Coupling reduced basis methods and the
Landweber method to solve inverse problems
Dominik Garmatter
[email protected]
Chair of Optimization and Inverse Problems, University of Stuttgart, Germany
Joint work with Bastian Harrach and Bernard Haasdonk
13th U.S. National Congress on Computational Mechanics
San Diego, California, July 26-30, 2015.
D. Garmatter: Coupling reduced basis methods and the Landweber method to solve inverse problems
Introduction
D. Garmatter: Coupling reduced basis methods and the Landweber method to solve inverse problems
Forward problem
Consider ∇ · (σ(x)∇u(x)) = 1, x ∈ Ω := (0, 1)2 , u(x) = 0, x ∈ ∂Ω.
Forward operator
F : D(F ) ⊂ Y −→ X , σ 7−→ u σ between Hilbert spaces with u σ ,
the detailed solution, solving
b(u σ , v ; σ) = f (v ; σ), for all v ∈ X , with
Z
Z
v dx.
σ∇u · ∇w dx, f (v ; σ) := −
b(u, w ; σ) :=
Ω
Ω
D. Garmatter: Coupling reduced basis methods and the Landweber method to solve inverse problems
(1a)
(1b)
Inverse problem and its difficulties
Inverse problem
For given solution u ∈ X of (1), find corresponding parameter
σ + ∈ D(F ) with F (σ + ) = u ( a-example“).
”
◮
◮
Naive inversion, i.e. solving σ + = F −1 (u) fails due to
ill-posedness of the problem (F −1 is discontinuous!)
; Small errors get amplified!
Typically only noisy data u δ with ku − u δ k < δ given
; F −1 (u δ ) 9 F −1 (u) for δ → 0.
; Remedy: Regularization methods (e.g. Landweber method) still
provide stable approximative solutions σ δ to σ + .
D. Garmatter: Coupling reduced basis methods and the Landweber method to solve inverse problems
Landweber method
Algorithm 1 Landweber(σstart , τ )
1:
2:
3:
4:
5:
6:
n := 0, σ0δ := σstart
while kF (σnδ ) − u δ k X > τ δ do
δ
σn+1
:= σnδ + ωF ′ (σnδ )∗ (u δ − F (σnδ ))
n := n + 1
end while
return σLW := σnδ
◮
Numerous evaluations of F for many different parameters.
◮
Detailed solution (e.g. FEM, FV, FD) is expensive.
; model order reduction
D. Garmatter: Coupling reduced basis methods and the Landweber method to solve inverse problems
Reduced basis method
Reduced basis space (RB-space) XN := span{φ1 , . . . , φN } ⊂ X
(dim XN ≪ dim X ) is given via e.g. φi = F (σi ), with meaningful
parameters σi ∈ D(F ), i = 1, . . . , N.
Reduced forward operator
For given F and XN ⊂ X , define FN : D(F ) ⊂ Y −→ XN ,
σ with u σ , the reduced solution, solving
σ 7−→ uN
N
σ
b(uN
, v ; σ) = f (v ; σ), for all v ∈ XN .
◮
◮
σ k ≤ ∆ (σ) := kvr k X with
Residual error estimator: ku σ − uN
X
N
α(σ)
σ , v ; σ), for all v ∈ X .
hvr , v iX := r (v ; σ) := f (v ; σ) − b(uN
Offline/online decomposition: enables efficient and cheap
evaluation of FN (σ).
D. Garmatter: Coupling reduced basis methods and the Landweber method to solve inverse problems
Combining RBM & Landweber
method
D. Garmatter: Coupling reduced basis methods and the Landweber method to solve inverse problems
Various approaches
Standard approach:
Construct global XN approximating whole R(F ) (providing
σ for all σ ∈ D(F )) ; offline-phase“.
good reduced solutions uN
”
◮ Using this space, quickly compute FN (σ) and substitute F (σ)
for FN (σ) in Algorithm 1 ; online-phase“.
”
Problem: Only feasible for low-dimensional parameter spaces
(≤ 30), not feasible for imaging.
◮
Our approach: Create problem-adapted RB-space by iterative
enrichment (inspired by Druskin & Zaslavski, 2007).
