相关论文: Quantitative Estimates for the Finite Section Meth…
We describe a novel optimization method for finite sums (such as empirical risk minimization problems) building on the recently introduced SAGA method. Our method achieves an accelerated convergence rate on strongly convex smooth problems.…
This paper is concerned with superconvergence properties of a class of finite volume methods of arbitrary order over rectangular meshes. Our main result is to prove {\it 2k-conjecture}: at each vertex of the underlying rectangular mesh, the…
In this paper, we analyze vertex-centered finite volume method (FVM) of any order for elliptic equations on rectangular meshes. The novelty is a unified proof of the inf-sup condition, based on which, we show that the FVM approximation…
We introduce a new Partition of Unity Method for the numerical homogenization of elliptic partial differential equations with arbitrarily rough coefficients. We do not restrict to a particular ansatz space or the existence of a finite…
This article is a review on basic concepts and tools devoted to a posteriori error estimation for problems solved with the Finite Element Method. For the sake of simplicity and clarity, we mostly focus on linear elliptic diffusion problems,…
We study the rate of convergence of an explicit and an implicit-explicit finite difference scheme for linear stochastic integro-differential equations of parabolic type arising in non-linear filtering of jump-diffusion processes. We show…
Two-time-scale stochastic approximation is a popular iterative method for finding the solution of a system of two equations. Such methods have found broad applications in many areas, especially in machine learning and reinforcement…
We explore the connection between fractional order partial differential equations in two or more spatial dimensions with boundary integral operators to develop techniques that enable one to efficiently tackle the integral fractional…
A semi-local analysis of Newton's method for solving nonlinear inclusion problems in Banach space is presented in this paper. Under a affine majorant condition on the nonlinear function which is associated to the inclusion problem, the…
Semiclassical methods are essential in analyzing quantum mechanical systems. Although they generally produce approximate results, relatively rare potentials exist for which these methods are exact. Such intriguing potentials serve as…
Well known to the machine learning community, the random feature model is a parametric approximation to kernel interpolation or regression methods. It is typically used to approximate functions mapping a finite-dimensional input space to…
This paper deals with a general form of variational problems in Banach spaces which encompasses variational inequalities as well as minimization problems. We prove a characterization of local error bounds for the distance to the…
We study universal approximation of continuous functionals on compact subsets of products of Hilbert spaces. We prove that any such functional can be uniformly approximated by models that first take finitely many continuous linear…
A new heuristic method for the evaluation of definite integrals is presented. This method of brackets has its origin in methods developed for theevaluation of Feynman diagrams. We describe the operational rules and illustrate the method…
Optimization problems with the objective function in the form of weighted sum and linear equality constraints are considered. Given that the number of local cost functions can be large as well as the number of constraints, a stochastic…
We consider adaptive finite element methods for second-order elliptic PDEs, where the arising discrete systems are not solved exactly. For contractive iterative solvers, we formulate an adaptive algorithm which monitors and steers the…
We analyze a semi-implicit finite volume scheme for the Gray--Scott system, a model for pattern formation in chemical and biological media. We prove unconditional well-posedness of the fully discrete problem and establish qualitative…
We investigate the use of piecewise linear systems, whose coefficient matrix is a piecewise constant function of the solution itself. Such systems arise, for example, from the numerical solution of linear complementarity problems and in the…
We construct a finite element method for the numerical solution of a fractional porous medium equation on a bounded open Lipschitz polytopal domain $\Omega \subset \mathbb{R}^{d}$, where $d = 2$ or $3$. The pressure in the model is defined…
We apply two families of novel fractional $\theta$-methods, the FBT-$\theta$ and FBN-$\theta$ methods developed by the authors in previous work, to the fractional Cable model, in which the time direction is approximated by the fractional…