Related papers: The Evolution of Pointwise Statistics in Hyperboli…
We consider the statistics for the vorticity field in two-dimensional homogeneous isotropic turbulence (HIT). First, we exploit the invariance properties to derive dimensionally reduced governing equations for the one-point and two-point…
A direct method for the computation of polynomial conservation laws of polynomial systems of nonlinear partial differential equations (PDEs) in multi-dimensions is presented. The method avoids advanced differential-geometric tools. Instead,…
Given a heterogeneous time-series sample, the objective is to find points in time (called change points) where the probability distribution generating the data has changed. The data are assumed to have been generated by arbitrary unknown…
This paper introduces a high-dimensional binary variate model that accommodates nonstationary covariates and factors, and studies their asymptotic theory. This framework encompasses scenarios where single indices are nonstationary or…
Developing algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the notoriously difficult problem known as the "curse of dimensionality". This paper…
We present a novel method for solving population density equations (PDEs), where the populations can be subject to non-Markov noise for arbitrary distributions of jump sizes. The method combines recent developments in two different…
Systems whose time evolutions are entirely deterministic can nevertheless be studied probabilistically, i.e. in terms of the evolution of probability distributions rather than individual trajectories. This approach is central to the…
We study the homogenization of a linear kinetic equation which models the evolution of the density of charged particles submitted to a highly oscillating electric field. The electric field and the initial density are assumed to be random…
Stochastic difference equations and a stochastic partial differential equation (SPDE) are simultaneously derived for the time-dependent neutron angular density in a general three-dimensional medium where the neutron angular density is a…
We investigate the longtime behavior of stochastic partial differential equations (SPDEs) with differential operators that depend on time and the underlying probability space. In particular, we consider stochastic parabolic evolution…
We study step-wise time approximations of non-linear hyperbolic initial value problems. The technique used here is a generalization of the minimizing movements method, using two time-scales: one for velocity, the other (potentially much…
This paper aims to investigate a full numerical approximation of non-autonomous semilnear parabolic partial differential equations (PDEs) with nonsmooth initial data. Our main interest is on such PDEs where the nonlinear part is stronger…
In probability density function (PDF) methods a transport equation is solved numerically to compute the time and space dependent probability distribution of several flow variables in a turbulent flow. The joint PDF of the velocity…
We give sufficient conditions for the uniform hyperbolicity of certain nonuniformly hyperbolic dynamical systems. In particular, we show that local diffeomorphisms that are nonuniformly expanding on sets of total probability are necessarily…
In this paper we introduce a numerical method for nonlinear parabolic PDEs that combines operator splitting with deep learning. It divides the PDE approximation problem into a sequence of separate learning problems. Since the computational…
A number of physical phenomena are described by nonlinear hyperbolic equations. Presence of discontinuous solutions motivates the necessity of development of reliable numerical methods based on the fundamental mathematical properties of…
We consider random hyperbolic graphs in hyperbolic spaces of any dimension $d+1\geq 2$. We present a rescaling of model parameters that casts the random hyperbolic graph model of any dimension to a unified mathematical framework, leaving…
The paper concerns boundary value problems for general nonautonomous first order quasilinear hyperbolic systems in a strip. We construct small global classical solutions, assuming that the right hand sides are small. In the case that all…
Many large scale problems in computational fluid dynamics such as uncertainty quantification, Bayesian inversion, data assimilation and PDE constrained optimization are considered very challenging computationally as they require a large…
We derive diffusive macroscopic equations for the particle and energy density of a system whose time evolution is described by a kinetic equation for the one particle position and velocity function f(r,v,t) that consists of a part that…