Regularization of linear and nonlinear ill-posed problems by mollification
Walter Cedric Simo Tao Lee
备注：
This paper is a premature version of a work performed while the author was a PhD student under the supervision of Pierre Maréchal and Anne Vanhems, at the University of Toulouse, and it benefited from the directions provided by them. The paper has to be re-written
In this paper, we address the problem of approximating solutions of ill-posed problems using mollification. We quickly review existing mollification regularization methods and provide two new approximate solutions to a general ill-posed equation $T(f) =g$ where $T$ can be nonlinear. The regularized solutions we define extend the work of Bonnefond and Maréchal \cite{xapi}, and trace their origins in the variational formulation of mollification, which to the best of our knowledge, was first introduced by Lannes et al. \cite{lannes}. In addition to consistency results, for the first time, we provide some convergence rates for a mollification method defined through a variational formulation.
This paper addresses a backward heat conduction problem with fractional Laplacian and time-dependent coefficient in an unbounded domain. The problem models generalized diffusion processes and is well-known to be severely ill-posed. We investigate a simple and powerful variational regularization technique based on mollification. Under classical Sobolev smoothness conditions, we derive order-optimal convergence rates between the exact solution and regularized approximation in the practical case where both the data and the operator are noisy. Moreover, we propose an order-optimal a-posteriori par…
This paper focuses on the regularization of backward time-fractional diffusion problem on unbounded domain. This problem is well-known to be ill-posed, whence the need of a regularization method in order to recover stable approximate solution. For the problem under consideration, we present a unified framework of regularization which covers some techniques such as Fourier regularization [19], mollification [12] and approximate-inverse [7]. We investigate a regularization technique with two major advantages: the simplicity of computation of the regularized solution and the avoid of truncation o…
In this paper, we study a fractional-order variant of the asymptotical regularization method, called {\it Fractional Asymptotical Regularization (FAR)}, for solving linear ill-posed operator equations in a Hilbert space setting. We assign the method to the general linear regularization schema and prove that under certain smoothness assumptions, FAR with fractional order in the range $(1,2)$ yields an acceleration with respect to comparable order optimal regularization methods. Based on the one-step Adams-Moulton method, a novel iterative regularization scheme is developed for the numerical rea…
We consider a regularization concept for the solution of ill--posed operator equations, where the operator is composed of a continuous and a discontinuous operator. A particular application is level set regularization, where we develop a novel concept of minimizers. The proposed level set regularization is capable of handling changing topologies. A functional analytic framework explaining the splitting of topologies is given. The asymptotic limit of the level set regularization method is an evolution process, which is implemented numerically and the quality of the proposed algorithm is demonst…
Many inverse problems are concerned with the estimation of non-negative parameter functions. In this paper, in order to obtain non-negative stable approximate solutions to ill-posed linear operator equations in a Hilbert space setting, we develop two novel non-negativity preserving iterative regularization methods. They are based on fixed point iterations in combination with preconditioning ideas. In contrast to the projected Landweber iteration, for which only weak convergence can be shown for the regularized solution when the noise level tends to zero, the introduced regularization methods e…
In this paper, we consider the $α\| \cdot\|_{\ell_1}-β\| \cdot\|_{\ell_2}$ sparsity regularization with parameter $α\geqβ\geq0$ for nonlinear ill-posed inverse problems. We investigate the well-posedness of the regularization. Compared to the case where $α>β\geq0$, the results for the case $α=β\geq0$ are weaker due to the lack of coercivity and Radon-Riesz property of the regularization term. Under certain condition on the nonlinearity of $F$, we prove that every minimizer of $ α\| \cdot\|_{\ell_1}-β\| \cdot\|_{\ell_2}$ regularization is sparse. For the case $α>β\geq0$, if the exact solution i…
In this article we develop and analyze novel iterative regularization techniques for the solution of systems of nonlinear ill--posed operator equations. The basic idea consists in considering separately each equation of this system and incorporating a loping strategy. The first technique is a Kaczmarz-type method, equipped with a novel stopping criteria. The second method is obtained using an embedding strategy, and again a Kaczmarz-type approach. We prove well-posedness, stability and convergence of both methods.
