An entropic Landweber method for linear ill-posed problems

被引：4

作者：

Burger, M. ^{[1
]}

Resmerita, E. ^{[2
]}

Benning, M. ^{[3
]}

机构：

[1] Friedrich Alexander Univ Erlangen Nurnberg, Dept Math, Cauerstr 11, D-91058 Erlangen, Germany

[2] Alpen Adria Univ Klagenfurt, Inst Math, Univ Str 65-67, A-9020 Klagenfurt, Austria

[3] Queen Mary Univ London, Sch Math Sci, Mile End Rd, London E1 4NS, England

来源：

INVERSE PROBLEMS | 2020年 / 36卷 / 01期

基金：

欧盟地平线“2020”;

关键词：

regularization; Shannon entropy; Landweber method; integral equation; MIRROR DESCENT; CONVERGENCE; REGULARIZATION; MINIMIZATION; ALGORITHM; ITERATION; EQUATIONS;

D O I：

10.1088/1361-6420/ab5c49

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

The aim of this paper is to investigate the use of a Landweber-type method involving the Shannon entropy for the regularization of linear ill-posed problems. We derive a closed form solution for the iterates and analyze their convergence behaviour both in a case of reconstructing general nonnegative unknowns as well as for the sake of recovering probability distributions. Moreover, we discuss several variants of the algorithm and relations to other methods in the literature. The effectiveness of the approach is studied numerically in several examples.

引用

页数：22

共 32 条

[1] MAXIMUM-ENTROPY REGULARIZATION OF FREDHOLM INTEGRAL-EQUATIONS OF THE 1ST KIND [J].

AMATO, U ;

HUGHES, W .

INVERSE PROBLEMS, 1991, 7 (06) :793-808

[2]

[Anonymous], 1983, Wiley-Interscience Series in Discrete Mathematics

[3] Sampling the posterior: An approach to non-Gaussian data assimilation [J].

Apte, A. ;

Hairer, M. ;

Stuart, A. M. ;

Voss, J. .

PHYSICA D-NONLINEAR PHENOMENA, 2007, 230 (1-2) :50-64

[4] On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations [J].

Arnold, A ;

Markowich, P ;

Toscani, G ;

Unterreiter, A .

COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS, 2001, 26 (1-2) :43-100

[5]

Arnold A, 2008, Communications on Stochastic Analysis, V2, P11

[6] Iterative total variation schemes for nonlinear inverse problems [J].

Bachmayr, Markus ;

Burger, Martin .

INVERSE PROBLEMS, 2009, 25 (10)

[7]

Bauschke H, 2016, MATH OPERATIONS RES

[8] Mirror descent and nonlinear projected subgradient methods for convex optimization [J].

Beck, A ;

Teboulle, M .

OPERATIONS RESEARCH LETTERS, 2003, 31 (03) :167-175

[9]

Beck A., 2017, MOS SIAM SERIES OPTI

[10] ITERATIVE BREGMAN PROJECTIONS FOR REGULARIZED TRANSPORTATION PROBLEMS [J].

Benamou, Jean-David ;

Carlier, Guillaume ;

Cuturi, Marco ;

Nenna, Luca ;

Peyre, Gabriel .

SIAM JOURNAL ON SCIENTIFIC COMPUTING, 2015, 37 (02) :A1111-A1138

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