Related papers: On the convergence of spectral deferred correction…

This paper is concerned with the adaptive numerical treatment of stochastic partial differential equations. Our method of choice is Rothe's method. We use the implicit Euler scheme for the time discretization. Consequently, in each step, an…

For many nonlinear physical systems, approximate solutions are pursued by conventional perturbation theory in powers of the non-linear terms. Unfortunately, this often produces divergent asymptotic series, collectively dismissed by Abel as…

We provide improved parallel approximation algorithms for the important class of packing and covering linear programs. In particular, we present new parallel $\epsilon$-approximate packing and covering solvers which run in…

In this paper, we revisit the backward Euler method for numerical approximations of random periodic solutions of semilinear SDEs with additive noise. Improved $L^{p}$-estimates of the random periodic solutions of the considered SDEs are…

The Obreshkov method is a single-step multi-derivative method used in the numerical solution of differential equations and has been used in recent years in efficient circuit simulation. It has been shown that it can be made of arbitrary…

Over the last few decades, the numerical methods for stochastic differential delay equations (SDDEs) have been investigated and developed by many scholars. Nevertheless, there is still little work to be completed. By virtue of the novel…

In this paper, we develop an optimization-based framework for solving coupled forward-backward stochastic differential equations. We introduce an integral-form objective function and prove its equivalence to the error between consecutive…

The accuracy of information transmission while solving domain decomposed problems is crucial to smooth transition of a solution around the interface/overlapping region. This paper describes a systematical study on an accuracy enhancing…

An alternative first step approximation based on subgrid artificial viscosity modeling (SAV) is proposed for defect-deferred correction method (DDC) for incompresible Navier-Stokes equation at high Reynolds number. This new approach not…

In recent work of Hairer, Hutzenthaler and Jentzen, see [9], a stochastic differential equation (SDE) with infinitely often differentiable and bounded coefficients was constructed such that the Monte Carlo Euler method for approximation of…

The spectral bundle method developed by Helmberg and Rendl is well-established for solving large-scale semidefinite programs (SDPs) in the dual form, especially when the SDPs admit $\textit{low-rank primal solutions}$. Under mild regularity…

Machine learned partial differential equation (PDE) solvers trade the reliability of standard numerical methods for potential gains in accuracy and/or speed. The only way for a solver to guarantee that it outputs the exact solution is to…

Stochastic gradient descent (SGD) is a workhorse algorithm for solving large-scale optimization problems in data science and machine learning. Understanding the convergence of SGD is hence of fundamental importance. In this work we examine…

Stellar convection poses two main gargantuan challenges for astrophysical fluid solvers: low-Mach number flows and minuscule perturbations over steeply stratified hydrostatic equilibria. Most methods exhibit excessive numerical diffusion…

Semi-implicit multilevel spectral deferred correction (SI-MLSDC) methods provide a promising approach for high-order time integration for nonlinear evolution equations including conservation laws. However, existing methods lack robustness…

In this article we investigate the numerical solution of a scalar semilinear stochastic delay differential equation (SDDE) where the linear instantaneous feedback and nonlinear delayed feedback terms are perturbed by a pair of standard…

We construct a high-order adaptive time stepping scheme for vesicle suspensions with viscosity contrast. The high-order accuracy is achieved using a spectral deferred correction (SDC) method, and adaptivity is achieved by estimating the…

Sampling from Diffusion Models can alternatively be seen as solving differential equations, where there is a challenge in balancing speed and image visual quality. ODE-based samplers offer rapid sampling time but reach a performance limit,…

The state-of-the-art methods for solving optimization problems in big dimensions are variants of randomized coordinate descent (RCD). In this paper we introduce a fundamentally new type of acceleration strategy for RCD based on the…

We study in this paper three variants of the high-order Discontinuous Galerkin (DG) method with Runge-Kutta (RK) time integration for the induction equation, analysing their ability to preserve the divergence free constraint of the magnetic…