Related papers: Adaptive Finite Element Method Assisted by Stochas…
The aim of this contribution is to study the particle dynamics in a storage ring under the influence of noise. Some simplified stochastic beam dynamics problems are treated by solving the corresponding Fokker-Planck equations numerically.
The Fokker-Plank-Kolmogorov (FPK) equation is an idealized model representing many stochastic systems commonly encountered in the analysis of stochastic structures as well as many other applications. Its solution thus provides an invaluable…
We analyze optimal complexity of adaptive finite element methods (AFEMs) for general second-order linear elliptic partial differential equations (PDEs) in the Lax-Milgram setting. To this end, we formulate an adaptive algorithm which steers…
Experiments show that when a monolayer of cells cultured on an elastic substrate is subject to a cyclic stretch, cells tend to re-orient either perpendicularly or at an oblique angle with respect to the main direction of the stretch. Due to…
The macroscopic behavior of dissipative stochastic partial differential equations usually can be described by a finite dimensional system. This article proves that a macroscopic reduced model may be constructed for stochastic…
Solving the Fokker-Planck equation for high-dimensional complex turbulent dynamical systems is an important and practical issue. However, most traditional methods suffer from the curse of dimensionality and have difficulties in capturing…
Intracellular biomolecular systems exhibit intrinsic stochasticity due to low molecular copy numbers, leading to multimodal probability distributions that play a crucial role in probabilistic differentiation and cellular decision-making.…
The normalization constraint on probability density poses a significant challenge for solving the Fokker-Planck equation. Normalizing Flow, an invertible generative model leverages the change of variables formula to ensure probability…
Functionally graded materials (FGMs) are two-phase composites with continuously changing microstructure adapted to performance requirements. Traditionally, the overall behavior of FGMs has been determined using local averaging techniques or…
This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. We discuss analytical and numerical methods for the solution of master equations, keeping our focus on…
Particle-based stochastic approximations of the Boltzmann equation are popular tools for simulations of non-equilibrium gas flows, for which the Navier-Stokes-Fourier equations fail to provide accurate description. However, these numerical…
In an experimental study of single enzyme reactions, it has been proposed that the rate constants of the enzymatic reactions fluctuate randomly, according to a given distribution. To quantify the uncertainty arising from random rate…
This article initiates the study of space-time adaptive mesh refinements for time-dependent boundary element formulations of wave equations. Based on error indicators of residual type, we formulate an adaptive boundary element procedure for…
This paper interprets the stabilized finite element method via residual minimization as a variational multiscale method. We approximate the solution to the partial differential equations using two discrete spaces that we build on a…
Stochastic mathematical models are essential tools for understanding and predicting complex phenomena. The purpose of this work is to study the exit times of a stochastic dynamical system-specifically, the mean exit time and the…
In this paper we focus on the construction of numerical schemes for nonlinear Fokker-Planck equations that preserve the structural properties, like non negativity of the solution, entropy dissipation and large time behavior. The methods…
The Finite Element Method (FEM) is a well-established procedure for computing approximate solutions to deterministic engineering problems described by partial differential equations. FEM produces discrete approximations of the solution with…
The aim of this paper is to develop a general method for constructing approximation schemes for viscosity solutions of fully nonlinear pathwise stochastic partial differential equations, and for proving their convergence. Our results apply…
We propose an approach to directly estimate the moments or marginals for a high-dimensional equilibrium distribution in statistical mechanics, via solving the high-dimensional Fokker-Planck equation in terms of low-order cluster moments or…
We derive non-linear stochastic Fokker-Planck equation from stochastic systems particles with individual and environmental noise via relative entropy method, with pathwise quantitative bounds. Moreover, we prove the existence of a unique…