A Direct Method for reconstructing inclusions from Electrostatic Data

Transcription

1 A Direct Method for reconstructing inclusions from Electrostatic Data Isaac Harris Texas A&M University, Department of Mathematics College Station, Texas Joint work with: W. Rundell PDE Seminar, University of Houston - April 2017 Reconstructing inclusions from Electrostatic Data 1 / 38

2 Electrical Impedance Tomography Electrical impedance tomography (EIT) is a nondestructive type of imaging method in which the physical parameters of a material is recovered from surface electrode measurements. Figure: Picture taken from Reconstructing inclusions from Electrostatic Data 2 / 38

3 Formulation of the Problem Formulation of the Problem Reconstructing inclusions from Electrostatic Data 3 / 38

4 Formulation of the Problem The Inverse Electrostatic Problem Inverse Problem 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. Reconstructing inclusions from Electrostatic Data 4 / 38

5 Formulation of the Problem In this talk we focus on B(u 0 ) = ν u 0 + γ(x)u 0 Assumptions: We consider the Electrostatic Problem in R 2 and R 3 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 Remark: The results in section can be extended for the case when is replaced by A(x) with symmetric A(x) C 1 (D, C d d ), where the Re(A) is positive definite and Im(A) is non-positive. Reconstructing inclusions from Electrostatic Data 5 / 38

6 Formulation of the Problem Some Previous Works Uniqueness for the inverse problem for 2 Cauchy pairs. 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 6 / 38

7 Formulation of the Problem Iterative Reconstruction Methods 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 = u (j) 0 and g j = ν u (j) 0 on D. Λ 0 f j : ( Ω, γ) g j for j = 1, 2 Then you would solve a non-linear and (exponentially) ill-posed posed problem using a iterative scheme. Reconstructing inclusions from Electrostatic Data 7 / 38

8 Formulation of the Problem Figure: Reconstruction of boundary and impedance via an iterative method based on non-linear BIE for the boundary and impedance. W. Rundell, Recovering an obstacle and its impedance from Cauchy data, Inverse Problems 24 (2008), Reconstructing inclusions from Electrostatic Data 8 / 38

9 Formulation of the Problem Restrictions for Iterative Methods Good initial guest is usually needed to insure Convergence This requires a prior 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 prior information? A lack of information cannot be remedied by any mathematical trickery -C. Lanczos Reconstructing inclusions from Electrostatic Data 9 / 38

10 Formulation of the Problem 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 10 / 38

11 Formulation of the Problem Useful Monographs for Elliptic IBVP: F. Cakoni and D. Colton, A Qualitative Approach to Inverse Scattering Theory. Applied Mathematical Sciences, Vol 188, Springer, Berlin A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems. Oxford University Press, Oxford Qualitative Methods in the Time Domain: H. Haddar, A. Lechleiter, S. Marmorat, An improved time domain linear sampling method for Robin and Neumann obstacles, App Analysis, 93 (2014) pp H. Heck, G. Nakamura and H. Wang, Linear sampling method for identifying cavities in a heat conductor, Inverse Problems 28 (2012) Reconstructing inclusions from Electrostatic Data 11 / 38

12 The Linear Sampling Method A Sampling Method to Recover the Boundary Reconstructing inclusions from Electrostatic Data 12 / 38

13 The Linear Sampling Method 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 Gh = ν w D where w = 0 in D \ Ω with w D = 0 and ν w Ω = h Using that G(x, z) as x z and the Range of G consists of smooth functions we have that following result. Theorem (Range Test) ν G(, z) D Range(G) if and only if z Ω. Where the operator G : H 1/2 ( Ω) H 1/2 ( D). Reconstructing inclusions from Electrostatic Data 13 / 38

14 The Linear Sampling Method Theorem (Properties of the Data Operator) The difference of the voltage to current operators (Λ Λ 0 ) : H 1/2 ( D) H 1/2 ( D) is compact, injective with a dense range (under some assumptions). Moreover, satisfies that Range(Λ Λ 0 ) Range(G). Techniques used in the proof 1. Derive the factorization (Λ Λ 0 ) = G(L L 0 ) 2. Compactness follows from elliptic regularity 3. Injectivity is a consequence on unique continuation 4. Green s 1st identity gives that (Λ Λ 0 ) is symmetric Reconstructing inclusions from Electrostatic Data 14 / 38