D. Garmatter: Coupling reduced basis methods and the Landweber method to solve inverse problems
Motivation - Combining RBM & LW
X
Y
F
D(F )
D. Garmatter: Coupling reduced basis methods and the Landweber method to solve inverse problems
R(F )
Motivation - Combining RBM & LW
X
Y
σ+
D(F )
uδ
F
u σstart
σstart
D. Garmatter: Coupling reduced basis methods and the Landweber method to solve inverse problems
uσ
+
R(F )
Motivation - Combining RBM & LW
X
Y
σ+
uδ
F
u σstart
D(F )
LW Iterates
σstart
D. Garmatter: Coupling reduced basis methods and the Landweber method to solve inverse problems
uσ
+
R(F )
Motivation - Combining RBM & LW
X
Y
σ+
uδ
F
u σstart
D(F )
+
R(F )
LW Iterates
σstart
uσ
XN
D. Garmatter: Coupling reduced basis methods and the Landweber method to solve inverse problems
Procedure - adaptive space enrichment
1. Start with initial guess σstart and initial RB-spaces XN,1 , XN,2
(Landweber method requires adjoint of the derivative!).
2. Update XN,1 , XN,2 using current iterate (first update via σstart ).
3. Solve the inverse problem up to a certain accuracy using the
nonlinear Landweber method projected onto XN,1 and XN,2 .
4. If resulting iterate is accepted by the discrepancy principle,
terminate, else go to step 2.
D. Garmatter: Coupling reduced basis methods and the Landweber method to solve inverse problems
Reduced Basis Landweber (RBL) method
Algorithm 2 RBL(σstart , τ, XN,1 , XN,2 )
1:
2:
3:
4:
5:
6:
7:
8:
9:
10:
11:
12:
n := 0, σ0δ := σstart
while kF (σnδ ) − u δ k X > τ δ do
enrich XN,1 , XN,2
i := 1, σiδ := σnδ
repeat
σiδ+1 := σiδ + ωFN′ (σiδ )∗ (u δ − FN (σiδ ))
i := i + 1
until kFN (σiδ ) − u δ k X ≤ τ δ or ∆N (σiδ ) > (τ − 2)δ
δ
σn+1
:= σiδ
n := n + 1
end while
return σRBL := σnδ
D. Garmatter: Coupling reduced basis methods and the Landweber method to solve inverse problems
Dual problem
For σ, κ ∈ D(F ) and l ∈ X , one can show
Z
κ∇u σ · ∇ulσ dx,
hκ, F ′ (σ)∗ l iY =
(2)
Ω
with ulσ ∈ X the unique solution of the dual problem
Z
l v dx.
b(u, v ; σ) = m(v ; l ), for all v ∈ X , m(v ) := −
Ω
In Algorithm 2
◮
◮
σδ
enrich XN,1 with F (σnδ ) and XN,2 with ul n solving (3) for
l := u δ − F (σnδ ).
evaluate FN′ (σiδ )∗ (u δ − FN (σiδ )) using (2) and associated
reduced solutions.
D. Garmatter: Coupling reduced basis methods and the Landweber method to solve inverse problems
(3)
Numerics - compare reconstructions
Setting: 900 pixels, τ = 2.5, δ ≈ 0.8% and ω = 12 (kF ′ (σstart )k)−1 .
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
5
4.5
4
3.5
0
0
0.5
1
0
0
0.5
1
3
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
2.5
2
1.5
0
0
0.5
1
0
0
0.5
1
1
Figure: Exact solution (top right), σstart (top left). Reconstruction via
Algorithm 2 (bottom left) and Algorithm 1 (bottom right).
D. Garmatter: Coupling reduced basis methods and the Landweber method to solve inverse problems
Numerics - time comparison
◮
◮
Outer iteration: space enrichment, projection ( offline“).
”
Inner iteration: one iteration of repeat loop ( online“).
”
Algorithm
time (s)
Landweber
266958
# Iterations
789304
time per Iteration (s)
0.338
# forward solves
1578608
RBL
22014
outer
25
inner 789318
outer 8.961
inner
0.028
50
Table: Time comparison of Algorithms 1 & 2.
D. Garmatter: Coupling reduced basis methods and the Landweber method to solve inverse problems
Conclusion & Outlook
◮
◮
◮
◮
Solving inverse coefficient problem requires many PDE solves.
Reduced basis (RB) approach can speed up PDE solution.
But standard RB approach is only applicable for low
dimensional parameter spaces.
Using adaptive problem-specific RB enrichment, we can handle
high-dimensional parameter spaces, e.g. for imaging problems.
; RBL method outperforms standard Landweber by an order
without loss of accuracy.
Outlook
◮ Convergence theory for RBL method.
◮ Apply methodology to other inverse problems and iterative
regularization algorithms of Gauß-Newton type.
D. Garmatter: Coupling reduced basis methods and the Landweber method to solve inverse problems
Thank you for your attention!
Preprint available:
A Reduced Basis Landweber method for nonlinear inverse problems
(arXiv ; 1507.05434).
D. Garmatter: Coupling reduced basis methods and the Landweber method to solve inverse problems