Reduced Basis Methods for inverse problems

Transcription

1 Reduced Basis Methods for inverse problems Dominik Garmatter Chair of Optimization and Inverse Problems, University of Stuttgart, Germany Joint work with Bastian Harrach and Bernard Haasdonk Distinguished Lectures on Inverse Problems Helsinki, Finland, August 4-8, 2014.

2 Reduced Basis Methods

3 Motivation l Linear elliptic parametrized PDE of the form Lu(x;σ) = f, in Ω u(x;σ) = g, on Ω (1a) (1b) Solution of (1) for many different parameters in a small amount of time (e.g. design optimization, optimal control, online-simulation, inverse problems) Detailed solution (e.g. FEM, FV, FD) is rather expensive model order reduction

4 Motivation ll Approximating the solution manifold M := {u(σ) σ P}, P R p a bounded parameter domain, with a low dimensional subspace X N X Construction of X N with so called snapshots, i.e. X N span{u(σ 1 ),...,u(σ N )} with meaningful parameters σ 1,...,σ N P M u(σ 1 ) u(σ 2 )

5 The detailed and reduced problem Detailed problem: Let X be a Hilbert space, P R p a bounded parameter domain, b : X X P Racontinuous, coercive, symmetricbilinearform, f : X P R a continuous linearform. For a given parameter σ P, we seek the detailed solution u(σ) X of b(u(σ),v;σ) = f(v;σ), v X. (P)

6 The detailed and reduced problem Detailed problem: Let X be a Hilbert space, P R p a bounded parameter domain, b : X X P Racontinuous, coercive, symmetricbilinearform, f : X P R a continuous linearform. For a given parameter σ P, we seek the detailed solution u(σ) X of b(u(σ),v;σ) = f(v;σ), v X. (P) Reduced problem: Let a Problem (P) be given and let X N X be a reduced basis space (RB-space). For a given parameter σ P, we seek the reduced solution u N (σ) X N of b(u N (σ),v;σ) = f(v;σ), v X N. (PN)

7 The reduced basis space X N Error estimator u u N X N (σ) := v r X α(σ), with v r,v X := r(v;σ) := f(v;σ) b(u N (σ),v;σ), v X Algorithm 1 Greedy-Procedure 1: X N := {0}, Φ N :=, M train, ε tol, N (σ) 2: repeat 3: σ := argmax σ Mtrain N (σ) 4: φ := u(σ ), Φ N := Φ N φ, X N := X N +span(φ) 5: ε := max σ Mtrain N (σ) 6: until ε ε tol 7: return Φ N, X N

8 Offline/Online Decomposition Assumption: b and f are affine in parameter, i.e. b(u,v;σ) = Q b q=1 Θq b (σ)bq (u,v), f(v;σ) = Q f q=1 Θq f (σ)f q (v), u,v X. Offline-phase (once) Compute RB Φ N = {φ 1,...,φ N } and RB-space X N Compute parameter-independant components B q N := (bq (φ i,φ j )) N i,j=1 RN N and f q N := (f q (φ i )) N i=1 RN Online-phase (for each new σ) Evaluate parameter-dependant coefficients Θ q b (σ), Θq f (σ) Assemble B N (σ), f N (σ) and solve B N u N = f N Reconstruct solution of (PN) via u N = i u N,i φ i

9 Conclusion RBM Can accelerate forward solver of a PDE in a multi-query context due to efficient offline/online decomposition t k t det Greedy-scheme to construct RB-space X N Approximates highdimensional solution u (FEM, FD, FV) t off k t off +k t on k

10 RBM for inverse problems

11 Model problem Forward problem: Non-linear operator F : D(F) Y X between Hilbert spaces maps a parameter σ D(F) to a solution u of a given PDE: F(σ) = u. Inverse problem: For a given PDE-solution u X find the corresponding parameter σ D(F) with F(σ) = u.

12 Numerical approach Given u and exact solution σ + (s.t. F(σ + ) = u) and noisy data u δ with F(σ + ) u δ δ Consider σ δ n+1 = σδ n +s δ n with starting value σ δ 0 D(F) Want sn δ to approximate sn e := σ + σn δ If F is Fréchet-differentiable sn e solves the linear system F (σ δ n)s e n = u F(σ δ n) E(σ +,σ δ n) Compute s δ n as a solution of F (σ δ n )s = uδ F(σ δ n )

13 REGINN(REGularisation based on INexact Newtoniteration) 1 Algorithm 2 REGINN(σ start,τ,{θ n }) 1: n := 0, σ δ 0 := σ start 2: while F(σ δ n) u δ > τδ do 3: i := 1 4: repeat 5: i := 2i 6: s n,i := ( F (σ δ n ) F (σ δ n )+ 1 i Id ) 1 F (σ δ n ) (u δ F(σ δ n )) 7: until F (σ δ n )s n,i +F(σ δ n ) uδ < Θ n F(σ δ n ) uδ 8: σ δ n+1 := σδ n +s n,i 9: n := n+1 10: end while 11: σ REGINN := σ δ n 1 see A.Rieder: Keine Probleme mit Inversen Problemen, Seite 251.

