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In this paper, we study a novel second-order energy stable Backward Differentiation Formula (BDF) finite difference scheme for the epitaxial thin film equation with slope selection (SS). One major challenge for the higher oder in time…
In this paper we consider an initial-boundary value problem with a Caputo time derivative of order $\alpha\in(0,1)$. The solution typically exhibits a weak singularity near the initial time and this causes a reduction in the orders of…
In this article, mixed finite element methods are discussed for a class of hyperbolic integro-differential equations (HIDEs). Based on a modification of the nonstandard energy formulation of Baker, both semidiscrete and completely discrete…
We study a second order BDF (Backward Differentiation Formula) scheme for the numerical approximation of parabolic HJB (Hamilton-Jacobi-Bellman) equations. The scheme under consideration is implicit, non-monotone, and second order accurate…
In this paper the numerical approximation of stochastic differential equations satisfying a global monotonicity condition is studied. The strong rate of convergence with respect to the mean square norm is determined to be $\frac{1}{2}$ for…
The Feynman-Kac equation governs the distribution of the statistical observable -- functional, having wide applications in almost all disciplines. After overcoming challenges from the time-space coupled nonlocal operator and the possible…
We develop a multilevel approach to compute approximate solutions to backward differential equations (BSDEs). The fully implementable algorithm of our multilevel scheme constructs sequential martingale control variates along a sequence of…
We introduce an inexact variant of Stochastic Mirror Descent (SMD), called Inexact Stochastic Mirror Descent (ISMD), to solve nonlinear two-stage stochastic programs where the second stage problem has linear and nonlinear coupling…
This paper establishes and analyzes a second-order accurate numerical scheme for the nonlinear partial integrodifferential equation with a weakly singular kernel. In the time direction, we apply the Crank-Nicolson method for the time…
The parareal algorithm is a powerful parallel-in-time integration method that accelerates the numerical solution of evolution equations by iteratively combining a fine propagator and a coarse propagator. Although the convergence of the…
Scale-resolving simulations of high Reynolds number incompressible flows are often limited by the Courant-Friedrichs-Lewy (CFL) stability restriction imposed by explicit time-stepping schemes, resulting in small time step sizes and long…
We consider the problem of learning the optimal policy for infinite-horizon Markov decision processes (MDPs). For this purpose, some variant of Stochastic Mirror Descent is proposed for convex programming problems with Lipschitz-continuous…
High-dimensional parabolic partial integro-differential equations (PIDEs) appear in many applications in insurance and finance. Existing numerical methods suffer from the curse of dimensionality or provide solutions only for a given…
In this paper we propose a numerical scheme for the class of backward doubly stochastic (BDSDEs) with possible path-dependent terminal values. We prove that our scheme converge in the strong $L^2$-sense and derive its rate of convergence.…
In numerical time-integration with implicit-explicit (IMEX) methods, a within-step adaptable decomposition called residual balanced decomposition is introduced. With this decomposition, the requirement of a small enough residual in the…
The numerical analysis of stochastic time fractional evolution equations presents considerable challenges due to the limited regularity of the model caused by the nonlocal operator and the presence of noise. The existing time-stepping…
We present a new approach to parallelization of the first-order backward difference discretization (BDF1) of the time derivative in partial differential equations, such as the nonlinear heat and viscous Burgers equations. The time…
In this work, we study the numerical approximation of a class of singular fully coupled forward backward stochastic differential equations. These equations have a degenerate forward component and non-smooth terminal condition. They are…
In this paper we consider fully discrete approximations with inf-sup stable mixed finite element methods in space to approximate the Navier-Stokes equations. A continuous downscaling data assimilation algorithm is analyzed in which…
We propose a variational form of the BDF2 method as an alternative to the commonly used minimizing movement scheme for the time-discrete approximation of gradient flows in abstract metric spaces. Assuming uniform semi-convexity --- but no…