Institut für Numerische und Angewandte Mathematik, Westfälische Wilhelms-Universität (WWU) Münster（应用与数值数学研究所，明斯特西法利西亚·威廉大学）；Department of Mathematics and Statistics, University of Helsinki（赫尔辛基大学数学与统计系）；Cantab Capital institute, Faculty of Mathematics, University of Cambridge（剑桥大学数学学院坎塔资本研究所）
In this paper we consider variational regularization methods for inverse problems with large noise that is in general unbounded in the image space of the forward operator. We introduce a Banach space setting that allows to define a reasonable notion of solutions for more general noise in a larger space provided one has sufficient mapping properties of the forward operators.
A key observation, which guides us through the subsequent analysis, is that such a general noise model can be understood with the same setting as approximate source conditions (while a standard model of bounded noise is related directly to classical source conditions). Based on this insight we obtain a quite general existence result for regularized variational problems and derive error estimates in terms of Bregman distances. The latter are specialized for the particularly important cases of one- and p-homogeneous regularization functionals.
As a natural further step we study stochastic noise models and in particular white noise, for which we derive error estimates in terms of the expectation of the Bregman distance. The finiteness of certain expectations leads to a novel class of abstract smoothness conditions on the forward operator, which can be easily interpreted in the Hilbert space case. We finally exemplify the approach and in particular the conditions for popular examples of regularization functionals given by squared norm, Besov norm and total variation, respectively.
Friedrich-Alexander Universität Erlangen-Nürnberg（埃尔朗根-纽伦堡大学）
This paper discusses basic results and recent developments on variational regularization methods, as developed for inverse problems. In a typical setup we review basic properties needed to obtain a convergent regularization scheme and further discuss the derivation of quantitative estimates respectively needed ingredients such as Bregman distances for convex functionals. In addition to the approach developed for inverse problems we will also discuss variational regularization in machine learning and work out some connections to the classical regularization theory. In particular we will discuss…
In the present work, we discuss variational regularization for ill-posed nonlinear problems with focus on an oversmoothing penalty term. This means in our model that the searched-for solution of the considered nonlinear operator equation does not belong to the domain of definition of the penalty functional. In the past years, such variational regularization has been investigated comprehensively in Hilbert scales. Our present study tries to continue and to extend those investigations to Banach scales. This new study includes convergence rates results for a priori choices of the regularization p…
We present a new approach to convergence rate results for variational regularization. Avoiding Bregman distances and using image space approximation rates as source conditions we prove a nearly minimax theorem showing that the modulus of continuity is an upper bound on the reconstruction error up to a constant. Applied to Besov space regularization we obtain convergence rate results for $0,2,q$- and $0,p,p$-penalties without restrictions on $p,q\in (1,\infty).$ Finally we prove equivalence of Hölder-type variational source conditions, bounds on the defect of the Tikhonov functional, and image …
Heuristic parameter-choice rules for convex variational regularization based on error estimates
Zentrum für Technomathematik, Universität Bremen（不来梅大学技术数学中心）；Institute for Analysis and Algebra, Pockelsstr. 14, D-38118 Braunschweig, Germany（不伦瑞克工业大学分析与代数研究所）
We consider a family of variational regularization functionals for a generic inverse problem, where the data fidelity and regularization term are given by powers of a Hilbert norm and an absolutely one-homogeneous functional, respectively, and the regularization parameter is interpreted as artificial time. We investigate the small and large time behavior of the associated solution paths and, in particular, prove finite extinction time for a large class of functionals. Depending on the powers, we also show that the solution paths are of bounded variation or even Lipschitz continuous. In additio…
Many methods for processing scalar and vector valued images, volumes and other data in the context of inverse problems are based on variational formulations. Such formulations require appropriate regularization functionals that model expected properties of the object to reconstruct. Prominent examples of regularization functionals in a vector-space context are the total variation (TV) and the Mumford-Shah functional, as well as higher-order schemes such as total generalized variation models. Driven by applications where the signals or data live in nonlinear manifolds, there has been quite some…
Variational Regularization of Inverse Problems for Manifold-Valued Data
Image Analysis and Learning Group, Interdisciplinary Center for Scientific Computing, Universität Heidelberg, Germany（图像分析与学习组，跨学科科学计算中心，海德堡大学，德国）；Department of Mathematics and Natural Sciences, Hochschule Darmstadt, and Institute of Computational Biology, Helmholtz Zentrum München, Germany（数学与自然科学系，达姆施塔特应用科学大学，以及计算生物学研究所，海德堡研究中心，德国）
Ground States and Singular Vectors of Convex Variational Regularization Methods
Magnetic Resonance Research Centre, Department of Chemical Engineering and Biotechnology（磁共振研究中心，化学工程与生物系）；Westfälische Wilhelms-Universität Münster, Institut für Numerische und Angewandte Mathematik（明斯特西法利希威廉大学，数值与应用数学研究所）
Variational regularization of complex deautoconvolution and phase retrieval in ultrashort laser pulse characterization
