct Ω via a Sampling Method 2. Once Ω is known reconstruct (non-negative) γ(x) Reconstructing inclusions from Electrostatic Data 13 / 35

14 Solution to the Inverse Shape Problem Reconstructing inclusions from Electrostatic Data 14 / 35

15 The Green s Function & Range Test Let G(x, z) be the solution to the problem G(, z) = δ( z) in D and G(, z) = 0 on D. Now definite operator such that Th = ν w D where w = 0 in D \ Ω with w D = 0 and ν w Ω = h Theorem (I.Harris and W.Rundell) ν G(, z) D Range(T ) if and only if z Ω. Where the operator T : H 1/2 ( Ω) H 1/2 ( D). Main Idea: Using that G(x, z) as x z and the Range of T consists of smooth functions gives the result. Reconstructing inclusions from Electrostatic Data 15 / 35

16 Theorem (I.Harris and W.Rundell) The difference of the DtN operators (Λ Λ 0 ) : H 1/2 ( D) H 1/2 ( D) is compact, injective with a dense range (under some assumptions). Moreover, Range(Λ Λ 0 ) Range(T ). Techniques used in the proof 1. Compactness follows from elliptic regularity 2. Injectivity is a consequence on unique continuation 3. Green s 1st identity gives that (Λ Λ 0 ) is symmetric 4. Derive the factorization (Λ Λ 0 ) = TH Reconstructing inclusions from Electrostatic Data 16 / 35

17 Since (Λ Λ 0 ) has a dense range that there is a sequence such that (Λ Λ 0 )f z,ε ν G(, z) H 0 as ε 0. 1/2 ( D) Assume that f z,ε H 1/2 ( D) is bounded as ε 0. Since f z,ε f z,0 we have that (Λ Λ 0 )f z,ε (Λ Λ 0 )f z,0 as ε 0. Therefore, we have that ν G(, z) D Range(Λ Λ 0 ) Range(T ) which is a contradiction provided that z / Ω. Reconstructing inclusions from Electrostatic Data 17 / 35

18 Determining Ω via Sampling Theorem (I.Harris and W.Rundell) If the sequence { f z,ε }ε>0 H1/2 ( D) is such that (Λ Λ 0 )f z,ε ν G(, z) H 0 as ε 0 1/2 ( D) then f z,ε H 1/2 ( D) as ε 0 for all z / Ω. Reconstructing the boundary Ω via Sampling: 1. Choose a grid of points in D 2. For each grid point solve (Λ Λ 0 )f z,ε = ν G(, z) D 3. Compute the indicator W (z) = f z,ε 1 H 1/2 ( D) 4. Then Ω δ = { z D : W (z) = δ 1 } approximates Ω Reconstructing inclusions from Electrostatic Data 18 / 35

19 Reconstruction of an Ellipse Ellipse: x(θ) = ( 0.5 cos(θ), 0.3 sin(θ) ) 1 Contour Plot Reconstruction 1 Level Curve Reconstruction W(z)=0.005 Reconstruction Inclusion Unit Circle Figure: Reconstruction via the Sampling Method with impedance parameter γ ( x(θ) ) = 2 sin 4 (θ) with cut-off parameter Reconstructing inclusions from Electrostatic Data 19 / 35

20 Reconstruction of an Cardioid Cardioid: x(θ) = cos(θ) sin(2θ) cos(θ) ( cos(θ), sin(θ) ) Contour Plot Reconstruction Level Curve Reconstruction W(z)=0.005 Reconstruction Inclusion Unit Circle Figure: Reconstruction via the Sampling Method with impedance parameter γ ( x(θ) ) = sin3 (θ) with cut-off parameter Reconstructing inclusions from Electrostatic Data 20 / 35

21 Solution to the Inverse Impedance Problem Reconstructing inclusions from Electrostatic Data 21 / 35

22 Theorem (I.Harris and W.Rundell) The DtN mapping Λ 0 : H 1/2 ( D) H 1/2 ( D) uniquely determines the impedance parameter γ(x) L + ( Ω). Techniques used in the proof 1. Assume γ 1 and γ 2 give that same DtN mapping 2. Unique continuation implies that (γ 1 γ 2 )u Ω 0 = 0 for all f H 1/2 ( D) 3. Then prove that the set { } u Ω 0 : for all f H 1/2 ( D) is a dense in L 2 ( Ω) Reconstructing inclusions from Electrostatic Data 22 / 35

23 Data Completion for Recovering γ(x) We wish to derive a non-iterative method for recovering the impedance parameter using Data Completion, ( f, Λ0 f ) D ( u f 0, ν u f 0) Ω. Using the Boundary Condition on Ω we have that (Direct Solution) γ(x n ) = νu f 0 (x n) u f 0 (x n) for n = 1,, N with x n = x(θ n ) Ω. Reconstructing inclusions from Electrostatic Data 23 / 35

