12 research outputs found
Multilevel Hierarchical Decomposition of Finite Element White Noise with Application to Multilevel Markov Chain Monte Carlo
In this work we develop a new hierarchical multilevel approach to generate
Gaussian random field realizations in an algorithmically scalable manner that
is well-suited to incorporate into multilevel Markov chain Monte Carlo (MCMC)
algorithms. This approach builds off of other partial differential equation
(PDE) approaches for generating Gaussian random field realizations; in
particular, a single field realization may be formed by solving a
reaction-diffusion PDE with a spatial white noise source function as the
righthand side. While these approaches have been explored to accelerate forward
uncertainty quantification tasks, e.g. multilevel Monte Carlo, the previous
constructions are not directly applicable to multilevel MCMC frameworks which
build fine scale random fields in a hierarchical fashion from coarse scale
random fields. Our new hierarchical multilevel method relies on a hierarchical
decomposition of the white noise source function in which allows us to
form Gaussian random field realizations across multiple levels of
discretization in a way that fits into multilevel MCMC algorithmic frameworks.
After presenting our main theoretical results and numerical scaling results to
showcase the utility of this new hierarchical PDE method for generating
Gaussian random field realizations, this method is tested on a four-level MCMC
algorithm to explore its feasibility
Multilevel Monte Carlo methods for the Dean-Kawasaki equation from Fluctuating Hydrodynamics
Stochastic PDEs of Fluctuating Hydrodynamics are a powerful tool for the
description of fluctuations in many-particle systems. In this paper, we develop
and analyze a Multilevel Monte Carlo (MLMC) scheme for the Dean-Kawasaki
equation, a pivotal representative of this class of SPDEs. We prove
analytically and demonstrate numerically that our MLMC scheme provides a
significant speed-up (with respect to a standard Monte Carlo method) in the
simulation of the Dean-Kawasaki equation. Specifically, we quantify how the
speed-up factor increases as the average particle density increases, and show
that sizeable speed-ups can be obtained even in regimes of low particle
density. Numerical simulations are provided in the two-dimensional case,
confirming our theoretical predictions.
Our results are formulated entirely in terms of the law of distributions
rather than in terms of strong spatial norms: this crucially allows for MLMC
speed-ups altogether despite the Dean-Kawasaki equation being highly singular.Comment: 23 pages, 9 figure
A probabilistic finite element method based on random meshes: Error estimators and Bayesian inverse problems
We present a novel probabilistic finite element method (FEM) for the solution
and uncertainty quantification of elliptic partial differential equations based
on random meshes, which we call random mesh FEM (RM-FEM). Our methodology
allows to introduce a probability measure on standard piecewise linear FEM. We
present a posteriori error estimators based uniquely on probabilistic
information. A series of numerical experiments illustrates the potential of the
RM-FEM for error estimation and validates our analysis. We furthermore
demonstrate how employing the RM-FEM enhances the quality of the solution of
Bayesian inverse problems, thus allowing a better quantification of numerical
errors in pipelines of computations
Generalized averaged Gaussian quadrature and applications
A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal
MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications
Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described
[Research activities in applied mathematics, fluid mechanics, and computer science]
This report summarizes research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, fluid mechanics, and computer science during the period April 1, 1995 through September 30, 1995
Efficient white noise sampling and coupling for multilevel Monte Carlo with nonnested meshes
When solving stochastic partial differential equations (SPDEs) driven by additive spatial white noise, the efficient sampling of white noise realizations can be challenging. Here, we present a new sampling technique that can be used to efficiently compute white noise samples in a finite element method (FEM) and multilevel Monte Carlo (MLMC) setting. The key idea is to exploit the finite element matrix assembly procedure and factorize each local mass matrix independently, hence avoiding the factorization of a large matrix. Moreover, in an MLMC framework, the white noise samples must be coupled between subsequent levels. We show how our technique can be used to enforce this coupling even in the case of nonnested mesh hierarchies. We demonstrate the efficacy of our method with numerical experiments. We observe optimal convergence rates for the finite element solution of the elliptic SPDEs of interest in 2D and 3D and we show convergence of the sampled field covariances. In an MLMC setting, a good coupling is enforced and the telescoping sum is respected
Efficient white noise sampling and coupling for multilevel Monte Carlo with nonnested meshes
When solving stochastic partial differential equations (SPDEs) driven by additive spatial white noise, the efficient sampling of white noise realizations can be challenging. Here, we present a new sampling technique that can be used to efficiently compute white noise samples in a finite element method (FEM) and multilevel Monte Carlo (MLMC) setting. The key idea is to exploit the finite element matrix assembly procedure and factorize each local mass matrix independently, hence avoiding the factorization of a large matrix. Moreover, in an MLMC framework, the white noise samples must be coupled between subsequent levels. We show how our technique can be used to enforce this coupling even in the case of nonnested mesh hierarchies. We demonstrate the efficacy of our method with numerical experiments. We observe optimal convergence rates for the finite element solution of the elliptic SPDEs of interest in 2D and 3D and we show convergence of the sampled field covariances. In an MLMC setting, a good coupling is enforced and the telescoping sum is respected