Related papers: Modified equations for weak stochastic symplectic …

Transport in Hamiltonian systems with weak chaotic perturbations has been much studied in the past. In this paper, we introduce a new class of problems: transport in Hamiltonian systems with slowly changing phase space structure that are…

A computational tool for coarse-graining nonlinear systems of ordinary differential equations in time is discussed. Three illustrative model examples are worked out that demonstrate the range of capability of the method. This includes the…

We leverage commutative hypercomplex analysis to find closed-form solutions of some systems of stochastic differential equations. Specifically, we obtain necessary and sufficient conditions under which a system of stochastic differential…

Stochastic optimization methods have been hugely successful in making large-scale optimization problems feasible when computing the full gradient is computationally prohibitive. Using the theory of modified equations for numerical…

In recent work, the authors have developed a generic methodology for calibrating the noise in fluid dynamics stochastic partial differential equations where the stochasticity was introduced to parametrize subgrid-scale processes. The…

The long-time behavior of stochastic Hamilton-Jacobi equations is analyzed, including the stochastic mean curvature flow as a special case. In a variety of settings, new and sharpened results are obtained. Among them are (i) a…

This paper focuses on the numerical approximation of the linearized shallow water equations using hybridizable discontinuous Galerkin (HDG) methods, leveraging the Hamiltonian structure of the evolution system. First, we propose an…

This paper presents a new stochastic finite element method for computing structural stochastic responses. The method provides a new expansion of stochastic response and decouples the stochastic response into a combination of a series of…

This paper proposes a fully discrete method called the symplectic dG full discretization for stochastic Maxwell equations driven by additive noises, based on a stochastic symplectic method in time and a discontinuous Galerkin (dG) method…

Reduced basis methods are popular for approximately solving large and complex systems of differential equations. However, conventional reduced basis methods do not generally preserve conservation laws and symmetries of the full order model.…

Many stochastic time series can be described by a Langevin equation composed of a deterministic and a stochastic dynamical part. Such a stochastic process can be reconstructed by means of a recently introduced nonparametric method, thus…

Geometric numerical integration has recently been exploited to design symplectic accelerated optimization algorithms by simulating the Lagrangian and Hamiltonian systems from the variational framework introduced in Wibisono et al. In this…

This paper presents a stochastic model predictive control (SMPC) algorithm for linear systems subject to additive Gaussian mixture disturbances, with the goal of satisfying chance constraints. We focus on a special case where each Gaussian…

The effect of weak multiplicative colored noise on the dynamics of a Hamiltonian system is studied by means of asymptotic methods, in the vicinity of homoclinic or heteroclinic trajectories. A general expression for the probability of…

We propose an algorithm for optimizations in which the gradients contain stochastic noise. This arises, for example, in structural optimizations when computations of forces and stresses rely on methods involving Monte Carlo sampling, such…

A class of stochastic parabolic equations with singular potentials is analysed in the chaos expansion setting where the Wick product is used to give sense to the product of generalized stochastic processes. For the analysis of such…

Accelerated gradient methods have had significant impact in machine learning -- in particular the theoretical side of machine learning -- due to their ability to achieve oracle lower bounds. But their heuristic construction has hindered…

Symplectic integration methods based on operator splitting are well established in many branches of science. For Hamiltonian systems which split in more than two parts, symplectic methods of higher order have been studied in detail only for…

In this paper, a simulation-based method for the analysis and design of abstracted models for a stochastic hybrid system is proposed. The accuracy of a model is evaluated in terms of its capability to reproduce the system output for all the…

The paper concerns semidiscretizations in time of stochastic Maxwell equations driven by additive noise. We show that the equations admit physical properties and mathematical structures, including regularity, energy and divergence evolution…