1,573 research outputs found
Multi-level Monte Carlo Finite Element method for elliptic PDEs with stochastic coefficients
In Monte Carlo methods quadrupling the sample size halves the error. In simulations of stochastic partial differential equations (SPDEs), the total work is the sample size times the solution cost of an instance of the partial differential equation. A Multi-level Monte Carlo method is introduced which allows, in certain cases, to reduce the overall work to that of the discretization of one instance of the deterministic PDE. The model problem is an elliptic equation with stochastic coefficients. Multi-level Monte Carlo errors and work estimates are given both for the mean of the solutions and for higher moments. The overall complexity of computing mean fields as well as k-point correlations of the random solution is proved to be of log-linear complexity in the number of unknowns of a single Multi-level solve of the deterministic elliptic problem. Numerical examples complete the theoretical analysi
Application of quasi-Monte Carlo methods to PDEs with random coefficients -- an overview and tutorial
This article provides a high-level overview of some recent works on the
application of quasi-Monte Carlo (QMC) methods to PDEs with random
coefficients. It is based on an in-depth survey of a similar title by the same
authors, with an accompanying software package which is also briefly discussed
here. Embedded in this article is a step-by-step tutorial of the required
analysis for the setting known as the uniform case with first order QMC rules.
The aim of this article is to provide an easy entry point for QMC experts
wanting to start research in this direction and for PDE analysts and
practitioners wanting to tap into contemporary QMC theory and methods.Comment: arXiv admin note: text overlap with arXiv:1606.0661
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
Hot new directions for quasi-Monte Carlo research in step with applications
This article provides an overview of some interfaces between the theory of
quasi-Monte Carlo (QMC) methods and applications. We summarize three QMC
theoretical settings: first order QMC methods in the unit cube and in
, and higher order QMC methods in the unit cube. One important
feature is that their error bounds can be independent of the dimension
under appropriate conditions on the function spaces. Another important feature
is that good parameters for these QMC methods can be obtained by fast efficient
algorithms even when is large. We outline three different applications and
explain how they can tap into the different QMC theory. We also discuss three
cost saving strategies that can be combined with QMC in these applications.
Many of these recent QMC theory and methods are developed not in isolation, but
in close connection with applications
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
Multilevel Methods for Uncertainty Quantification of Elliptic PDEs with Random Anisotropic Diffusion
We consider elliptic diffusion problems with a random anisotropic diffusion
coefficient, where, in a notable direction given by a random vector field, the
diffusion strength differs from the diffusion strength perpendicular to this
notable direction. The Karhunen-Lo\`eve expansion then yields a parametrisation
of the random vector field and, therefore, also of the solution of the elliptic
diffusion problem. We show that, given regularity of the elliptic diffusion
problem, the decay of the Karhunen-Lo\`eve expansion entirely determines the
regularity of the solution's dependence on the random parameter, also when
considering this higher spatial regularity. This result then implies that
multilevel collocation and multilevel quadrature methods may be used to lessen
the computation complexity when approximating quantities of interest, like the
solution's mean or its second moment, while still yielding the expected rates
of convergence. Numerical examples in three spatial dimensions are provided to
validate the presented theory
Polynomial Chaos Expansion of random coefficients and the solution of stochastic partial differential equations in the Tensor Train format
We apply the Tensor Train (TT) decomposition to construct the tensor product
Polynomial Chaos Expansion (PCE) of a random field, to solve the stochastic
elliptic diffusion PDE with the stochastic Galerkin discretization, and to
compute some quantities of interest (mean, variance, exceedance probabilities).
We assume that the random diffusion coefficient is given as a smooth
transformation of a Gaussian random field. In this case, the PCE is delivered
by a complicated formula, which lacks an analytic TT representation. To
construct its TT approximation numerically, we develop the new block TT cross
algorithm, a method that computes the whole TT decomposition from a few
evaluations of the PCE formula. The new method is conceptually similar to the
adaptive cross approximation in the TT format, but is more efficient when
several tensors must be stored in the same TT representation, which is the case
for the PCE. Besides, we demonstrate how to assemble the stochastic Galerkin
matrix and to compute the solution of the elliptic equation and its
post-processing, staying in the TT format.
We compare our technique with the traditional sparse polynomial chaos and the
Monte Carlo approaches. In the tensor product polynomial chaos, the polynomial
degree is bounded for each random variable independently. This provides higher
accuracy than the sparse polynomial set or the Monte Carlo method, but the
cardinality of the tensor product set grows exponentially with the number of
random variables. However, when the PCE coefficients are implicitly
approximated in the TT format, the computations with the full tensor product
polynomial set become possible. In the numerical experiments, we confirm that
the new methodology is competitive in a wide range of parameters, especially
where high accuracy and high polynomial degrees are required.Comment: This is a major revision of the manuscript arXiv:1406.2816 with
significantly extended numerical experiments. Some unused material is remove
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