137 research outputs found
Finite element approximations for second order stochastic differential equation driven by fractional Brownian motion
We consider finite element approximations for a one dimensional second order
stochastic differential equation of boundary value type driven by a fractional
Brownian motion with Hurst index . We make use of a sequence of
approximate solutions with the fractional noise replaced by its piecewise con-
stant approximations to construct the finite element approximations for the
equation. The error estimate of the approximations is derived through rigorous
convergence analysis.Comment: To appear in IMA Journal of Numerical Analysis; the time-dependent
case such as stochastic heat equation and stochastic wave equation driven by
fractional Brownian sheet with temporal Hurst index and spatial Hurst
index has been considered by arXiv:1601.02085 for spatially
Galerkin approximations and a forthcoming paper for fully discrete
approximation
Kernel-based Methods for Stochastic Partial Differential Equations
This article gives a new insight of kernel-based (approximation) methods to
solve the high-dimensional stochastic partial differential equations. We will
combine the techniques of meshfree approximation and kriging interpolation to
extend the kernel-based methods for the deterministic data to the stochastic
data. The main idea is to endow the Sobolev spaces with the probability
measures induced by the positive definite kernels such that the Gaussian random
variables can be well-defined on the Sobolev spaces. The constructions of these
Gaussian random variables provide the kernel-based approximate solutions of the
stochastic models. In the numerical examples of the stochastic Poisson and heat
equations, we show that the approximate probability distributions are
well-posed for various kinds of kernels such as the compactly supported kernels
(Wendland functions) and the Sobolev-spline kernels (Mat\'ern functions)
Weak convergence of Galerkin approximations of stochastic partial differential equations driven by additive L\'evy noise
This work considers weak approximations of stochastic partial differential
equations (SPDEs) driven by L\'evy noise. The SPDEs at hand are parabolic with
additive noise processes. A weak-convergence rate for the corresponding
Galerkin approximation is derived. The convergence result is derived by use of
the Malliavin derivative rather then the common approach via the Kolmogorov
backward equation
A Domain-Decomposition Model Reduction Method for Linear Convection-Diffusion Equations with Random Coefficients
We develop a domain-decomposition model reduction method for linear
steady-state convection-diffusion equations with random coefficients. Of
particular interest to this effort are the diffusion equations with random
diffusivities, and the convection-dominated transport equations with random
velocities. We investigate the equations with two types of random fields, i.e.,
colored noises and discrete white noises, both of which can lead to
high-dimensional parametric dependence. The motivation is to use domain
decomposition to exploit low-dimensional structures of local problems in the
sub-domains, such that the total number of expensive PDE solves can be greatly
reduced. Our objective is to develop an efficient model reduction method to
simultaneously handle high-dimensionality and irregular behaviors of the
stochastic PDEs under consideration. The advantages of our method lie in three
aspects: (i) online-offline decomposition, i.e., the online cost is independent
of the size of the triangle mesh; (ii) operator approximation for handling
non-affine and high-dimensional random fields; (iii) effective strategy to
capture irregular behaviors, e.g., sharp transitions of the PDE solution. Two
numerical examples will be provided to demonstrate the advantageous performance
of our method
Resolve the multitude of microscale interactions to model stochastic partial differential equations
Constructing numerical models of noisy partial differential equations is very
delicate. Our long term aim is to use modern dynamical systems theory to derive
discretisations of dissipative stochastic partial differential equations. As a
second step we here consider a small domain, representing a finite element, and
apply stochastic centre manifold theory to derive a one degree of freedom model
for the dynamics in the element. The approach automatically parametrises the
microscale structures induced by spatially varying stochastic noise within the
element. The crucial aspect of this work is that we explore how many noise
processes may interact in nonlinear dynamics. We see that noise processes with
coarse structure across a finite element are the significant noises for the
modelling. Further, the nonlinear dynamics abstracts effectively new noise
sources over the macroscopic time scales resolved by the model.Comment: extended results, better proof
Some recent progress in singular stochastic PDEs
Stochastic PDEs are ubiquitous in mathematical modeling. Yet, many such
equations are too singular to admit classical treatment. In this article we
review some recent progress in defining, approximating and studying the
properties of a few examples of such equations. We focus mainly on the
dynamical Phi4 equation, KPZ equation and Parabolic Anderson Model, as well as
touch on a few other equations which arise mainly in physics.Comment: 41 page
Numerical solution of fractional elliptic stochastic PDEs with spatial white noise
The numerical approximation of solutions to stochastic partial differential
equations with additive spatial white noise on bounded domains in
is considered. The differential operator is given by the
fractional power , , of an integer order elliptic
differential operator and is therefore non-local. Its inverse
is represented by a Bochner integral from the Dunford-Taylor functional
calculus. By applying a quadrature formula to this integral representation, the
inverse fractional power operator is approximated by a weighted
sum of non-fractional resolvents at certain quadrature
nodes . The resolvents are then discretized in space by a standard
finite element method.
This approach is combined with an approximation of the white noise, which is
based only on the mass matrix of the finite element discretization. In this
way, an efficient numerical algorithm for computing samples of the approximate
solution is obtained. For the resulting approximation, the strong mean-square
error is analyzed and an explicit rate of convergence is derived. Numerical
experiments for , , with homogeneous Dirichlet
boundary conditions on the unit cube in spatial dimensions
for varying attest the theoretical results.Comment: 21 pages, 1 figur
Simulation of SPDE's for Excitable Media using Finite Elements
In this paper, we address the question of the discretization of Stochastic
Partial Differential Equations (SPDE's) for excitable media. Working with
SPDE's driven by colored noise, we consider a numerical scheme based on finite
differences in time (Euler-Maruyama) and finite elements in space. Motivated by
biological considerations, we study numerically the emergence of reentrant
patterns in excitable systems such as the Barkley or Mitchell-Schaeffer models.Comment: 24 page
Wiener chaos vs stochastic collocation methods for linear advection-diffusion equations with multiplicative white noise
We compare Wiener chaos and stochastic collocation methods for linear
advection-reaction-diffusion equations with multiplicative white noise. Both
methods are constructed based on a recursive multi-stage algorithm for
long-time integration. We derive error estimates for both methods and compare
their numerical performance. Numerical results confirm that the recursive
multi-stage stochastic collocation method is of order (time step size)
in the second-order moments while the recursive multi-stage Wiener chaos method
is of order ( is the order of Wiener
chaos) for advection-diffusion-reaction equations with commutative noises, in
agreement with the theoretical error estimates. However, for non-commutative
noises, both methods are of order one in the second-order moments
Strong convergence of some Euler-type schemes for the finite element discretization of time-fractional SPDE driven by standard and fractional Brownian motion
In this work, we provide the first strong convergence result of numerical
approximation of a general second order semilinear stochastic fractional order
evolution equation involving a Caputo derivative in time of order
and driven by Gaussian and non-Gaussian noises
simultaneously more useful in concrete applications. The Gaussian noise
considered here is a Hilbert space valued Q-Wiener process and the non-Gaussian
noise is defined through compensated Poisson random measure associated to a
L\'evy process. The linear operator is not necessary self-adjoint. The
fractional stochastic partial differential equation is discretized in space by
the finite element method and in time by a variant of the exponential
integrator scheme. We investigate the mean square error estimate of our fully
discrete scheme and the result shows how the convergence orders depend on the
regularity of the initial data and the power of the fractional derivative
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