Related papers: Topological sensitivities via a Lagrangian approac…
This article proposes a new numerical algorithm for second order elliptic equations in non-divergence form. The new method is based on a discrete weak Hessian operator locally constructed by following the weak Galerkin strategy. The…
In insurance mathematics optimal control problems over an infinite time horizon arise when computing risk measures. Their solutions correspond to solutions of deterministic semilinear (degenerate) elliptic partial differential equations. In…
This paper is devoted to studying an augmented Lagrangian method for solving a class of manifold optimization problems, which have nonsmooth objective functions and nonlinear constraints. Under the constant positive linear dependence…
Semi-Lagrangian methods are numerical methods designed to find approximate solutions to particular time-dependent partial differential equations (PDEs) that describe the advection process. We propose semi-Lagrangian one-step methods for…
We study iterative finite element approximations for the numerical approximation of semilinear elliptic boundary value problems with monotone nonlinear reactions of subcritical growth. The focus of our contribution is on an optimal a priori…
This paper aims to devise an adaptive neural network basis method for numerically solving a second-order semilinear partial differential equation (PDE) with low-regular solutions in two/three dimensions. The method is obtained by combining…
In this paper, neural network approximation methods are developed for elliptic partial differential equations with multi-frequency solutions. Neural network work approximation methods have advantages over classical approaches in that they…
We present a priori and a posteriori error analysis of a high order hybridizable discontinuous Galerkin (HDG) method applied to a semi-linear elliptic problem posed on a piecewise curved, non polygonal domain. We approximate $\Omega$ by a…
This work is devoted to the development and analysis of a linearization algorithm for microscopic elliptic equations, with scaled degenerate production, posed in a perforated medium and constrained by the homogeneous Neumann-Dirichlet…
In this work, we investigate the numerical reconstruction of inclusions in a semilinear elliptic equation arising in the mathematical modeling of cardiac ischemia. We propose an adaptive finite element method for the resulting constrained…
For a general formulation of linearised hybrid inverse problems in impedance tomography, the qualitative properties of the solutions are analysed. Using an appropriate scalar pseudo-differential formulation, the problems are shown to permit…
Convolutional layers within graph neural networks operate by aggregating information about local neighbourhood structures; one common way to encode such substructures is through random walks. The distribution of these random walks evolves…
We put together a general framework to deal with elliptic and parabolic equations associated with (nonlinear) nonlocal (fractional order) operators. Many well-known nonlocal operators enter into our framework, and in addition one may…
Often, when solving forward, inverse or data assimilation problems, only a part of the solution is needed. As a model, we consider the stationary diffusion problem. We demonstrate an algorithm that can compute only a part or a functional of…
We present an introduction to periodic and stochastic homogenization of ellip- tic partial differential equations. The first part is concerned with the qualitative theory, which we present for equations with periodic and random coefficients…
In this paper we study the asymptotic behavior of the solutions of a class of nonlinear elliptic problems posed in a 2-dimensional domain that degenerates into a line segment (a thin domain) when a positive parameter $\varepsilon$ goes to…
This article develops a general framework for continuous deterministic data assimilation for semilinear parabolic equations by means of evolution equations. Introducing a nudged model driven by partial observations, the global…
In this short note, we investigate simultaneous recovery inverse problems for semilinear elliptic equations with partial data. The main technique is based on higher order linearization and monotonicity approaches. With these methods at…
We construct sub-grid scale models of incompressible fluids by considering expectations of semi-martingale Lagrangian particle trajectories. Our construction is based on the Lagrangian decomposition of flow maps into mean and fluctuation…
This paper studies a recovery task of finding a low multilinear-rank tensor that fulfills some linear constraints in the general settings, which has many applications in computer vision and graphics. This problem is named as the low…