499 research outputs found
On local Fourier analysis of multigrid methods for PDEs with jumping and random coefficients
In this paper, we propose a novel non-standard Local Fourier Analysis (LFA)
variant for accurately predicting the multigrid convergence of problems with
random and jumping coefficients. This LFA method is based on a specific basis
of the Fourier space rather than the commonly used Fourier modes. To show the
utility of this analysis, we consider, as an example, a simple cell-centered
multigrid method for solving a steady-state single phase flow problem in a
random porous medium. We successfully demonstrate the prediction capability of
the proposed LFA using a number of challenging benchmark problems. The
information provided by this analysis helps us to estimate a-priori the time
needed for solving certain uncertainty quantification problems by means of a
multigrid multilevel Monte Carlo method
Multilevel Sparse Grid Methods for Elliptic Partial Differential Equations with Random Coefficients
Stochastic sampling methods are arguably the most direct and least intrusive
means of incorporating parametric uncertainty into numerical simulations of
partial differential equations with random inputs. However, to achieve an
overall error that is within a desired tolerance, a large number of sample
simulations may be required (to control the sampling error), each of which may
need to be run at high levels of spatial fidelity (to control the spatial
error). Multilevel sampling methods aim to achieve the same accuracy as
traditional sampling methods, but at a reduced computational cost, through the
use of a hierarchy of spatial discretization models. Multilevel algorithms
coordinate the number of samples needed at each discretization level by
minimizing the computational cost, subject to a given error tolerance. They can
be applied to a variety of sampling schemes, exploit nesting when available,
can be implemented in parallel and can be used to inform adaptive spatial
refinement strategies. We extend the multilevel sampling algorithm to sparse
grid stochastic collocation methods, discuss its numerical implementation and
demonstrate its efficiency both theoretically and by means of numerical
examples
Multi-index Stochastic Collocation convergence rates for random PDEs with parametric regularity
We analyze the recent Multi-index Stochastic Collocation (MISC) method for
computing statistics of the solution of a partial differential equation (PDEs)
with random data, where the random coefficient is parametrized by means of a
countable sequence of terms in a suitable expansion. MISC is a combination
technique based on mixed differences of spatial approximations and quadratures
over the space of random data and, naturally, the error analysis uses the joint
regularity of the solution with respect to both the variables in the physical
domain and parametric variables. In MISC, the number of problem solutions
performed at each discretization level is not determined by balancing the
spatial and stochastic components of the error, but rather by suitably
extending the knapsack-problem approach employed in the construction of the
quasi-optimal sparse-grids and Multi-index Monte Carlo methods. We use a greedy
optimization procedure to select the most effective mixed differences to
include in the MISC estimator. We apply our theoretical estimates to a linear
elliptic PDEs in which the log-diffusion coefficient is modeled as a random
field, with a covariance similar to a Mat\'ern model, whose realizations have
spatial regularity determined by a scalar parameter. We conduct a complexity
analysis based on a summability argument showing algebraic rates of convergence
with respect to the overall computational work. The rate of convergence depends
on the smoothness parameter, the physical dimensionality and the efficiency of
the linear solver. Numerical experiments show the effectiveness of MISC in this
infinite-dimensional setting compared with the Multi-index Monte Carlo method
and compare the convergence rate against the rates predicted in our theoretical
analysis
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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