Related papers: Locally-averaged McCormick relaxations for discret…
McCormick envelopes are a standard tool for deriving convex relaxations of optimization problems that involve polynomial terms. Such McCormick relaxations provide lower bounds, for example, in branch-and-bound procedures for mixed-integer…
In this work we consider the regularization of a supervised learning problem by partial differential equations (PDEs) and derive error bounds for the obtained approximation in terms of a PDE error term and a data error term. Assuming that…
We consider the ill-posed inverse problem of identifying a nonlinearity in a time-dependent PDE model. The nonlinearity is approximated by a neural network, and needs to be determined alongside other unknown physical parameters and the…
This paper is concerned with approximations and related discretization error estimates for the normal derivatives of solutions of linear elliptic partial differential equations. In order to illustrate the ideas, we consider the Poisson…
This paper studies an optimal control problem governed by a semilinear elliptic equation, in which the control acts in a multiplicative or bilinear way as the reaction coefficient of the equation. We focus on the numerical discretization of…
We consider the inverse conductivity problem with discontinuous conductivities. We show in a rigorous way, by a convergence analysis, that one can construct a completely discrete minimization problem whose solution is a good approximation…
In this paper we investigate an adaptive discretization strategy for ill-posed linear prob- lems combined with a regularization from a class of semiiterative methods. We show that such a discretization approach in combination with a…
We present a new approach for computing approximate global minimizers to a large class of non-local pairwise interaction problems defined over probability distributions. The approach predicts candidate global minimizers, with a recovery…
The problem of the logarithmic discretization of an arbitrary positive function (such as the density of states) is studied in general terms. Logarithmic discretization has arbitrary high resolution around some chosen point (such as Fermi…
We present a method for finding lower bounds on the global infima of integral variational problems, wherein $\int_\Omega f(x,u(x),\nabla u(x)){\rm d}x$ is minimized over functions $u\colon\Omega\subset\mathbb{R}^n\to\mathbb{R}^m$ satisfying…
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 consider linear inverse problems under white noise. These types of problems can be tackled with, e.g., iterative regularisation methods and the main challenge is to determine a suitable stopping index for the iteration. Convergence…
PDE-constrained optimal control problems require regularisation to ensure well-posedness, introducing small perturbations that make the solutions challenging to approximate accurately. We propose a finite element approach that couples both…
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…
This paper is concerned with finite element error estimates for Neumann boundary control problems posed on convex and polyhedral domains. Different discretization concepts are considered and for each optimal discretization error estimates…
Binary optimization is a central problem in mathematical optimization and its applications are abundant. To solve this problem, we propose a new class of continuous optimization techniques which is based on Mathematical Programming with…
The numerical solution of partial differential equations (PDEs) is challenging because of the need to resolve spatiotemporal features over wide length and timescales. Often, it is computationally intractable to resolve the finest features…
In this paper we the formulation of inverse problems as constrained minimization problems and their iterative solution by gradient or Newton type. We carry out a convergence analysis in the sense of regularization methods and discuss…
This paper explores a fully discrete approximation for a nonlinear hyperbolic PDE-constrained optimization problem (P) with applications in acoustic full waveform inversion. The optimization problem is primarily complicated by the…
This paper explores the role of regularization in data-driven predictive control (DDPC) through the lens of convex relaxation. Using a bi-level optimization framework, we model system identification as an inner problem and predictive…