Related papers: Notes on the discretization of TV-norm regularized…
We consider a pointwise tracking optimal control problem for a semilinear elliptic partial differential equation. We derive the existence of optimal solutions and analyze first and, necessary and sufficient, second order optimality…
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…
Ill-posed linear inverse problems (ILIP), such as restoration and reconstruction, are a core topic of signal/image processing. A standard approach to deal with ILIP uses a constrained optimization problem, where a regularization function is…
Inverse problems are ubiquitous in science and engineering. Many of these are naturally formulated as a PDE-constrained optimization problem. These non-linear, large-scale, constrained optimization problems know many challenges, of which…
The formulation of norms on continuous-domain Banach spaces with exact pixel-based discretization is advantageous for solving inverse problems (IPs). In this paper, we investigate a new regularization that is a convex combination of a TV…
We consider inverse problems with large null spaces, which arise in important applications such as in inverse ECG and EEG procedures. Standard regularization methods typically produce solutions in or near the orthogonal complement of the…
We consider a control-constrained optimal control problem subject to time-harmonic Maxwell's equations; the control variable belongs to a finite-dimensional set and enters the state equation as a coefficient. We derive existence of optimal…
We consider the variational discretization of a linear-quadratic optimal control problem with pointwise control and state constraints. In order to allow for a Fr\'echet smooth norm, the problem is reformulated by means of a reflexive…
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…
We consider inverse problems estimating distributed parameters from indirect noisy observations through discretization of continuum models described by partial differential or integral equations. It is well understood that the errors…
We present a parameter estimation method in Ordinary Differential Equation (ODE) models. Due to complex relationships between parameters and states the use of standard techniques such as nonlinear least squares can lead to the presence of…
Optimization problems with convex quadratic cost and polyhedral constraints are ubiquitous in signal processing, automatic control and decision-making. We consider here an enlarged problem class that allows to encode logical conditions and…
Iterative regularization exploits the implicit bias of an optimization algorithm to regularize ill-posed problems. Constructing algorithms with such built-in regularization mechanisms is a classic challenge in inverse problems but also in…
In this paper, we develop a numerical method for determining the potential in one and two dimensional fractional Calder\'{o}n problems with a single measurement. Finite difference scheme is employed to discretize the fractional Laplacian,…
Common computational problems, such as parameter estimation in dynamic models and PDE constrained optimization, require data fitting over a set of auxiliary parameters subject to physical constraints over an underlying state. Naive…
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…
In this paper, we propose a general approach for explicit a posteriori error representation for convex minimization problems using basic convex duality relations. Exploiting discrete orthogonality relations in the space of element-wise…
Block-coordinate descent (BCD) is a popular framework for large-scale regularized optimization problems with block-separable structure. Existing methods have several limitations. They often assume that subproblems can be solved exactly at…
A multi-convex optimization problem is one in which the variables can be partitioned into sets over which the problem is convex when the other variables are fixed. Multi-convex problems are generally solved approximately using variations on…
The choice of the parameter value for regularized inverse problems is critical to the results and remains a topic of interest. This article explores a criterion for selecting a good parameter value by maximizing the probability of the data,…