Magnet Resonance Electrical Impedance Tomography (MREIT): convergence of the Harmonic B z Algorithm

Transcription

1 Magnet Resonance Electrical Impedance Tomography (MREIT): convergence of the Harmonic B z Algorithm Dominik Garmatter garmatter@math.uni-frankfurt.de Group for Numerics of PDEs, Goethe University Frankfurt, Germany Joint work with Bastian Harrach OCIP 2017, Munich, Germany, April 5-7, 2017.

2 MREIT: the Harmonic B z Algorithm 1 1 Based on Seo, Woo, et al. since 2003

3 Motivation: Electrical Impedance Tomography (EIT) Setting: Imaging obect O R 3 with electrode pairs attached Aim: reconstruct cross-sectional (Ω = O {z = z 0 } R 2 ) image of electrical conductivity inside Ω Ω In EIT Data: Current-Voltage measurements on boundary Difficulty: highly ill-posed low spatial resolution

4 MREIT: Setting Aim: achieve higher resolution of conductivity σ. E + 2 Ω E 1 E + 1 E 2 I 2 Obect Ω with two electrode pairs E ± attached Place obect inside MRI-Scanner Apply current I between electrode pair Generates magnetic flux density B (z-comp. measurable with MRI scanner) Full internal data set to overcome ill-posedness

5 Forward problem Consider ( = 1, 2 determines active electrode pair) (σ u ) = 0 in Ω, I = σ u n ds = u n = 0, on E + E + E E, σ u ( n = 0 on Ω\ E + σ u n ds E ) (1a) (1b) P C 1 ±( Ω) := {σ C 1 ( Ω) 0 < σ σ σ < } Unique solution of (1) up to an additive constant

6 Inverse Problem Aim: Determine σ from B. Maxwell Equations σ u = 1 µ 0 B (Ampère s law) and B = 0 of Ampère s law u σ = 1 µ 0 2 B Only B z measurable and two different electrode pairs/currents Core-relation (point-wise in Ω) ( σ x σ y ) = 1 µ 0 A[σ] 1 ( 2 B 1 z 2 B 2 z ), A[σ] := ( u1 u 1 y x u 2 u 2 y x ) (CR)

7 Recovery of σ from σ So far: σ obtainable in Ω from 2 B z via (CR). How to get σ? Fundamental solution: for r = (x, y), r = (x, y ) Ω R 2 Φ(r r ) := 1 2π ln r r fulfilling 2 Φ(r r ) = δ(r r ) For σ C 1 (Ω) one can show σ(r) = r Φ(r r ) σ(r )dr Ω + ν(r ) r Φ(r r )σ(r )dl r Ω (2)

8 Harmonic B z Algorithm 2 Assume: σ P with σ Ω\ Ω = σ b, Ω Ω, σ b > 0 known Idea: iteration for (2) + (CR) + ln-formulation for positivity Algorithm 1 BZ(σ b, µ 0, tol) 1: n := 0, σ 0 := σ b 2: Calculate BI(r) := Ω ν(r ) r Φ(r r ) ln σ (r )dl r 3: repeat ( ) 4: F n+1 (r) := 1 1 µ 0 σ A[σ n ] 1 2 Bz, 1 n 2 Bz, 2 (r) 5: ln σ n+1 (r) := BI(r) Ω r Φ(r r ) F n+1 (r )dr 6: n := n + 1 7: until ln σ n ln σ n 1 P tol 8: return σ BZ := σ n 2 Seo, Woo 2003

9 Harmonic B z Algorithm: convergence Define Ξ(σ b, ɛ 0 ) := {σ P σ Ω\ Ω = σ b, ln σ C(Ω) < ɛ 0 }. Existing result 3 There exists 0 < ɛ < ɛ 0, such that for each σ Ξ(σ b, ɛ 0 ) with ln σ C(Ω) ɛ the sequence {σ n } generated by the Harmonic B z Algorithm with initial guess σ b satisfies σ n σ b in Ω \ Ω, ln σ n ln σ C1 ( Ω) K ( 1 2) n ɛ, with K := diam(ω) Seo, Woo, Liu, 2010

10 Harmonic B z Algorithm: our approach Idea: Replace u σn (see A[σ n ]) in Algo. 1 by approximations u σn N, Assumptions (for all n = 0, 1, 2,... ) u σn N, C 1 ( Ω) u σn N, u σn C( Ω) ɛn+1 C, ɛ from Theorem and C > 0 Assumptions fulfilled Existing result is replicated Applications Fineness of FEM-mesh (actual numerical convergence) Our scope: approximations u σn N, via Reduced Basis Methods also speed-up the method

11 Reduced Basis Methods & MREIT 4 4 Based on G./Haasdonk/Harrach, A Reduced Basis Landweber method for nonlinear inverse problems, 2016 Inverse Problems 32 (3) (doi).

12 Reduced Basis Methods (RBM): Idea X N X u σ 1 M u σ u σ N u σ 2 Solution manifold M := {u σ σ P} Construction of X N via carefully chosen snapshots u σ i

13 RBM: problem formulation & properties (for =1,2) Assume: X N, := span{u σ 1,..., u σ N } is given. Reduced forward problem (Galerkin proection) For σ P, find u σ N, X N, X, the reduced solution, of b(u σ N,, v; σ) = f (v), v X N,. Reproduction of solutions: u σ X N, u σ N, = u σ Offline/Online-decomposition: rapid computation of u σ N, Certification: u σ u σ N, X N, (σ) := v r, X α(σ)

14 Reduced Basis Harmonic B z Algorithm Algorithm 2 RBZ(σ b, µ 0, tol 1, tol 2, X N,1, X N,2 ) 1: n := 0, σ 0 := σ b 2: Calculate BI(r) := Ω ν(r ) r Φ(r r ) ln σ (r )dl r 3: repeat 4: enrich X N,1, X N,2 using u σn 1, uσn 2 5: i := 1, σ i := σ n 6: repeat ( 7: F n+1 N (r) := 1 1 µ 0 A σ i N [σ i ] 1 2 Bz, 1 2 Bz, 2 ) (r) 8: ln σ i+1 (r) := BI(r) Ω r Φ(r r ) F n+1 N (r )dr 9: i := i : until σ i σ i 1 P tol 1 or N,1 (σ i )> tol 2 or N,2 (σ i )> tol 2 11: σ n+1 := σ i, n := n : until σ i σ i 1 P tol 1 13: return σ RBZ := σ n

15 Numerics - Setting Ω := [ 1, 1] [ 2, 2] E ± 1 := {(±1, y) y < 0.1}, E± 2 := {(x, ±2) x < 0.1} For r = x 2 + y 2, x, y Ω { ( ) σ 10 cos(r) 3 + 2, 0 r π/6 2 σ(r) := 2, otherwise using rectangles (piecewise constant approximation) σ b = 2, µ 0 = 1, tol 1 = 10 5, tol 2 = 1, no noise 100

16 Numerics - Comparison Figure: σ b (top left), σ (top right), σ BZ (bottom left), σ RBZ (bottom right). BZ: 3.45s and 16 PDE solves RBZ: 2.21s and 6 PDE solves σ σ BZ P 0.05, σ BZ σ RBZ P

17 Conclusion Summary: Presented Harmonic B z Algorithm to solve MREIT imaging problem Extended existing convergence result to numerical applicability Applied adaptive RB approach to also speed-up the algorithm Ongoing work: Improve theoretical and numerical results Publish paper Thank you for your attention!