J. Radon Institute for Computational and Applied Mathematics（J. Radon计算与应用数学研究所）；Faculty of Mathematics, TU Chemnitz（开姆尼茨工业大学数学学院）；Max-Born-Institute（马克斯·玻恩研究所）
Ordinary differential equations for regularized variational problems involving semi-discrete optimal transport
Université Paris-Saclay（巴黎萨克雷大学）；CNRS（法国国家科学研究中心）；Inria Saclay（法国国家信息与自动化研究所萨克莱分中心）；Institut Universitaire de France (IUF)（法兰西学院）；University of Alberta（阿尔伯塔大学）
Discretization of variational regularization in Banach spaces
Universitat Pompeu Fabra（庞培法布拉大学）；Johannes Kepler University（约翰内斯·开普勒大学）；University of Vienna（维也纳大学）；Radon Institute of Computational and Applied Mathematics（拉德纳计算与应用数学研究所）
Focus of this work is solving a non-smooth constraint minimization problem by a primal-dual splitting algorithm involving proximity operators. The problem is penalized by the Bregman divergence associated with the non-smooth total variation (TV) functional. We analyse two aspects: Firstly, the convergence of the regularized solution of the minimization problem to the minimum norm solution. Second, the convergence of the iteratively regularized minimizer to the minimum norm solution by a primal-dual algorithm. For both aspects, we use the assumption of a variational source condition (VSC). This…
Reconstructing the structure of the soil using non-invasive techniques is a very relevant problem in many scientific fields, like geophysics and archaeology. This can be done, for instance, with the aid of Frequency Domain Electromagnetic (FDEM) induction devices. Inverting FDEM data is a very challenging inverse problem, as the problem is extremely ill-posed, i.e., sensible to the presence of noise in the measured data, and non-linear. Regularization methods substitute the original ill-posed problem with a well-posed one whose solution is an accurate approximation of the desired one. In this …
Non-uniqueness and instability are characteristic features of image reconstruction processes. As a result, it is necessary to develop regularization methods that can be used to compute reliable approximate solutions. A regularization method provides of a family of stable reconstructions that converge to an exact solution of the noise-free problem as the noise level tends to zero. The standard regularization technique is defined by variational image reconstruction, which minimizes a data discrepancy augmented by a regularizer. The actual numerical implementation makes use of iterative methods, …
Lifting methods allow to transform hard variational problems such as segmentation and optical flow estimation into convex problems in a suitable higher-dimensional space. The lifted models can then be efficiently solved to a global optimum, which allows to find approximate global minimizers of the original problem. Recently, these techniques have also been applied to problems with values in a manifold. We provide a review of such methods in a refined framework based on a finite element discretization of the range, which extends the concept of sublabel-accurate lifting to manifolds. We also gen…
CMAP, CNRS, École Polytechnique, Institut Polytechnique de Paris（巴黎综合理工学院综合理工学院应用数学中心、法国国家科学研究中心）；University of Bologna（博洛尼亚大学）
In this paper we present a new regularization term for variational image restoration which can be regarded as a space-variant anisotropic extension of the classical isotropic Total Variation (TV) regularizer. The proposed regularizer comes from the statistical assumption that the gradients of the target image distribute locally according to space-variant bivariate Laplacian distributions. The highly flexible variational structure of the corresponding regularizer encodes several free parameters which hold the potential for faithfully modelling the local geometry in the image and describing loca…
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…
A Nonlinear Variational Approach to Motion-Corrected Reconstruction of Density Images
University of Münster（明斯特大学）；Institute for Computational and Applied Mathematics, University of Münster（明斯特大学计算与应用数学研究所）；Cells in Motion Cluster of Excellence, University of Münster（明斯特大学运动细胞卓越集群）；University of Lübeck（吕贝克大学）；Institute of Mathematics and Image Computing, University of Lübeck（吕贝克大学数学与图像计算研究所）；Fraunhofer MEVIS Project Group Image Registration（弗劳恩霍夫 MEVIS 图像配准项目组）
A main drawback of classical Tikhonov regularization is that often the parameters required to apply theoretical results, e.g., the smoothness of the sought-after solution and the noise level, are unknown in practice. In this paper we investigate in new detail the residuals in Tikhonov regularization viewed as functions of the regularization parameter. We show that the residual carries, with some restrictions, the information on both the unknown solution and the noise level. By calculating approximate solutions for a large range of regularization parameters, we can extract both parameters from …
Choosing an appropriate regularization term is necessary to obtain a meaningful solution to an ill-posed linear inverse problem contaminated with measurement errors or noise. The $\ell_p$ norm covers a wide range of choices for the regularization term since its behavior critically depends on the choice of $p$ and since it can easily be combined with a suitable regularization matrix. We develop an efficient algorithm that simultaneously determines the regularization parameter and corresponding $\ell_p$ regularized solution such that the discrepancy principle is satisfied. We project the problem…
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 prov…
Generalized Variational Source Condition Associated with the Bregman Distance-I: Verification of the Variational Source Condition and Stability of the Total Error Estimation
Institute for Numerical and Applied Mathematics, University of Göttingen（哥廷根大学应用数值数学研究所）
