1,906 research outputs found
Weak order for the discretization of the stochastic heat equation
In this paper we study the approximation of the distribution of
Hilbert--valued stochastic process solution of a linear parabolic stochastic
partial differential equation written in an abstract form as driven by a Gaussian
space time noise whose covariance operator is given. We assume that
is a finite trace operator for some and that is
bounded from into for some . It is not required
to be nuclear or to commute with . The discretization is achieved thanks to
finite element methods in space (parameter ) and implicit Euler schemes in
time (parameter ). We define a discrete solution and for
suitable functions defined on , we show that |\E \phi(X^N_h) - \E
\phi(X_T) | = O(h^{2\gamma} + \Delta t^\gamma) \noindent where . Let us note that as in the finite dimensional case the rate of
convergence is twice the one for pathwise approximations
Efficient white noise sampling and coupling for multilevel Monte Carlo with non-nested 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 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 a 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 non-nested 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 a MLMC setting, a good coupling is enforced and the telescoping sum is
respected.Comment: 28 pages, 10 figure
Stochastic finite differences and multilevel Monte Carlo for a class of SPDEs in finance
In this article, we propose a Milstein finite difference scheme for a
stochastic partial differential equation (SPDE) describing a large particle
system. We show, by means of Fourier analysis, that the discretisation on an
unbounded domain is convergent of first order in the timestep and second order
in the spatial grid size, and that the discretisation is stable with respect to
boundary data. Numerical experiments clearly indicate that the same convergence
order also holds for boundary-value problems. Multilevel path simulation,
previously used for SDEs, is shown to give substantial complexity gains
compared to a standard discretisation of the SPDE or direct simulation of the
particle system. We derive complexity bounds and illustrate the results by an
application to basket credit derivatives
On the well-posedness of the stochastic Allen-Cahn equation in two dimensions
White noise-driven nonlinear stochastic partial differential equations
(SPDEs) of parabolic type are frequently used to model physical and biological
systems in space dimensions d = 1,2,3. Whereas existence and uniqueness of weak
solutions to these equations are well established in one dimension, the
situation is different for d \geq 2. Despite their popularity in the applied
sciences, higher dimensional versions of these SPDE models are generally
assumed to be ill-posed by the mathematics community. We study this discrepancy
on the specific example of the two dimensional Allen-Cahn equation driven by
additive white noise. Since it is unclear how to define the notion of a weak
solution to this equation, we regularize the noise and introduce a family of
approximations. Based on heuristic arguments and numerical experiments, we
conjecture that these approximations exhibit divergent behavior in the
continuum limit. The results strongly suggest that a series of published
numerical studies are problematic: shrinking the mesh size in these simulations
does not lead to the recovery of a physically meaningful limit.Comment: 21 pages, 4 figures; accepted by Journal of Computational Physics
(Dec 2011
Weak convergence of Galerkin approximations for fractional elliptic stochastic PDEs with spatial white noise
The numerical approximation of the solution to a stochastic partial
differential equation with additive spatial white noise on a bounded domain is
considered. The differential operator is assumed to be a fractional power of an
integer order elliptic differential operator. The solution is approximated by
means of a finite element discretization in space and a quadrature
approximation of an integral representation of the fractional inverse from the
Dunford-Taylor calculus.
For the resulting approximation, a concise analysis of the weak error is
performed. Specifically, for the class of twice continuously Fr\'echet
differentiable functionals with second derivatives of polynomial growth, an
explicit rate of weak convergence is derived, and it is shown that the
component of the convergence rate stemming from the stochasticity is doubled
compared to the corresponding strong rate. Numerical experiments for different
functionals validate the theoretical results.Comment: 22 pages, 1 figur
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