Related papers: Minimal positive stencils in meshfree finite diffe…
A method is introduced for the construction of meshless discretization schemes which preserve Lie symmetries of the differential equations that these schemes approximate. The method exploits the fact that equivariant moving frames provide a…
The Fast Multipole Method (FMM) provides a highly efficient computational tool for solving constant coefficient partial differential equations (e.g. the Poisson equation) on infinite domains. The solution to such an equation is given as the…
We prove lower bounds on the complexity of finding $\epsilon$-stationary points (points $x$ such that $\|\nabla f(x)\| \le \epsilon$) of smooth, high-dimensional, and potentially non-convex functions $f$. We consider oracle-based complexity…
This paper presents a unified Least-Squares framework for solving nonlinear partial differential equations by recasting the governing system as a residual minimisation problem. A Least-Squares functional is formulated and the corresponding…
We survey structure-preserving discretizations of minimal surfaces in Euclidean space. Our focus is on a discretization defined via parallel face offsets of polyhedral surfaces, which naturally leads to a notion of vanishing mean curvature…
We prove that the finite-difference based derivative-free descent (FD-DFD) methods have a capability to find the global minima for a class of multiple minima problems. Our main result shows that, for a class of multiple minima objectives…
An important task in machine learning and statistics is the approximation of a probability measure by an empirical measure supported on a discrete point set. Stein Points are a class of algorithms for this task, which proceed by…
Finite difference based micromagnetic simulations are a powerful tool for the computational investigation of magnetic structures. In this paper, we demonstrate how the discretization of continuous micromagnetic equations introduces a…
In this paper, we consider the problem of minimizing a smooth function, given as finite sum of black-box functions, over a convex set. In order to advantageously exploit the structure of the problem, for instance when the terms of the…
This paper investigates a discretization scheme for mean curvature motion on point cloud varifolds with particular emphasis on singular evolutions. To define the varifold a local covariance analysis is applied to compute an approximate…
In the present work, a novel particle merging scheme is proposed for PIC-DSMC simulations, based on the solution of a Non-negative Least Squares problem. The merging algorithm conserves arbitrary moments of the velocity distribution…
Solutions of partial differential equations (PDEs) on manifolds have provided important applications in different fields in science and engineering. Existing methods are majorly based on discretization of manifolds as implicit functions,…
The strong-form asymmetric kernel-based collocation method, commonly referred to as the Kansa method, is easy to implement and hence is widely used for solving engineering problems and partial differential equations despite the lack of…
We develop a new spatial semidiscrete multiscale method based upon the edge multiscale methods to solve semilinear parabolic problems with heterogeneous coefficients and smooth initial data. This method allows for a cheap spatial…
Standard approaches to stochastic gradient estimation, with only noisy black-box function evaluations, use the finite-difference method or its variants. While natural, it is open to our knowledge whether their statistical accuracy is the…
We propose a new fictitious domain finite element method, well suited for elliptic problems posed in a domain given by a level-set function without requiring a mesh fitting the boundary. To impose the Dirichlet boundary conditions, we…
Motivated by the need for efficient estimation of conditional expectations, we consider a least-squares function approximation problem with heavily polluted data. Existing methods that are effective in the small-noise regime are suboptimal…
We analyze, from the viewpoint of positivity preservation, certain discretizations of a fundamental partial differential equation, the one-dimensional advection equation with periodic boundary condition. The full discretization is obtained…
Steady and unsteady Poisson and Stokes equations are solved using mesh dependent Finite Element Method and meshless Radial Basis Function Collocation Method to compare the performances of these two numerical techniques across several…
Several results on constrained spline smoothing are obtained. In particular, we establish a general result, showing how one can constructively smooth any monotone or convex piecewise polynomial function (ppf) (or any $q$-monotone ppf,…