In this paper, we consider the minimization of a Tikhonov functional with an $\ell_1$ penalty for solving linear inverse problems with sparsity constraints. One of the many approaches used to solve this problem uses the Nemskii operator to transform the Tikhonov functional into one with an $\ell_2$ penalty term but a nonlinear operator. The transformed problem can then be analyzed and minimized using standard methods. However, by the nature of this transform, the resulting functional is only once continuously differentiable, which prohibits the use of second order methods. Hence, in this paper, we propose a different transformation, which leads to a twice differentiable functional that can now be minimized using efficient second order methods like Newton's method. We provide a convergence analysis of our proposed scheme, as well as a number of numerical results showing the usefulness of our proposed approach.
For the Tikhonov regularization of ill-posed nonlinear operator equations, convergence is studied in a Hilbert scale setting. We include the case of oversmoothing penalty terms, which means that the exact solution does not belong to the domain of definition of the considered penalty functional. In this case, we try to close a gap in the present theory, where Hölder-type convergence rates results have been proven under corresponding source conditions, but assertions on norm convergence of regularized solutions without source conditions are completely missing. A result of the present work is to …
We investigate Tikhonov regularization methods for nonlinear ill-posed problems in Banach spaces, where the penalty term is described by Bregman distances. We prove convergence and stability results. Moreover, using appropriate source conditions, we are able to derive rates of convergence in terms of Bregman distances. We also analyze an iterated Tikhonov method for nonlinear problems, where the penalization is given by an appropriate convex functional.
A global minimization algorithm for Tikhonov functionals with sparsity constraints
Jiaxing University（嘉兴学院）；Harbin Institute of Technology（哈尔滨工业大学）；Technische Universität Chemnitz（开姆尼茨工业大学）；Johannes Kepler University（约翰·开普勒大学）；Industrial Mathematics Institute（工业数学研究所）；Johann Radon Institute for Computational and Appied Mathematics（约翰·拉登计算与应用数学研究所）
An Analysis of Finite Element Approximation in Electrical Impedance Tomography
Center for Industrial Mathematics, University of Bremen（不来梅大学工业数学中心）；Department of Mathematics, University of California, Riverside（加州大学河滨分校数学系）；School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）
Given a compact linear operator $\K$, the (pseudo) inverse $\K^\dagger$ is usually substituted by a family of regularizing operators $\R_α$ which depends on $\K$ itself. Naturally, in the actual computation we are forced to approximate the true continuous operator $\K$ with a discrete operator $\K^{(n)}$ characterized by a finesses discretization parameter $n$, and obtaining then a discretized family of regularizing operators $\R_α^{(n)}$. In general, the numerical scheme applied to discretize $\K$ does not preserve, asymptotically, the full spectrum of $\K$. In the context of a generalized Ti…
Tikhonov regularization with square-norm penalty for linear forward operators has been studied extensively in the literature. However, the results on convergence theory are based on technical proofs and difficult to interpret. It is also often not clear how those results translate into the discrete, numerical setting. In this paper we present a new strategy to study the properties of a regularization method on the example of Tikhonov regularization. The technique is based on the observation that Tikhonov regularization approximates the unknown exact solution in the range of the adjoint of the …
This paper discusses weighted tensor Golub-Kahan-type bidiagonalization processes using the t-product. This product was introduced in [M. E. Kilmer and C. D. Martin, Factorization strategies for third order tensors, Linear Algebra Appl., 435 (2011), pp.~641--658]. A few steps of a bidiagonalization process with a weighted least squares norm are carried out to reduce a large-scale linear discrete ill-posed problem to a problem of small size. The weights are determined by symmetric positive definite (SPD) tensors. Tikhonov regularization is applied to the reduced problem. An algorithm for tensor…
Inverse initial data reconstruction for a memory convection-diffusion equation via Legendre spatial reduction and Tikhonov regularization
University of North Carolina at Charlotte（北卡罗来纳大学夏洛特分校）；International University - Vietnam National University（越南国立大学国际大学）；Texas A&M University（德克萨斯农工大学）
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）
