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In this paper, it is shown that three-dimensional stochastic Maxwell equations with multiplicative noise are stochastic Hamiltonian partial differential equations possessing a geometric structure (i.e. stochastic mutli-symplectic…
For stochastic models with intractable likelihood functions, approximate Bayesian computation offers a way of approximating the true posterior through repeated comparisons of observations with simulated model outputs in terms of a small set…
A set of algorithms is presented for efficient numerical calculation of the time evolution of classical dynamical systems. Starting with a first approximation for solving the differential equations that has a "reversible" character, we show…
Invariant manifolds of unstable periodic orbits organize the dynamics of chaotic orbits in phase space. They provide insight into the mechanisms of transport and chaotic advection and have important applications in physical situations…
Many interesting functions arising in applications map into Riemannian manifolds. We present an algorithm, using the manifold exponential and logarithm, for approximating such functions. Our approach extends approximation techniques for…
A multiscale numerical method is proposed for the solution of semi-linear elliptic stochastic partial differential equations with localized uncertainties and non-linearities, the uncertainties being modeled by a set of random parameters. It…
In this paper, an analytic approximation method for highly nonlinear equations, namely the homotopy analysis method (HAM), is employed to solve some backward stochastic differential equations (BSDEs) and forward-backward stochastic…
Sticky diffusion models a Markovian particle experiencing reflection and temporary adhesion phenomena at the boundary. Numerous numerical schemes exist for approximating stopped or reflected stochastic differential equations (SDEs), but…
The calculation of physical quantities by lattice QCD simulations requires in some important cases the determination of the inverse of a very large matrix. In this article we describe how stochastic estimator methods can be applied to this…
Stochastic gradient methods (SGMs) are predominant approaches for solving stochastic optimization. On smooth nonconvex problems, a few acceleration techniques have been applied to improve the convergence rate of SGMs. However, little…
Using lattice approximations of Euclidean space, we develop a way to approximate stable processes that are represented by stochastic integrals over Euclidean space. Via a stable version of the Lindeberg-Feller Theorem we show that the…
In this paper, we propose an inertial forward backward splitting algorithm to compute a zero of the sum of two monotone operators, with one of the two operators being co-coercive. The algorithm is inspired by the accelerated gradient method…
In this paper we present the theoretical framework needed to justify the use of a kernel-based collocation method (meshfree approximation method) to estimate the solution of high-dimensional stochastic partial differential equations…
We examine a wide class of stochastic approximation algorithms for solving (stochastic) nonlinear problems on Riemannian manifolds. Such algorithms arise naturally in the study of Riemannian optimization, game theory and optimal transport,…
This paper investigates the two-dimensional stochastic steady-state Navier-Stokes(NS) equations with additive random noise. We introduce an innovative splitting method that decomposes the stochastic NS equations into a deterministic NS…
Stochastic differential equations projected onto manifolds occur in physics, chemistry, biology, engineering, nanotechnology and optimization, with interdisciplinary applications. Intrinsic coordinate stochastic equations on the manifold…
We address a three-tier data-driven approach to solve the inverse problem in complex systems modelling from spatio-temporal data produced by microscopic simulators using machine learning. In the first step, we exploit manifold learning and…
In the infectious disease literature, significant effort has been devoted to studying dynamics at a single scale. For example, compartmental models describing population-level dynamics are often formulated using differential equations. In…
In this paper we discuss backward stochastic differential equations with Markov chain noise, having continuous drivers. We obtain the existence of a solution which is possibly not unique. Moreover, we show there is a minimal solution for…
This article proposes and analyzes explicit and easily implementable temporal numerical approximation schemes for additive noise-driven stochastic partial differential equations (SPDEs) with polynomial nonlinearities such as, e.g.,…