14 REGINN utilizing RB Algorithm 3 REGINN utilizing RB(σ start,τ,{θ n },X N ) 1: n := 0, σ δ 0 := σ start 2: while F N (σ δ n ) uδ > τδ do 3: i := 1 4: repeat 5: i := 2i 6: s n,i := ( F (σ δ n) F (σ δ n)+ 1 i Id ) 1 F (σ δ n) (u δ F N (σ δ n)) 7: until F (σ δ n)s n,i +F N (σ δ n) u δ < Θ n F N (σ δ n) u δ 8: σn+1 δ := σδ n +s n,i 9: n := n+1 10: end while 11: σ REGINN,RB := σ δ n

15 Numerical toy problem Find a solution u H 1 0 (Ω) L2 (Ω) of div(σ u) = 1, in Ω = [0,1] 2, with σ L +(Ω). Reduced Basis requires P R p Restrict to σ = 9 i=1 σ iχ Ωi with σ i [0.1,10] Ω 7 Ω 8 Ω 9 Ω 4 Ω 5 Ω 6 Ω 1 Ω 2 Ω 3

16 Numerical results - How good is RB? 10 5 σ REGINN σ REGINN,RB σ REGINN σ REGINN,RB RB size N Figure : Comparison of Algorithm 2 and 3 with δ = 0.01, depending on the size N of the used RB. Target parameter was σ + = (2,2,2,2,9,2,2,2,2), σ start = (1,1,1,1,1,1,1,1,1).

17 Numerical results - When does RB pay off? Average time of REGINN without RB (Alg. 2): s size RB REGINN RB (s) Offline-time RB (s) querys required Table : Average time of Algorithm 3 (using 200 random parameters with starting value σ start = (1,1,1,1,1,1,1,1,1) and δ = 0.01).

18 Limitations of Algorithm 3 Increasing dimension of parameter space p very problematic For large p (> 40) RB-space covering the whole variety in parameter cannot be constructed (in our case) computation of Fréchet-derivative becomes more and more expensive Current approach not feasible for large p

19 New approach (inspired by V. Druskin and M. Zaslavsky, 2007) Don t construct an a-priori RB-space approximating the whole solution manifold Construct a small, problem-oriented RB-space X N,1 while solving the inverse problem Use a second RB-space X N,2 containing informtion about the derivative Utilize the property of the regularization algorithm to determine a new meaningful parameter to enrich the RB-space

20 New approach - Pseudocode Algorithm 4 new reduced Landweber(σ start,τ,φ N,1,Φ N,2 ) 1: n := 0, σ δ 0 := σ start, X N,1 := span(φ N,1 ), X N,2 := span(φ N,2 ) 2: while F(σ δ n ) uδ > τδ do 3: Φ N,1 := Φ N,1 F(σ δ n ), Φ N,2 := Φ N,2 F (σ δ n ) 4: X N,1 = span{φ N,1 }, X N,2 = span{φ N,2 } 5: i := 1, σi δ := σn δ 6: repeat 7: σi+1 δ := σδ i +F N (σδ i ) (u δ F N (σi δ)) 8: i := i +1 9: until F N (σi δ) uδ > τδ 10: σ δ n+1 := σδ i 11: n := n+1 12: end while 13: σ final := σ δ n

21 First Results: comparing Algorithm 2 & 4 p # DOFs t Alg. 2 (s) t Alg. 4 (s) σ Alg2 σ Alg Table : Average time and normdifference of Algorithm 2 & 4 (using 20 random parameters with starting value σ start = (1,1,1,1,1,1,1,1,1) and δ = 0.01) for different settings.

22 Outlook Investigate Algorithm 4 and provide further numerical results Provide theoretical background Extend to more realistic examples problems with an actual application (e.g. EIT) parameter functions

23 Outlook Investigate Algorithm 4 and provide further numerical results Provide theoretical background Extend to more realistic examples problems with an actual application (e.g. EIT) parameter functions Thank you for your attention!