15 The Linear Sampling Method 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(G) which is a contradiction provided that z / Ω. Reconstructing inclusions from Electrostatic Data 15 / 38

16 The Linear Sampling Method Determining Ω via Sampling Theorem (The Sampling Method) 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 16 / 38

17 The Linear Sampling Method Numerical Experiments in the Unit Disk 1. Approximate the function u 0 via a System of BIE 2. Approximate the mappings Λ 0 e inθ and Λe inθ = ne inθ 3. Let f z n=0 n=0 f z n e inθ which gives the equation fn z (Λ Λ 0 )e inθ = 1 1 z 2 2π z z cos(θ θ z ) 4. Solve the above equation by a spectral cut-off 5. To visualize the inclusion we let W cutoff (z) = [ 31 n=0 f z n ] 1/2 2 and plot W (z) = W cutoff(z) W cutoff (z) Reconstructing inclusions from Electrostatic Data 17 / 38

18 The Linear Sampling Method 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 18 / 38

19 The Linear Sampling Method 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 19 / 38

20 Recovering the BC via Integral Equations A Direct Method to Recover the Impedance Reconstructing inclusions from Electrostatic Data 20 / 38

21 Recovering the BC via Integral Equations Theorem (Uniqueness) The voltage to current mapping Λ 0 : H 1/2 ( D) H 1/2 ( D) uniquely determines the inclusion Ω (provided Ω = Lipshitz) and the impedance parameter γ(x) L + ( Ω). Techniques used in the proof 1. Using the fact that (Λ Λ 0 ) has a dense range gives uniqueness for the inclusion 2. Then prove that the set { } u Ω 0 : for all f H 1/2 ( D) is a dense in L 2 ( Ω) which gives uniqueness for the impedance Reconstructing inclusions from Electrostatic Data 21 / 38

22 Recovering the BC via Integral Equations Data Completion and Recovery of γ(x) Let D = Γ m (measurements) and Ω = Γ i (impedance) both be class C 2 boundaries 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 22 / 38

23 Recovering the BC via Integral Equations Data Completion and Recovery of γ(x) 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, g) = (u 0, ν u 0 ) 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 23 / 38

24 Recovering the BC via Integral Equations Data Completion and Recovery of γ(x) Define the operator [ I K A = mm K mi K im I + K ii ] which represents the integral operator associated with the system of BIE on the previous slide. Theorem (Inversion of A) 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. R. Kress, Linear Integral Equations, Springer, New York, 3rd edition, Reconstructing inclusions from Electrostatic Data 24 / 38

25 Recovering the BC via Integral Equations Data Completion and Recovery of γ(x) Using the fact the we know the normal derivative on we have that [ I K g = [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. Solve for the impedance by using the boundary condition ν u 0 + γ(x)u 0 = 0 for x Γ i. Reconstructing inclusions from Electrostatic Data 25 / 38

26 Recovering the BC via Integral Equations Data Completion and Recovery of γ(x) One can consider two methods for solving for the impedance: (Naive) Direct Solution γ(x n ) = νu 0 (x n ) u 0 (x n ) for n = 1,, N with x n = x(θ n ) Ω and Least Squares Method min a,b N ν u 0 (x n ) + γ(x n )u 0 (x n ) n=1 2 with the impedance function given by γ ( x(θ) ) M a 0 + a m cos(mθ) + b m sin(mθ) m=1 Reconstructing inclusions from Electrostatic Data 26 / 38

27 Recovering the BC via Integral Equations Reconstruction of γ ( x(θ) ) = 5 4.8exp ( 10 sin 4 (θ 3.5) ) sin(θ) Perturbed Ellipse: x(θ) = ( cos(θ), sin(θ) ) (6, 7) 6 5 Exact Reconstructed Figure: Reconstruction for a perturbed ellipse from 10 Cauchy pairs via the Directly Solve Method. Reconstructing inclusions from Electrostatic Data 27 / 38