Recently, inverse problems have attracted more and more attention in computational mathematics and become increasingly important in engineering applications. After the discretization, many of inverse problems are reduced to linear systems. Due to the typical ill-posedness of inverse problems, the reduced linear systems are often ill-posed, especially when their scales are large. This brings great computational difficulty. Particularly, a small perturbation in the right side of an ill-posed linear system may cause a dramatical change in the solution. Therefore, regularization methods should be …
This paper provides a new regularization method which is particularly suitable for linear exponentially ill-posed problems. Under logarithmic source conditions (which have a natural interpretation in terms of Sobolev spaces in the aforementioned context), concepts of qualifications as well as order optimal rates of convergence are presented. Optimality results under general source conditions expressed in terms of index functions are also studied. Finally, numerical experiments on three test problems attest the better performance of the new method compared to the well known Tikhonov method in i…
Risk Estimators for Choosing Regularization Parameters in Ill-Posed Problems - Properties and Limitations
University College London（伦敦大学学院）；Georg-August-Universität Göttingen（哥廷根大学）；University of Twente（特温特大学）；Ruhr-Universität Bochum（波鸿鲁尔大学）；Westfälische Wilhelms-Universität (WWU) Münster（明斯特大学）
In part I we introduced modified Landweber-Kaczmarz methods and have established a convergence analysis. In the present work we investigate three applications: an inverse problem related to thermoacoustic tomography, a nonlinear inverse problem for semiconductor equations, and a nonlinear problem in Schlieren tomography. Each application is considered in the framework established in the previous part. The novel algorithms show robustness, stability, computational efficiency and high accuracy.
This paper describes and compares some structure preserving techniques for the solution of linear discrete ill-posed problems with the t-product. A new randomized tensor singular value decomposition (R-tSVD) with a t-product is presented for low tubal rank tensor approximations. Regularization of linear inverse problems by truncated tensor eigenvalue decomposition (T-tEVD), truncated tSVD (T-tSVD), randomized T-tSVD (RT-tSVD), t-product Golub-Kahan bidiagonalization (tGKB) process, and t-product Lanczos (t-Lanczos) process are considered. A solution method that is based on reusing tensor Krylo…
Error Estimates for Arnoldo-Tikhonov Regularization for Ill-Posed Operator Equations
Institute for Industrial Mathematics, Kepler University（工业数学研究所，开普勒大学）；Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences（约翰·拉德纳计算与应用数学研究所，奥地利科学院）；Department of Mathematical Sciences, Kent State University（凯特州立大学数学科学系）
These lecture notes for a graduate class present the regularization theory for linear and nonlinear ill-posed operator equations in Hilbert spaces. Covered are the general framework of regularization methods and their analysis via spectral filters as well as the concrete examples of Tikhonov regularization, Landweber iteration, regularization by discretization for linear inverse problems. In the nonlinear setting, Tikhonov regularization and iterative regularization (Landweber, Levenberg-Marquardt, and iteratively regularized Gauß-Newton methods) are discussed. The necessary background from fu…
In this article we investigate the connection between regularization theory for inverse problems and dynamic programming theory. This is done by developing two new regularization methods, based on dynamic programming techniques. The aim of these methods is to obtain stable approximations to the solution of linear inverse ill-posed problems. We follow two different approaches and derive a continuous and a discrete regularization method. Regularization properties for both methods are proved as well as rates of convergence. A numerical benchmark problem concerning integral operators with convolut…
Shallow neural network yields regularization for ill-posed inverse problems
School of Mathematics and Statistics, Beijing Institute of Technology（北京理工大学数学与统计学院）；MSU-BIT-SMBU Joint Research Center of Applied Mathematics, Shenzhen MSU-BIT University（深圳北理莫斯科大学应用数学联合研究中心）
Two-parameter regularization of ill-posed spherical pseudo-differential equations in the space of continuous functions
Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University（中山大学）；Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences（奥地利科学院约翰·拉顿计算与应用数学研究所）；School of Mathematics and Statistics, University of New South Wales（新南威尔士大学数学与统计学院）
Beyond convergence rates: Exact recovery with Tikhonov regularization with sparsity constraints
TU Braunschweig, Institute for Analysis and Algebra（布伦瑞克工业大学，分析与代数研究所）；Zentrum für Technomathematik, Universität Bremen, Fachbereich Mathematik/Informatik（不来梅大学，技术与数学中心，数学/信息学系）
Regularization matrices for discrete ill-posed problems in several space-dimensions
University School（大学学校）；Geomathematics Key Laboratory of Sichuan, College of Management Science, Chengdu University of Technology（四川地球数学重点实验室，成都理工大学管理学院）；Department of Mathematics, SAPIENZA Università di Roma（罗马智慧大学数学系）；Department of Mathematical Sciences, Kent State University（肯特州立大学数理科学系）
Mathematical modeling of fluid flow in a porous medium is usually described by a continuity equation and a chosen constitutive law. The latter, depending on the problem at hand, may be a nonlinear relation between the fluid's pressure gradient and velocity. The actual shape of this relation is normally chosen at the outset of the problem, even though, in practice, the fluid may experience velocities outside of its range of applicability. We propose here an adaptive model, so that the most appropriate law is locally selected depending on the computed velocity. From the analytical point of view,…
We investigate continuous regularization methods for linear inverse problems of static and dynamic type. These methods are based on dynamic programming approaches for linear quadratic optimal control problems. We prove regularization properties and also obtain rates of convergence for our methods. A numerical example concerning a dynamical electrical impedance tomography (EIT) problem is used to illustrate the theoretical results.
