Iterative regularization with a general penalty term—theory and application toL1andTVregularization

Boţ, Radu Ioan; Hein, Torsten

doi:10.1088/0266-5611/28/10/104010

Cited by 39 publications

(49 citation statements)

References 30 publications

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“…Note that, although the Bregman projection x k+1 is unique, t k,ϕ might not be unique. If (10) is not fulfilled, we define an update inspired from [23] by setting…”

Section: Realizations Of the Methodsmentioning

confidence: 99%

“…At first, we characterize fixed points of Algorithm 1 and Algorithm 2 and provide necessary lemmas for the subsequent analysis. In Section 4.1, we prove that for nonnegative star-convex functions f i , condition (10) is always fulfilled and the step size (11) is better than the relaxed step size (12) in the sense that it results in an iterate with a smaller Bregman distance to all solutions of (1). Finally, we present convergence results for Algorithm 1 for this setting.…”

Section: Convergencementioning

confidence: 98%

“…Bregman projections were combined with the nonlinear Landweber iteration the first time in [54]. Later, [10] employed Bregman projections for L 1 -and TVregularization. A different nonlinear Landweber iteration with Bregman projections for sparse solutions of inverse problems was investigated in [39].…”

Section: Related Work and Our Contributionsmentioning

confidence: 99%

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A Bregman-Kaczmarz method for nonlinear systems of equations

Gower¹,

Lorenz²,

Winkler³

2023

Preprint

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We propose a new randomized method for solving systems of nonlinear equations, which can find sparse solutions or solutions under certain simple constraints. The scheme only takes gradients of component functions and uses Bregman projections onto the solution space of a Newton equation. In the special case of euclidean projections, the method is known as nonlinear Kaczmarz method. Furthermore if the component functions are nonnegative, we are in the setting of optimization under the interpolation assumption and the method reduces to SGD with the recently proposed stochastic Polyak step size. For general Bregman projections, our method is a stochastic mirror descent with a novel adaptive step size. We prove that in the convex setting each iteration of our method results in a smaller Bregman distance to exact solutions as compared to the standard Polyak step. Our generalization to Bregman projections comes with the price that a convex one-dimensional optimization problem needs to be solved in each iteration. This can typically be done with globalized Newton iterations. Convergence is proved in two classical settings of nonlinearity: for convex nonnegative functions and locally for functions which fulfill the tangential cone condition. Finally, we show examples in which the proposed method outperforms similar methods with the same memory requirements.

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“…Note that, although the Bregman projection x k+1 is unique, t k,ϕ might not be unique. If (10) is not fulfilled, we define an update inspired from [23] by setting…”

Section: Realizations Of the Methodsmentioning

confidence: 99%

Section: Convergencementioning

confidence: 98%

See 1 more Smart Citation

A Bregman-Kaczmarz method for nonlinear systems of equations

Gower¹,

Lorenz²,

Winkler³

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…where F x δ n is the Fréchet derivative of F at x δ n . Assuming the nth iterates ξ δ n , x δ n are available, by employing the Landweber iteration in [9] to (4), the inexact Newton-Landweber iteration in [6] generates the inner iterates {(ξ δ n,k , x δ n,k )} by…”

Section: Introductionmentioning

confidence: 99%

Generalized Inexact Newton-Landweber Iteration for Possibly Non-Smooth Inverse Problems in Banach Spaces

Wang

2023

Mathematics

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In this paper, we consider a generalized inexact Newton-Landweber iteration to solve nonlinear ill-posed inverse problems in Banach spaces, where the forward operator might not be Gâteaux differentiable. The method is designed with non-smooth convex penalty terms, including L1-like and total variation-like penalty functionals, to capture special features of solutions such as sparsity and piecewise constancy. Furthermore, the inaccurate inner solver is incorporated into the minimization problem in each iteration step. Under some assumptions, based on ε-subdifferential, we establish the convergence analysis of the proposed method. Finally, some numerical simulations are provided to illustrate the effectiveness of the method for solving both smooth and non-smooth nonlinear inverse problems.

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“…Some key references for gradient, i.e., Landweber type iterative methods are [ 5 ] on projected Landweber iteration for linear inverse problems, [ 9 ] on (unconstrained) nonlinear Landweber iteration, and more recently [ 23 ] on gradient type methods under very general conditions on the cost function or the forward operator, respectively. Extensions with a penalty term (also allowing for the incorporation of constraints) for linear inverse problems can be found in [ 4 ]; For nonlinear problems we also point to [ 13 , 33 ], however, they do not seem to be applicable to constrained problems, since the penalty term is assumed to be p -convex and thus cannot be an indicator function.…”

Section: Introductionmentioning

confidence: 99%

Iterative regularization for constrained minimization formulations of nonlinear inverse problems

Kaltenbacher

Huynh

2021

Comput Optim Appl

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In this paper we study the formulation of inverse problems as constrained minimization problems and their iterative solution by gradient or Newton type methods. We carry out a convergence analysis in the sense of regularization methods and discuss applicability to the problem of identifying the spatially varying diffusivity in an elliptic PDE from different sets of observations. Among these is a novel hybrid imaging technology known as impedance acoustic tomography, for which we provide numerical experiments.

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Iterative regularization with a general penalty term—theory and application toL¹andTVregularization

Cited by 39 publications

References 30 publications

A Bregman-Kaczmarz method for nonlinear systems of equations

A Bregman-Kaczmarz method for nonlinear systems of equations

Generalized Inexact Newton-Landweber Iteration for Possibly Non-Smooth Inverse Problems in Banach Spaces

Iterative regularization for constrained minimization formulations of nonlinear inverse problems

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