24 Data Completion Algorithm I Let D = Γ m (measurements) and Ω = Γ i (impedance) and define the boundary integral operators (D m ϕ)(x) = 2 ϕ(y) ν(y) Φ(x, y) ds y for x R d \ Γ m Γ m and ( D i ψ)(x) = 2 [ ψ(y) ν(y) Φ(x, y) + x d 2] ds y for x R d \ Γ i Γ i where Φ(x, y) is given by Φ(x, y) = 1 2π ln x y in R2 and Φ(x, y) = 1 1 4π x y in R 3. Reconstructing inclusions from Electrostatic Data 24 / 35

25 Data Completion Algorithm II We make the ansatz that u 0 (x) = (D m ϕ)(x) + ( D i ψ)(x) for x D \ Ω. Now assume that we have ( f, Λ 0 f ) on Γ m. Therefore (I K mm ) ϕ K im ψ = f on Γ m K mi ϕ + (I + K ii ) ψ = u Γi 0 on Γ i. Here the double layer operators are given by K pq ϕ = (D p ϕ)(x) for Kpq ψ = ( D p ψ)(x) for x Γ q with the index p,q =m,i. Reconstructing inclusions from Electrostatic Data 25 / 35

26 Data Completion Algorithm III Define the operator associated with the system of BIE [ I K A = mm K ] im K mi I + K ii Theorem (I.Harris and W.Rundell) The mapping A is a bounded linear operator from H 1/2 (Γ m ) H 1/2 (Γ i ) to itself and has a bounded inverse. Techniques used in the proof 1. Prove that A is Fredholm of index zero 2. Prove injectivity using the jump-relations Reconstructing inclusions from Electrostatic Data 26 / 35

27 Data Completion Algorithm IV Using the fact the we know the normal derivative on we have that [ I K Λ 0 f = [T mm Tim ] mm K mi K ] 1 [ ] im f I + K ii u Γi for x Γ m 0 where the hyper singular operators are given by T mm ϕ = ν(x) (D m ϕ)(x) and Tim ψ = ν(x) ( D i ψ)(x) for x Γ m. Notice: This reduces to solving a linear system of equation that is independent of the boundary condition being recovered. Reconstructing inclusions from Electrostatic Data 27 / 35

28 Reconstruction of γ ( x(θ) ) = 2 sin 4 (θ) Figure: Left: Reconstructed boundary of the inclusion. Right: Reconstructed impedance parameterγ ( x(θ) ) = 2 sin 4 (θ). Reconstructing inclusions from Electrostatic Data 28 / 35

29 Current/Future Research Reconstructing inclusions from Electrostatic Data 29 / 35

30 The IBVP for Generalized Impedance condition Reconstruct the support of the inclusion Ω and the boundary condition B from the knowledge of the voltage to current maps Λf = ν u D and Λ 0 f = ν u 0 D. Where the Generalized Impedance Boundary Condition (GIBC) B(u 0 ) = ν u 0 d ds η d ds u 0 + γu 0. I. Harris Detecting inclusions with a generalized impedance condition from electrostatic data via sampling Submitted (arxiv: ) Reconstructing inclusions from Electrostatic Data 30 / 35

31 Some Current Work The direct problem is well-posed for η and γ L ( Ω) provided that the real-parts of the coefficients satisfy Re(η) η min > 0 and Re(γ) γ min > 0 where as the imaginary-parts satisfy Im(η) 0 and Im(γ) 0. Uniqueness for recovering Ω hold for a C 2 -boundary. Uniqueness holds for real-valued coefficients provided that η C( Ω) and γ L ( Ω). The Sampling Method is valid for recovering Ω. Reconstructing inclusions from Electrostatic Data 31 / 35

32 Reconstruction of an Inclusion with GIBC Let the coefficients be given by η ( x(θ) ) = 1 and γ ( x(θ) ) = sin 2 (θ). 1 Contour Plot Reconstruction 1 Level Curve W(z)=0.01 Reconstruction Reconstruction Inclusion Unit Circle Figure: Reconstruction for a inclusion with a GIBC via Sampling. Reconstructing inclusions from Electrostatic Data 32 / 35

33 Inverse Source Problem for the Diffusion ( t )u(x, t) = F (x) for (x, t) R d (0, ). with u(x, 0) = 0. We consider two types of sources F (x) = M c m δ(x z m ) and F (x) = m=1 M m=1 χ Dm with constants c m R where the points z m R d and D m R d are bounded open sets. Inverse Problem Recover the heat source F (x) such that Supp(F ) B R from the boundary measurements { u(x j, t) } j J where x j B R. Reconstructing inclusions from Electrostatic Data 33 / 35

34 Some Preliminary Results Three measurements { u(x j, t) } 3 uniquely determines the j=1 Point Sources F (x) = M c m δ(x z m ) for any M N m=1 Recovering the Center Points z m and D m is asymptotic to the point source problem for small regions Exact z m Recovered z m Exact D m Recovered D m (1, 1) (1.04, 0.98) (2, 2) (2.10, 2.10) Table: The reconstructed center points z m and area D m Reconstructing inclusions from Electrostatic Data 34 / 35

35 Figure: Questions? Reconstructing inclusions from Electrostatic Data 35 / 35