28 Recovering the BC via Integral Equations Reconstruction of γ ( x(θ) ) = 2 sin 4 (θ) Reconstructed Ellipse: x(θ) r(θ) ( cos(θ), sin(θ) ) r(θ) = Trig Polynomial determined by the level curve Exact Reconstructed Figure: Reconstruction for a reconstructed ellipse via the Least Squares Method with 16-Fourier Modes used. Reconstructing inclusions from Electrostatic Data 28 / 38

29 Current/Future Research An Extension to Other Inverse BVP Reconstructing inclusions from Electrostatic Data 29 / 38

30 Current/Future Research 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. F. Cakoni, Y. Hu and R. Kress Simultaneous reconstruction of shape and generalized impedance functions in electrostatic imaging Inverse Problems 30 (2014) Reconstructing inclusions from Electrostatic Data 30 / 38

31 Current/Future Research 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) 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 / 38

32 Current/Future Research Reconstruction of an Inclusion with GIBC Let the coefficients be given by η = 1/5 and γ = 2 + i where we use separation of variables to determine ( u 0 (r, θ) = a 0 + b 0 ln r + an r n + b n r n ) e inθ. n =1 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 / 38

33 Current/Future Research Reconstruction of (real-valued) Coefficients with GIBC Using Data Completion implies that the mapping ( f, Λ0 f ) ( Ω D u 0 (f ), ν u 0 (f )) is known. Multiplying the GIBC by u 0 and integrating given that Ω u 0 ν u 0 ds = Now assume that Ω η du 0 ds 2 + γ u 0 2 ds for all f H 1/2 ( D) η ( x(s) ) N n=1 η n Ψ (1) ( ) n x(s) and γ ( x(s) ) N n=1 γ n Ψ (2) ( ) n x(s) Notice that by taking f m for m = 1,, M the above equality gives a 2N M linear system of equations to recover η and γ. Reconstructing inclusions from Electrostatic Data 33 / 38

34 Current/Future Research The Inverse Scattering Problem The Physical Model: We consider the time harmonic Acoustic scattering in R 3 or Electromagnetic scattering in R 2 (scalar). Here we wish to use acoustic or electro-magnetic waves to recover physical parameters of a material. Reconstructing inclusions from Electrostatic Data 34 / 38

35 Current/Future Research The Direct Scattering Problem We consider the scattering by an inhomogeneous material with an interior impedance boundary. The total field u = u s + u i H 1 loc (R3 \ Ω) satisfies: u + k 2 (1 + q(x) ) u = 0 in R 3 \ Ω ν u + iγ(x)u = 0 on Ω r u s iku s = o ( r 1) as r. Let u i = e ikx ŷ then it is known that the scattering problem is well-posed for that radiating scattered fields u s (x, ŷ). Reconstructing inclusions from Electrostatic Data 35 / 38

36 Current/Future Research The Far Field Pattern The scattered field, has the following asymptotic expansion in R 3 (and similar in 2 dimensions) { ( )} u s (x, ŷ) = eik x 1 u (ˆx, ŷ) + O 4π x x as x. Using Green s Representation theorem one can show that the far field pattern is given by u (ˆx, ŷ) = u s (z, ŷ) ν e ik ˆx z ν u s (z, ŷ)e ik ˆx z ds(z) C where ˆx, ŷ S = {ˆp R 3 : ˆp = 1}. Reconstructing inclusions from Electrostatic Data 36 / 38

37 Current/Future Research Some Future Works Develop a Sampling Method to determine the inclusion. Uniqueness and Stability of determining Ω and γ(x). Derive Integral Equation to determine the Cauchy data on D. Derive a System of BIE to determine γ(x) on Ω. Some useful work for this problem: Y. Boukari and H. Haddar, A convergent data completion algorithm using surface integral equations, Inverse Problems 25, (2015) H. Qin and X. Liu, The linear sampling method for inhomogeneous media and buried objects from far field measurements, App. Num. Math, 105 (2016), pp Reconstructing inclusions from Electrostatic Data 37 / 38

38 Current/Future Research Figure: Questions? Reconstructing inclusions from Electrostatic Data 38 / 38