Mathematical models describing the behavior of viscoelastic materials are often based on evolution equations that measure the change in stress depending on its material parameters such as stiffness, viscosity or relaxation time. In this article, we introduce a Maxwell-based rheological model, define the associated forward operator and the inverse problem in order to determine the number of Maxwell elements and the material parameters of the underlying viscoelastic material. We perform a relaxation experiment by applying a strain to the material and measure the generated stress. Since the measu…
Ensemble Kalman inversion (EKI) is a derivative-free optimization method that lies between the deterministic and the probabilistic approaches for inverse problems. EKI iterates the Kalman update of ensemble-based Kalman filters, whose ensemble converges to a minimizer of an objective function. EKI regularizes ill-posed problems by restricting the ensemble to the linear span of the initial ensemble, or by iterating regularization with early stopping. Another regularization approach for EKI, Tikhonov EKI, penalizes the objective function using the $l_2$ penalty term, preventing overfitting in th…
A crucial issue in two-dimensional Nuclear Magnetic Resonance (NMR) is the speed and accuracy of the data inversion. This paper proposes a multi-penalty method with locally adapted regularization parameters for fast and accurate inversion of 2DNMR data. The method solves an unconstrained optimization problem whose objective contains a data-fitting term, a single $L1$ penalty parameter and a multiple parameter $L2$ penalty. We propose an adaptation of the Fast Iterative Shrinkage and Thresholding (FISTA) method to solve the multi-penalty minimization problem, and an automatic procedure to compu…
This paper is concerned with the convergence of a series associated with a certain version of the convexification method. That version has been recently developed by the research group of the first author for solving coefficient inverse problems. The convexification method aims to construct a globally convex Tikhonov-like functional with a Carleman Weight Function in it. In the previous works the construction of the strictly convex weighted Tikhonov-like functional assumes a truncated Fourier series (i.e. a finite series instead of an infinite one) for a function generated by the total wave fi…
A Convergent ADMM Algorithm for Grain Boundary Energy Minimization
The Hong Kong University of Science and Technology（香港科技大学）；Shenzhen University（深圳大学）；HKUST Shenzhen-Hong Kong Collaborative Innovation Research Institute（港深创新研究院）
In this article, we are concerned with the analysis on the numerical reconstruction of the spatial component in the source term of a time-fractional diffusion equation. This ill-posed problem is solved through a stabilized nonlinear minimization system by an appropriately selected Tikhonov regularization. The existence and the stability of the optimization system are demonstrated. The nonlinear optimization problem is approximated by a fully discrete scheme, whose convergence is established under a novel result verified in this study that the $H^1$-norm of the solution to the discrete forward …
Total Generalized Variation (TGV) has recently been introduced as penalty functional for modelling images with edges as well as smooth variations. It can be interpreted as a "sparse" penalization of optimal balancing from the first up to the $k$-th distributional derivative and leads to desirable results when applied to image denoising, i.e., $L^2$-fitting with TGV penalty. The present paper studies TGV of second order in the context of solving ill-posed linear inverse problems. Existence and stability for solutions of Tikhonov-functional minimization with respect to the data is shown and appl…
Total Variation, Adaptive Total Variation and Nonconvex Smoothly Clipped Absolute Deviation Penalty for Denoising Blocky Images
School of Computing Sciences, VIT University, Vellore, TN, India（计算机科学学院，VIT大学，Vellore，印度）；Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, 637371（数学科学系，南洋理工大学，新加坡）
We propose two variants of Newton method for solving unconstrained minimization problem. Our method leverages optimization techniques such as penalty and augmented Lagrangian method to generate novel variants of the Newton method namely the Penalty Newton method and the Augmented Newton method. In doing so, we recover several well-known existing Newton method variants such as Damped Newton, Levenberg, and Levenberg-Marquardt methods as special cases. Moreover, the proposed Augmented Newton method can be interpreted as Newton method with adaptive heavy ball momentum. We provide global convergen…
