Analysis of boundary effects on PDE-based sampling of Whittle-Matérn random fields

09/20/2018
by   Ustim Khristenko, et al.
0

We consider the generation of samples of a mean-zero Gaussian random field with Matérn covariance function. Every sample requires the solution of a differential equation with Gaussian white noise forcing, formulated on a bounded computational domain. This introduces unwanted boundary effects since the stochastic partial differential equation is originally posed on the whole R^d, without boundary conditions. We use a window technique, whereby one embeds the computational domain into a larger domain, and postulates convenient boundary conditions on the extended domain. To mitigate the pollution from the artificial boundary it has been suggested in numerical studies to choose a window size that is at least as large as the correlation length of the Matérn field. We provide a rigorous analysis for the error in the covariance introduced by the window technique, for homogeneous Dirichlet, homogeneous Neumann, and periodic boundary conditions. We show that the error decays exponentially in the window size, independently of the type of boundary condition. We conduct numerical experiments in 1D and 2D space, confirming our theoretical results.

READ FULL TEXT

page 1

page 2

page 3

page 4

02/17/2022

Computation of Miura surfaces for general Dirichlet boundary conditions

A nonlinear partial differential equation (PDE) that models the possible...
03/28/2022

Dissipation-preserving discretization of the Cahn–Hilliard equation with dynamic boundary conditions

This paper deals with time stepping schemes for the Cahn–Hilliard equati...
03/26/2022

The Mean Field Fokker-Planck Equation with Nonlinear No-flux Boundary Conditions

We consider the mean field Fokker-Planck equation subject to nonlinear n...
10/21/2021

DeepBND: a Machine Learning approach to enhance Multiscale Solid Mechanics

Effective properties of materials with random heterogeneous structures a...
03/03/2017

Computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model

We present a computer-assisted proof of heteroclinic connections in the ...
09/21/2020

PDE-Constrained Optimization Models and Pseudospectral Methods for Multiscale Particle Dynamics

We derive novel algorithms for optimization problems constrained by part...
12/19/2019

A simple framework for arriving at bounds on effective moduli in heterogeneous anisotropic poroelastic solids

The concepts of representative volume element (RVE), statistical homogen...