744 research outputs found
Adaptive Multilevel Monte Carlo Methods for Stochastic Variational Inequalities
While multilevel Monte Carlo (MLMC) methods for the numerical approximation of partial differential equations with random coefficients enjoy great popularity, combinations with spatial adaptivity seem to be rare. We present an adaptive MLMC finite element approach based on deterministic adaptive mesh refinement for the arising “pathwise” problems and outline a convergence theory in terms of desired accuracy and required computational cost. Our theoretical and heuristic reasoning together with the efficiency of our new approach are confirmed by numerical experiments
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
Adaptive Multilevel Monte Carlo Methods for Random Elliptic Problems
In this thesis we introduce a novel framework for uncertainty quantification in problems with random coefficients. The developed framework utilizes the ideas of multilevel Monte Carlo (MLMC) methods and allows for exploiting the advantages of adaptive finite element techniques. In contrast to the standard MLMC method, where levels are characterized by a hierarchy of uniform meshes, we associate the MLMC levels with a chosen sequence of tolerances. Each deterministic problem corresponding to a MC sample on a given level is then approximated up to the corresponding accuracy. This can be done, for example, using pathwise a posteriori error estimation and adaptive mesh refinement techniques.
We further introduce an adaptive MLMC finite element method for random
linear elliptic problems based on a residual-based a posteriori error estimation technique.
We provide a careful analysis of the novel method based on a generalization
of existing results, for deterministic residual-based error estimation, to the random
setting. We complement our theoretical results by numerical simulations illustrating
the advantages of our approach compared to the standard MLMC finite element
method when applied to problems with random singularities
A fully adaptive multilevel stochastic collocation strategy for solving elliptic PDEs with random data
We propose and analyse a fully adaptive strategy for solving elliptic PDEs
with random data in this work. A hierarchical sequence of adaptive mesh
refinements for the spatial approximation is combined with adaptive anisotropic
sparse Smolyak grids in the stochastic space in such a way as to minimize the
computational cost. The novel aspect of our strategy is that the hierarchy of
spatial approximations is sample dependent so that the computational effort at
each collocation point can be optimised individually. We outline a rigorous
analysis for the convergence and computational complexity of the adaptive
multilevel algorithm and we provide optimal choices for error tolerances at
each level. Two numerical examples demonstrate the reliability of the error
control and the significant decrease in the complexity that arises when
compared to single level algorithms and multilevel algorithms that employ
adaptivity solely in the spatial discretisation or in the collocation
procedure.Comment: 26 pages, 7 figure
Multilevel quasi-Monte Carlo for random elliptic eigenvalue problems I: Regularity and error analysis
Random eigenvalue problems are useful models for quantifying the uncertainty
in several applications from the physical sciences and engineering, e.g.,
structural vibration analysis, the criticality of a nuclear reactor or photonic
crystal structures. In this paper we present a simple multilevel quasi-Monte
Carlo (MLQMC) method for approximating the expectation of the minimal
eigenvalue of an elliptic eigenvalue problem with coefficients that are given
as a series expansion of countably-many stochastic parameters. The MLQMC
algorithm is based on a hierarchy of discretisations of the spatial domain and
truncations of the dimension of the stochastic parameter domain. To approximate
the expectations, randomly shifted lattice rules are employed. This paper is
primarily dedicated to giving a rigorous analysis of the error of this
algorithm. A key step in the error analysis requires bounds on the mixed
derivatives of the eigenfunction with respect to both the stochastic and
spatial variables simultaneously. An accompanying paper [Gilbert and Scheichl,
2021], focusses on practical extensions of the MLQMC algorithm to improve
efficiency, and presents numerical results
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New Discretization Methods for the Numerical Approximation of PDEs
The construction and mathematical analysis of numerical methods for PDEs is a fundamental area of modern applied mathematics. Among the various techniques that have been proposed in the past, some – in particular, finite element methods, – have been exceptionally successful in a range of applications. There are however a number of important challenges that remain, including the optimal adaptive finite element approximation of solutions to transport-dominated diffusion problems, the efficient numerical approximation of parametrized families of PDEs, and the efficient numerical approximation of high-dimensional partial differential equations (that arise from stochastic analysis and statistical physics, for example, in the form of a backward Kolmogorov equation, which, unlike its formal adjoint, the forward Kolmogorov equation, is not in divergence form, and therefore not directly amenable to finite element approximation, even when the spatial dimension is low). In recent years several original and conceptionally new ideas have emerged in order to tackle these open problems.
The goal of this workshop was to discuss and compare a number of novel approaches, to study their potential and applicability, and to formulate the strategic goals and directions of research in this field for the next five years
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