1,258 research outputs found
Optimization of mesh hierarchies in Multilevel Monte Carlo samplers
We perform a general optimization of the parameters in the Multilevel Monte
Carlo (MLMC) discretization hierarchy based on uniform discretization methods
with general approximation orders and computational costs. We optimize
hierarchies with geometric and non-geometric sequences of mesh sizes and show
that geometric hierarchies, when optimized, are nearly optimal and have the
same asymptotic computational complexity as non-geometric optimal hierarchies.
We discuss how enforcing constraints on parameters of MLMC hierarchies affects
the optimality of these hierarchies. These constraints include an upper and a
lower bound on the mesh size or enforcing that the number of samples and the
number of discretization elements are integers. We also discuss the optimal
tolerance splitting between the bias and the statistical error contributions
and its asymptotic behavior. To provide numerical grounds for our theoretical
results, we apply these optimized hierarchies together with the Continuation
MLMC Algorithm. The first example considers a three-dimensional elliptic
partial differential equation with random inputs. Its space discretization is
based on continuous piecewise trilinear finite elements and the corresponding
linear system is solved by either a direct or an iterative solver. The second
example considers a one-dimensional It\^o stochastic differential equation
discretized by a Milstein scheme
IGA-based Multi-Index Stochastic Collocation for random PDEs on arbitrary domains
This paper proposes an extension of the Multi-Index Stochastic Collocation
(MISC) method for forward uncertainty quantification (UQ) problems in
computational domains of shape other than a square or cube, by exploiting
isogeometric analysis (IGA) techniques. Introducing IGA solvers to the MISC
algorithm is very natural since they are tensor-based PDE solvers, which are
precisely what is required by the MISC machinery. Moreover, the
combination-technique formulation of MISC allows the straight-forward reuse of
existing implementations of IGA solvers. We present numerical results to
showcase the effectiveness of the proposed approach.Comment: version 3, version after revisio
Application of Hierarchical Matrix Techniques To The Homogenization of Composite Materials
In this paper, we study numerical homogenization methods based on integral
equations. Our work is motivated by materials such as concrete, modeled as
composites structured as randomly distributed inclusions imbedded in a matrix.
We investigate two integral reformulations of the corrector problem to be
solved, namely the equivalent inclusion method based on the Lippmann-Schwinger
equation, and a method based on boundary integral equations. The fully
populated matrices obtained by the discretization of the integral operators are
successfully dealt with using the H-matrix format
Multi-Index Monte Carlo: When Sparsity Meets Sampling
We propose and analyze a novel Multi-Index Monte Carlo (MIMC) method for weak
approximation of stochastic models that are described in terms of differential
equations either driven by random measures or with random coefficients. The
MIMC method is both a stochastic version of the combination technique
introduced by Zenger, Griebel and collaborators and an extension of the
Multilevel Monte Carlo (MLMC) method first described by Heinrich and Giles.
Inspired by Giles's seminal work, we use in MIMC high-order mixed differences
instead of using first-order differences as in MLMC to reduce the variance of
the hierarchical differences dramatically. This in turn yields new and improved
complexity results, which are natural generalizations of Giles's MLMC analysis
and which increase the domain of the problem parameters for which we achieve
the optimal convergence, Moreover, in MIMC, the
rate of increase of required memory with respect to is independent
of the number of directions up to a logarithmic term which allows far more
accurate solutions to be calculated for higher dimensions than what is possible
when using MLMC.
We motivate the setting of MIMC by first focusing on a simple full tensor
index set. We then propose a systematic construction of optimal sets of indices
for MIMC based on properly defined profits that in turn depend on the average
cost per sample and the corresponding weak error and variance. Under standard
assumptions on the convergence rates of the weak error, variance and work per
sample, the optimal index set turns out to be the total degree (TD) type. In
some cases, using optimal index sets, MIMC achieves a better rate for the
computational complexity than the corresponding rate when using full tensor
index sets..
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