Reduced Basis Methods for MREIT

Transcription

1 Reduced Basis Methods for MREIT Dominik Garmatter Group for Numerics of Partial Differential Equations, Goethe University Frankfurt, Germany Joint work with Bastian Harrach IMA Conference on Inverse Problems from Theory to Application Cambridge, UK, September 21st, 2017.

2 Magnetic Resonance Electrical Impedance Tomography (MREIT) 1 1 Based on Seo, Woo, et al. since 2003

3 Motivation: Electrical Impedance Tomography (EIT) Setting: Imaging object 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 B z -based MREIT: Setting Aim: achieve higher resolution of conductivity σ. E + 2 Ω E 1 E + 1 E 2 I 2 Object Ω with E ± 1, E± 2 attached placed inside MRI-Scanner Apply current I between electrode pair Generates magnetic flux density B B z measurable with MRI scanner (full internal data set)

5 Forward problem (shunt model) P = {σ C 1,α ( Ω) σ(x) > 0, x Ω}, α (0, 1) Detailed problem (j {1, 2} determines active electrode pair) For σ P, find u σ j, the detailed solution of (σ u j ) = 0 in Ω, I j = u j n = 0, on E + j E + j σ u j n ds = σ u j E n ds j E j, σ u ( j n = 0 on Ω\ E + j E j ) (1a) (1b) Unique solution of (1) up to an additive constant

6 Inverse Problem Aim: Determine σ from Bz,, 1 Bz, 2. Assume: σ P with σ Ω\ Ω= σ b, σ b > 0, Ω Ω, σ b known. Maxwell Equations σ u σ j = 1 µ 0 B j (Ampère s law) and B j = 0 of Ampère s law u σ j σ = 1 µ 0 2 B j Relation for σ (logarithmic version, point-wise in Ω) ln σ = 1 µ 0 (σ A[σ ]) 1 ( 2 B 1 z, 2 B 2 z, ), A[σ ]:= u σ 1 y u σ 2 y uσ 1 x uσ 2 x

7 Reconstruction of σ Initial guess: σ 0 = σ b. Iteration sequence ( Harmonic B z Algorithm 2 ) calculate V n+1 (r) = 1 µ 0 (σ n A[σ n ]) 1 ( 2 B 1 z, 2 B 2 z, define ln σ n+1 as the solution of ) (r), r Ω 2 ln σ n+1 = V n+1 in Ω, ln σ n+1 = ln σ on Ω σ n+1 = exp(ln σ n+1 ) (ensures positivity) 2 Seo, Woo 2003, et al.

8 Approximative approach Given: for σ P let u σ j,n be an (unspecified) approximation to uσ j. calculate V n+1 N with A N [σ n ]:= Approximative iteration sequence ( (r) = 1 µ 0 (σ n A N [σ n ]) 1 2 Bz, 1 2 Bz, 2 u1,n σn y u σn 2,N y uσ n 1,N x n uσ 2,N x define ln σ n+1 as the solution of ) (r), r Ω, 2 ln σ n+1 = V n+1 N in Ω, ln σ n+1 = ln σ on Ω σ n+1 = exp(ln σ n+1 )

9 Convergence Preliminary result 3 There exists an ɛ > 0, such that if ln σ C 0,α (Ω) < ɛ and the approximations u σn j,n fulfill u σn j,n C1,α ( Ω), Ω Ω Ω uj,n σn uσn j C 0,α ( Ω) C 1ɛ n+1 throughout the approximative iteration, the resulting sequence of iterates σ n, n = 1, 2,..., with initial guess σ 0 = σ b satisfies ( ) n 1 ɛ, n = 1, 2,... ln σ n ln σ C 1,α ( Ω) C Extension of Seo, Woo, Liu, Preprint available soon.

