Exact shape-reconstruction by one-step linearization in electrical impedance tomography

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1 Exact shape-reconstruction by one-step linearization in electrical impedance tomography Bastian von Harrach Institut für Mathematik, Joh. Gutenberg-Universität Mainz, Germany Joint work with Jin Keun Seo, Yonsei University, Seoul, Korea 15th Inverse Days, Luosto, Finland, 16th - 18th December 2009

2 Mathematical Model Forward operator of EIT: Λ : σ Λ(σ), conductivity measurements Conductivity: σ L + () Continuum model: Λ(σ): Neumann-Dirichlet-operator Λ(σ) : g u, applied current measured voltage (σ u) = 0 in, σ ν u = g on. (1) Linear elliptic PDE theory: g L 2 ( )!u H () 1 solving (1). Λ(σ) : L 2 ( ) L 2 ( ) linear, compact, self-adjoint

3 Inverse problem Non-linear forward operator of EIT Λ : σ Λ(σ), L + () L(L 2 ( )) Inverse problem of EIT: Λ(σ) σ? Generic approach: Linearization Λ(σ) Λ(σ 0 ) Λ (σ 0 )(σ σ 0 ) σ 0 : known reference conductivity Λ (σ 0 ): Fréchet-Derivative / sensitivity matrix. Λ (σ 0 ) : L + () L(L 2 ( )). Often: supp(σ σ 0 ) compact. ( shape / inclusion )

4 Linearization Linear reconstruction method e.g. NOSER (Cheney et al., 1990), GREIT (Adler et al., 2009) Solve Λ (σ 0 )κ Λ(σ) Λ(σ 0 ), then κ σ σ 0. Multiple possibilities to measure residual norm and to regularize. No rigorous theory for single linearization step. Very little theory for Newton iteration. Seemingly, no rigorous results possible for single linearization step. Seemingly, only justifiable for small σ σ 0. In this talk: Rigorous and global(!) result about the linearization error.

5 Exact Linearization Theorem (H./Seo, 2009) Let κ, σ, σ 0 piecewise analytic and Λ (σ 0 )κ = Λ(σ) Λ(σ 0 ). Then (a) supp κ = supp (σ σ 0 ). (b) σ 0 σ (σ σ 0) κ σ σ 0 on the bndry of supp (σ σ 0 ). supp : outer support ( = supp, if supp is compact and has conn. complement) Exact solution of lin. equation yields correct (outer) shape. No assumptions on σ σ 0! Single-step linearization error does not affect shape reconstrution. Proof: Combination of monotony and localized potentials.

6 Monotony Monotony (in the sense of quadr. forms): Λ (σ 0 )(σ σ 0 ) Λ(σ) Λ(σ 0 ) } {{ } =Λ (σ 0 )κ Quadratic forms / energy formulation: gλ(σ 0 )g ds = gλ(σ)g ds = Λ (σ 0 ) g (Λ(σ 0 ) κ) g ds = σ 0 u 0 2 dx σ u 2 dx ( σ0 ) σ (σ σ 0). κ u 0 2 dx u 0 (resp. u): solution corresponding to σ 0 (resp. σ) and bndry current g.

7 Bounds on squares Exact linearization yields (σ σ 0 ) u 0 2 dx κ u 0 2 dx σ 0 σ (σ σ 0) u 0 2 dx. for all reference solutions u 0. Does this imply σ σ 0 κ σ 0 σ (σ σ 0)? Famous concept of inverse problems for PDEs: Completeness of products (of solutions of a PDE) Here: bounds on squares (of gradients of solutions of a PDE).

8 Localized potentials (σ σ 0 ) u 0 2 dx κ u 0 2 dx σ 0 σ (σ σ 0) u 0 2 dx. Localized potentials: (H. 2008) Make u 0 2 arbitrarily large in a region connected to the boundary but keep it small outside the connecting domain. u 0 2 small u 0 2 large supp σ 0 σ (σ σ 0) = supp (σ σ 0 ) supp κ = supp (σ σ 0 ) Also: σ 0 σ (σ σ 0) κ σ σ 0 on the bndry of supp (σ σ 0 )

9 Non-exact Linearization? Theorem (H./Seo, 2009) Let κ, σ, σ 0 piecewise analytic and Λ (σ 0 )κ = Λ(σ) Λ(σ 0 ). Then (a) supp κ = supp (σ σ 0 ). (b) σ 0 σ (σ σ 0) κ σ σ 0 on the bndry of supp (σ σ 0 ). Existence of exact solution is unknown! In practice: finite-dimensional, noisy measurements. Proof only requires Λ (σ 0 )(σ σ 0 ) Λ (σ 0 )κ Λ (σ 0 ) Solve linearized equation s.t. (*) is fulfilled. ( σ0 ) σ (σ σ 0). ( )

10 Non-exact Linearization Additional definiteness assumption: σ σ 0. Assume we are given Noisy data Λ m (σ) Λ m (σ 0 ) Λ(σ) Λ(σ 0 ) Noisy sensitivity Λ m(σ 0 ) Λ (σ 0 ). Finite-dim. subspace V 1 V 2... L 2 ( ) with dense union. Equip V k with norm g 2 (m) := ( Λ m (σ) Λ m (σ 0 ))g, g. Minimize (Galerkin approx. of) linearization residual Λ(σ) Λ(σ 0 ) Λ (σ 0 )κ m in the sense of quadratic forms on V k.

11 Non-exact Linearization Theorem (H./Seo, 2009) For appropriately chosen δ 1, δ 2 > 0, every V k and suff. large m, κ m : δ 1 Λ(σ) Λ(σ 0 ) Λ (σ 0 )κ m δ 2. (in the sense of quadr. forms on V k, κ m piecewise analytic) Every piecewise analytic L -limit κ of a converging subsequence fulfills (a) supp κ = supp (σ σ 0 ). (b) ( σ0 σ δ 1 ) (σ σ0 ) κ (δ 2 +1)(σ σ 0 ) on bndry of supp (σ σ 0 ). Convergence guaranteed if σ σ 0 belongs to fin-dim. ansatz space. Globally convergent shape reconstruction by one-step linearization.

12 Summary and open questions The linearization error in EIT does not affect the shape. With additional definiteness assumption, we derived a local one-step linearization algorithm with globally convergent shape reconstruction properties. Additional definiteness property is typical for shape reconstruction. Open questions Numerical implementation? Formulation as Tikhonov regularization with special norms? Definiteness only enters in V k -norm. Can this be replaced by other oszillation-preventing regularization?