Research Institute for Mathematical Sciences (RIMS), Kyoto University, Analysis of inverse
problems through partial differential equations and related topics, January 8-10, 2020
FAST SUBSPACE OPTIMIZATION METHOD FOR NONLINEAR INVERSE PROBLEMSIN BANACH SPACES WITH UNIFORMLY CONVEX PENALTY TERMS
Ruixue Gua , Bo Hana , Yong Chenaa

Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang Province,150001, PR China
guruixue@hit.edu.cn bohan@hit.edu.cn yongchen@hit.edu.cn
In this talk, we aim to solve the inverse problem of the following form
F (x) = y,(1)

where F : D(F ) ⊂ X → Y is a nonlinear operator between two Banach spaces X and Y. We are in particularinterested in the case that we only have noisy data y δ satisfyingy δ − y ≤ δ,
where the noise level δ > 0 is known. Due to the inherent ill-posedness of inverse problems, some regularization
methods should be used to produce a stable approximate solution of (1).
Landweber iteration is one of the most prominent regularization methods for solving nonlinear inverse
problems due to its simplicity, see [1] and reference therein. In order to capture the special features of thesought solutions, such as sparsity and discontinuities, the penalty term is allowed to be non-smooth to include
L1 and total variation (TV) like penalty functionals. Let θ : X → (−∞, ∞] be a proper, lower semi-continuous,convex function, then the method in [1] has the form of∗



δ
δ
,x
ξn+1= ξnδ − tδn F 0 xδn JrY F xδn − y δ , xδn+1 = arg min θ (x) − ξn+1(2)x∈Xwith the suitable chosen step size tδn , JrY denotes the duality mapping of Y with the gauge function t → tr−1 .
This method could give satisfactory reconstructed solutions, however, the slow convergence of (2) makes it
inefficient in practical applications. Hence, it is necessary to introduce some accelerate strategies to fasten its
convergence speed.
One starting point is to use the sequential subspace optimization (SESOP) method, which was first proposed
to solve the large scale well-posed optimization problems by utilizing multiple search directions per iteration.
Then it was extended to solve the linear inverse problems [2,3] and nonlinear inverse problems [4,5]. The SESOPsmethod in [5], however, only covers θ (x) = kxk /s with 1 < s < ∞ on uniformly convex and uniformly smooth
Banach spaces and thus excludes the use of the L1 and TV like penalty functionals. Also the duality mapping
JrY : Y → Y ∗ requires r = 2 to guarantee the convergence and regularization of the method in [5], which is notsuitable for the data containing non-Gaussian noise.
Aiming at accelerating the iteration (2), we formulate an extension of the SESOP method in the sprit of[5] to solve nonlinear inverse problems with both X and Y being Banach spaces and the non-smooth convex
function θ : X → (−∞, ∞] is incorporated as the penalty term so that the method can be used for sparsity
reconstruction and discontinuity detection. Then the proposed SESOP method with convex penalty can beformulated asP δ 0 δ ∗ Yδδ,x ,
ξn+1= ξnδ −tn,i F xi Jr F xδi − y δ , xδn+1 = arg min θ (x) − ξn+1(3)x∈Xi∈In

where JrY denotes the duality mapping of Y with 1 < r < ∞ and In ⊆ {n − N + 1, · · · , n} is a finite index
set with given N > 1. A detailed convergence analysis and regularization results are given when the proposed
method is terminated by the discrepancy principle. Finally, we present some numerical examples for parameter
identification problems to validate the theoretical analysis and to verify the efficiency of the proposed method.
One-dimensional Sparsity Reconstruction. In this subsection, we assume that the sought parameter issparse andR the data only Rcontains Gaussian noise with δ = 0.5%, 0.1%, 0.05%. For this situation, we take21
θ (x) = 2β|x (w)| dw + Ω |x (w)|dw with β > 0.
ΩThe numerical results by our proposed SESOP method with the number M of search directions, called
SESOP-M in the following, and Landweber iteration in [1] are summarized in Table 1. Observe that theSESOP-2 method could produce the same accurate solutions but requires much less iteration numbers andcomputation time, validating its acceleration effect.

1Research Institute for Mathematical Sciences (RIMS), Kyoto University, Analysis of inverse
problems through partial differential equations and related topics, January 8-10, 2020

Table 1: Numerical results of one-dimensional sparsity reconstruction for noisy data.
δ = 0.5%
δ = 0.1%
δ = 0.05%
MethodnδTime(s)REnδTime(s)REnδTime(s)RE

Landweber
SESOP-1
SESOP-2

2698
686
212

10.55143.42471.70040.80440.80890.8073

37721
10271
4091

142.1369
49.0497
27.83890.25530.25420.2553

47036
17861
6614

193.6245
86.401347.52400.11910.11920.1188One-dimensional Discontinuity Detection. In this subsection,
we consider that the sought parameter isR21
piecewise constant. For this case, we choose θ (x) = 2β|x(w)|dw + T V (x), where β > 0 and T V (x)Ωdenotes the total variation of x.
We first consider that the data only contains Gaussian noise with various noise levels. The numerical results
by SESOP-2, SESOP-1 and Landweber iteration are displayed in Table 2. It is shown that SESOP-2 could leadto a significant reduction of iteration numbers through comparison with Landweber type iteration.
Table 2: Numerical results of one-dimensional discontinuity detection under Gaussian noise.
δ = 0.5%
δ = 0.1%
δ = 0.05%
MethodnδTime(s)REnδTime(s)REnδTime(s)RE

Landweber
SESOP-1
SESOP-2

2973
1242
600

18.0963
14.6804
13.1969

0.11810.11810.1182

27518
17087
8430

207.8141145.4954
94.7757

0.0696
0.06970.0695

37927
26952
11862

308.1425
214.6641
130.0063

0.06530.06530.0653We also investigate the effect of Banach spaces when the data contains impulsive noise. The impulsive
noise is plotted in Figure 1(a). Figure 1(b)-(d) present the reconstructions by SESOP-2 using Banach space
Y = Lr [0, 1] with r = 2, r = 1.5 and r = 1.1, respectively. Observe that L1.1 misfit data terms could produce amore accurate solution, which verifies that the Lr misfit data terms with r > 1 close to 1 are suitable for thiskind of noise.

(a)

(b)

(c)

(d)

Figure 1: The one-dimensional discontinuity detection under Impulsive noise. (a) Impulsive noise; (b) Reconstruction by SESOP-2 with r = 1.1; (c) Reconstruction by SESOP-2 with r = 1.5; (d) Reconstruction bySESOP-2 with r = 2.
ACKNOWLEDGEMENT
This work is supported by the National Natural Science Foundation of China (No. 11871180).
REFERENCES
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2. F. Schöpfer, T. Schuster and A. K. Louis, Metric and Bregman projections onto affine subspaces and theircomputation via sequential subspace optimization methods, Journal of Inverse and Ill-posed Problems, 15(2007), 1-29.
3. F. Schöpfer and T. Schuster, Fast regularizing sequential subspace optimization in Banach spaces, InverseProblems, 25 (2009), 015013.
4. A. Wald and T. Schuster, Sequential subspace optimization for nonlinear inverse problems, Journal of Inverseand Ill-posed Problems, 25 (2016), 99-117.
5. A. Wald, A fast subspace optimization method for nonlinear inverse problems in Banach spaces with anapplication in parameter identification, Inverse Problems, 34 (2018), 085008.
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