10 Reduced Basis Methods (RBM) and MREIT

11 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

12 RBM: The detailed & reduced problem Detailed problem (shunt model) F : P X, σ u σ, the detailed solution of b(u σ, v; σ) = f (v) for all v X b, f associated to (1). Assume: X N := span{u σ 1,..., u σ N } X is given. Reduced problem (Galerkin projection) For σ P, find u σ N X N X, the reduced solution of b(un, σ v; σ) = f (v), v X N. (one detailed/reduced problem/rb-space per j = 1, 2)

13 RBM: Properties Reproduction of solutions: u σ X N u σ N = uσ Offline/Online-decomposition: rapid computation of u σ N Certification - rigorous a-posteriori error estimator u σ u σ N X N (σ) := v r X α(σ), with v r, v X := r(v; σ) := f (v) b(u σ N, v; σ), v X

14 RBM & Inverse Problems: various approaches Naive/direct approach Construct global X N (greedy, POD,...) approximating whole M Use u σ N instead of uσ in the inversion scheme Limitation: Only feasible for low-dimensional parameter spaces, not feasible for imaging. Our approach 4 : Create problem-adapted RB-space by iterative enrichment (inspired by Druskin & Zaslavski 2007, Zahr & Fahrhat 2015 and Lass 2014). 4 G./Haasdonk/Harrach, A Reduced Basis Landweber method for nonlinear inverse problems, 2016 Inverse Problems 32 (3) (doi).

15 Adaptive RBM & IP: Idea P F X M

16 Adaptive RBM & IP: Idea P σ σ 0 F X M u σ0 u σ

17 Adaptive RBM & IP: Idea P σ σ 0 Iterates F X M u σ0 u σ

18 Adaptive RBM & IP: Idea P σ σ 0 Iterates F X M u σ0 u σ0 N u σ

19 Adaptive RBM & IP: Idea P σ σ 0 Iterates F X M u σ0 u σ0 N u σ

20 Adaptive RBM & IP: Idea P σ σ 0 Iterates F X M u σ0 u σ0 N u σ

21 Adaptive RBM & IP: Idea P σ σ 0 Iterates F X M u σ0 u σ0 N u σ

22 Recall: approximative approach for MREIT Given: RB-spaces X N,1, X N,2 u σ j,n is respective RB-approximation. calculate V n+1 N with A N [σ n ]:= RB iteration sequence ( (r) = 1 µ 0 (σ n A N [σ n ]) 1 2 Bz, 1 2 Bz, 2 u1,n σn y u σn 2,N y uσ n 1,N x n uσ 2,N x define ln σ n+1 as the solution of ) (r), r Ω, 2 ln σ n+1 = V n+1 N in Ω, ln σ n+1 = ln σ on Ω σ n+1 = exp(ln σ n+1 )

23 Reduced Basis Harmonic B z Algorithm (RBZ) Algorithm 1 RBZ(σ b, µ 0, ε 1, ε 2, X N,1, X N,2 ) 1: σ 0 := σ b, n := 0 2: repeat 3: Enrich spaces X N,1, X N,2 using u1 σn, 2. 4: repeat ( 5: V n+1 N (r) := 1 µ 0 (σ n A N [σ n ]) 1 2 Bz, 1 2 Bz, 2 ) (r), r Ω 6: Calculate ln σ n+1 as the solution of 2 ln σ n+1 = V n+1 N in Ω, ln σ n+1 = ln σ on Ω 7: σ n+1 := exp(ln σ n+1 ) n := n + 1 8: until ln σ n ln σ n 1 C(Ω) ε 1 or min j=1,2 { N,j(σ n )} > ε 2 9: until ln σ n ln σ n 1 C(Ω) ε 1 10: return σ RBZ := σ n

24 Numerics (idealized) Setting: image, ε 1 = 10 6, ε 2 = 10 3, no noise. Figure: σ b (top left), σ (top right), σ BZ (bottom left), σ RBZ (bottom right). BZ: 10.10s and 28 PDE solves σ σ BZ C(Ω) σ BZ C(Ω) , σ RBZ σ BZ C(Ω) σ BZ C(Ω) RBZ: 8.18s and 8 PDE solves

25 Conclusion Summary Inverse problem of MREIT & Harmonic B z Algorithm as solution algorithm Any (sufficient) approximative forward solution convergence Reduced Basis Method (adaptive approach - high-dimensional parameter space) to speed-up existing algorithm Novel RBZ-Algorithm, including (idealized) numerics Future work Finalize and publish results Thank you for your attention!