Related papers: A rational Arnoldi approach for ill-conditioned li…
Many challenging image processing tasks can be described by an ill-posed linear inverse problem: deblurring, deconvolution, inpainting, compressed sensing, and superresolution all lie in this framework. Traditional inverse problem solvers…
Iterative regularization is a classic idea in regularization theory, that has recently become popular in machine learning. On the one hand, it allows to design efficient algorithms controlling at the same time numerical and statistical…
Accurate determination of the regularization parameter in inverse problems still represents an analytical challenge, owing mainly to the considerable difficulty to separate the unknown noise from the signal. We present a new approach for…
This paper investigates using the conjugate gradient iterative solver for ill-posed problems. We show that preconditioner and Tikhonov-regularization work in conjunction. In particular when they employ the same symmetric positive…
Given a full column rank matrix $A \in \mathbb{R}^{m\times n}$ ($m\geq n$), we consider a special class of linear systems of the form $A^\top Ax=A^\top b+c$ with $x, c \in \mathbb{R}^{n}$ and $b \in \mathbb{R}^{m}$. The occurrence of $c$ in…
Recent work in deep learning has opened new possibilities for solving classical algorithmic tasks using end-to-end learned models. In this work, we investigate the fundamental task of solving linear systems, particularly those that are…
We consider the problem of robust matrix completion, which aims to recover a low rank matrix $L_*$ and a sparse matrix $S_*$ from incomplete observations of their sum $M=L_*+S_*\in\mathbb{R}^{m\times n}$. Algorithmically, the robust matrix…
The joint bidiagonalization process of a matrix pair $\{A,L\}$ can be used to develop iterative regularization algorithms for large scale ill-posed problems in general-form Tikhonov regularization…
This document introduces a strategy to solve linear optimization problems. The strategy is based on the bounding condition each constraint produces on each one of the problem's dimension. The solution of a linear optimization problem is…
We introduce subgradient-based Lavrentiev regularisation of the form \begin{equation*} \mathcal{A}(u) + \alpha \partial \mathcal{R}(u) \ni f^\delta \end{equation*} for linear and nonlinear ill-posed problems with monotone operators…
Overdetermined systems of first kind integral equations appear in many applications. When the right-hand side is discretized, the resulting finite-data problem is ill-posed and admits infinitely many solutions. We propose a numerical method…
Various versions of the Dynamical Systems Method (DSM) are proposed for solving linear ill-posed problems with bounded and unbounded operators. Convergence of the proposed methods is proved. Some new results concerning discrepancy principle…
A matrix approach to continuous iteration is proposed for general formal series. It leads, in particular, to an order{to{order iteration of the exponential function, and consequently to an algorithmic approach to tetration. Lower{order…
Algorithmic approach to the problem of linearization by point transformation of ordinary differential equation of arbitrary order is presented. Test-linearization is purely algorithmic.
We use convex relaxation techniques to provide a sequence of solutions to the matrix completion problem. Using the nuclear norm as a regularizer, we provide simple and very efficient algorithms for minimizing the reconstruction error…
Data assisted reconstruction algorithms, incorporating trained neural networks, are a novel paradigm for solving inverse problems. One approach is to first apply a classical reconstruction method and then apply a neural network to improve…
This work investigates a new approach to find closed form analytical approximate solution of linear initial value problems. Classical Bernoulli polynomials have been used to derive a finite set of orthonormal polynomials and a finite…
We propose a computational framework for computing low-rank approximations to the ensemble of solutions of a parametrized system of the form $A(\xi)x(\xi)+g(x(\xi))=b(\xi)$ for multiple parameter values. The central idea is to reinterpret…
A version of the Dynamical Systems Method (DSM) for solving ill-conditioned linear algebraic systems is studied in this paper. An {\it a priori} and {\it a posteriori} stopping rules are justified. An algorithm for computing the solution…
We investigate generalized versions of the Iteratively Regularized Landweber Method, initially introduced in [Appl. Math. Optim., 38(1):45-68, 1998], to address linear and nonlinear ill-posed problems. Our approach is inspired by the…