Related papers: The Estimation of Approximation Error using the In…
In this paper we present a convergence rate analysis of inexact variants of several randomized iterative methods. Among the methods studied are: stochastic gradient descent, stochastic Newton, stochastic proximal point and stochastic…
This paper develops a discrete data-driven approach for solving the inverse source problem of the wave equation with final time measurements. Focusing on the $L^2$-Tikhonov regularization method, we analyze its convergence under two…
This article deals with error estimates for the finite element approximation of variational normal derivatives and, as a consequence, error estimates for the finite element approximation of Dirichlet boundary control problems with energy…
This paper presents an error analysis of classical and learned Tikhonov regularization schemes for inverse problems. We first demonstrate, both theoretically and numerically, that using a fixed regularization parameter across varying noise…
This paper develops a probabilistic numerical method for solution of partial differential equations (PDEs) and studies application of that method to PDE-constrained inverse problems. This approach enables the solution of challenging inverse…
We study weighted Tikhonov regularization for large-scale linear discrete ill-posed problems with random noise. Under a polynomial upper-bound assumption on the generalized eigenvalues of the discrete forward operator, we derive stochastic…
We study gradient-based regularization methods for neural networks. We mainly focus on two regularization methods: the total variation and the Tikhonov regularization. Applying these methods is equivalent to using neural networks to solve…
Conditional stability estimates require additional regularization for obtaining stable approximate solutions if the validity area of such estimates is not completely known. In this context, we consider ill-posed nonlinear inverse problems…
A comprehensive convergence and stability analysis of some probabilistic numerical methods designed to solve Cauchy-type inverse problems is performed in this study. Such inverse problems aim at solving an elliptic partial differential…
We describe a framework in which is possible to develop and implement algorithms for the approximation of invariant measures of dynamical systems with a given bound on the error of the approximation. Our approach is based on a general…
One fundamental problem when solving inverse problems is how to find regularization parameters. This article considers solving this problem using data-driven bilevel optimization, i.e. we consider the adaptive learning of the regularization…
In this paper, we develop approximation error estimates as well as corresponding inverse inequalities for B-splines of maximum smoothness, where both the function to be approximated and the approximation error are measured in standard…
We present a novel approach for the inverse problem in electrical impedance tomography based on regularized quadratic regression. Our contribution introduces a new formulation for the forward model in the form of a nonlinear integral…
We consider a general linear parabolic problem with extended time boundary conditions (including initial value problems and periodic ones), and approximate it by the implicit Euler scheme in time and the Gradient Discretisation method in…
We propose a novel numerical inversion algorithm for the coefficients of parabolic partial differential equations, based on model reduction. The study is motivated by the application of controlled source electromagnetic exploration, where…
Blind inverse problems arise in many experimental settings where both the signal of interest and the forward operator are (partially) unknown. In this context, methods developed for the non-blind case cannot be adapted in a straightforward…
In this work we analyze the inverse problem of recovering the space-dependent potential coefficient in an elliptic / parabolic problem from distributed observation. We establish novel (weighted) conditional stability estimates under very…
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…
Second-order partial differential equations in non-divergence form are considered. Equations of this kind typically arise as subproblems for the solution of Hamilton-Jacobi-Bellman equations in the context of stochastic optimal control, or…
Inverse optimization refers to the inference of unknown parameters of an optimization problem based on knowledge of its optimal solutions. This paper considers inverse optimization in the setting where measurements of the optimal solutions…