Related papers: The Random Feature Method for Solving Interface Pr…
Kernel methods give powerful, flexible, and theoretically grounded approaches to solving many problems in machine learning. The standard approach, however, requires pairwise evaluations of a kernel function, which can lead to scalability…
Trustworthy machine learning necessitates meticulous regulation of model reliance on non-robust features. We propose a framework to delineate and regulate such features by attributing model predictions to the input. Within our approach,…
We consider a class of inverse problems where it is possible to aggregate the results of multiple experiments. This class includes problems where the forward model is the solution operator to linear ODEs or PDEs. The tremendous size of such…
We consider an open, bounded, simply connected (Lipschitz) domain in $\mathbb{R}^d$, which contains a closed polyhedral surface or polygonal contour, referred to as the interface. From this interface, forces are exerted in the normal…
The sparse representation of signals defined on Euclidean domains has been successfully applied in signal processing. Bringing the power of sparse representations to non-regular domains is still a challenge, but promising approaches have…
A sliding window algorithm receives a stream of symbols and has to output at each time instant a certain value which only depends on the last $n$ symbols. If the algorithm is randomized, then at each time instant it produces an incorrect…
We study unique continuation over an interface using a stabilized unfitted finite element method tailored to the conditional stability of the problem. The interface is approximated using an isoparametric transformation of the background…
We propose a high order unfitted finite element method for solving timeharmonic Maxwell interface problems. The unfitted finite element method is based on a mixed formulation in the discontinuous Galerkin framework on a Cartesian mesh with…
This paper introduces an abstract framework for randomized subspace correction methods for convex optimization, which unifies and generalizes a broad class of existing algorithms, including domain decomposition, multigrid, and block…
We consider the problem of solving linear least squares problems in a framework where only evaluations of the linear map are possible. We derive randomized methods that do not need any other matrix operations than forward evaluations,…
Kernel methods form a powerful, versatile, and theoretically-grounded unifying framework to solve nonlinear problems in signal processing and machine learning. The standard approach relies on the kernel trick to perform pairwise evaluations…
When modelling discontinuities (interfaces) using the finite element method, the standard approach is to use a conforming finite-element mesh in which the mesh matches the interfaces. However, this approach can prove cumbersome if the…
A new higher-order accurate method is proposed that combines the advantages of the classical $p$-version of the FEM on body-fitted meshes with embedded domain methods. A background mesh composed by higher-order Lagrange elements is used.…
In this paper, we study the stability and convergence of a decoupled and linearized mixed finite element method (FEM) for incompressible miscible displacement in a porous media whose permeability and porosity are discontinuous across some…
In the context of numerical solution of PDEs, dynamic mesh redistribution methods (r-adaptive methods) are an important procedure for increasing the resolution in regions of interest, without modifying the connectivity of the mesh. Key to…
In this paper, we propose the unfitted spectral element method for solving elliptic interface and corresponding eigenvalue problems. The novelty of the proposed method lies in its combination of the spectral accuracy of the spectral element…
Triangulated meshes have become ubiquitous discrete-surface representations. In this paper we address the problem of how to maintain the manifold properties of a surface while it undergoes strong deformations that may cause topological…
We propose joining a flexible mesh design with an integrated residual transcription in order to improve the accuracy of numerical solutions to optimal control problems. This approach is particularly useful when state or input trajectories…
Random feature approximation is arguably one of the most popular techniques to speed up kernel methods in large scale algorithms and provides a theoretical approach to the analysis of deep neural networks. We analyze generalization…
In this paper a novel method to solve the constant coefficient wave equation, subject to interface jump conditions, is presented. In general, such problems pose issues for standard finite difference solvers, as the inherent discontinuity in…