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21
QuasiMonte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients (2012)
, 2011
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A hierarchical multilevel markov chain monte carlo algorithm with applications to uncertainty quantification in subsurface flow. arXiv preprint arXiv:1303.7343
, 2013
"... In this paper we address the problem of the prohibitively large computational cost of existing Markov chain Monte Carlo methods for large–scale applications with high dimensional parameter spaces, e.g. in uncertainty quantification in porous media flow. We propose a new multilevel MetropolisHastin ..."
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Cited by 7 (2 self)
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In this paper we address the problem of the prohibitively large computational cost of existing Markov chain Monte Carlo methods for large–scale applications with high dimensional parameter spaces, e.g. in uncertainty quantification in porous media flow. We propose a new multilevel MetropolisHastings algorithm, and give an abstract, problem dependent theorem on the cost of the new multilevel estimator based on a set of simple, verifiable assumptions. For a typical model problem in subsurface flow, we then provide a detailed analysis of these assumptions and show significant gains over the standard MetropolisHastings estimator. Numerical experiments confirm the analysis and demonstrate the effectiveness of the method with consistent reductions of a factor of O(10–50) in the εcost of the multilevel estimator over the standard MetropolisHastings algorithm for tolerances ε around 10−3.
Optimization of mesh hierarchies in Multilevel Monte Carlo samplers. ArXiv eprints:1403.2480
, 2014
"... Abstract 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. Moreover, we discuss extensions to nonuniform discretizations based on a prio ..."
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Cited by 5 (1 self)
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Abstract 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. Moreover, we discuss extensions to nonuniform discretizations based on a priori refinements and the effect of imposing constraints on the largest and/or smallest mesh sizes. We optimize geometric and nongeometric hierarchies and compare them to each other, concluding that the geometric hierarchies, when optimized, are nearly optimal and have the same asymptotic computational complexity. We discuss how enforcing domain constraints on parameters of MLMC hierarchies affects the optimality of these hierarchies. These domain constraints include an upper and 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 [13]
A Priori Error Analysis of Stochastic Galerkin Mixed Approximations of Elliptic PDEs with Random Data
, 2011
"... Abstract. We construct stochastic Galerkin approximations to the solution of a firstorder system of PDEs with random coefficients. Under the standard finitedimensional noise assumption, we transform the variational saddle point problem to a parametric deterministic one. Approximations are construc ..."
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Cited by 4 (1 self)
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Abstract. We construct stochastic Galerkin approximations to the solution of a firstorder system of PDEs with random coefficients. Under the standard finitedimensional noise assumption, we transform the variational saddle point problem to a parametric deterministic one. Approximations are constructed by combining mixed finite elements on the computational domain with Mvariate tensor product polynomials. We study the infsup stability and wellposedness of the continuous and finitedimensional problems, the regularity of solutions with respect to the M parameters describing the random coefficients, and establish a priori error estimates for stochastic Galerkin finite element approximations.
Multilevel monte carlo methods for highly hetereogeneous media
 Proceedings of the Winter Simulation Conference
, 2012
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Multilevel Monte Carlo methods
"... An outline history inspired by undergraduate numerical projects course at Cambridge, and summer projects at RollsRoyce this was one of my first textbooks after 25 years working on CFD, 10 years ago I switched to Monte Carlo methods for computational finance and other application areas ..."
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Cited by 2 (1 self)
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An outline history inspired by undergraduate numerical projects course at Cambridge, and summer projects at RollsRoyce this was one of my first textbooks after 25 years working on CFD, 10 years ago I switched to Monte Carlo methods for computational finance and other application areas
Multilevel higher order QMC Galerkin discretization for affine parametric operator equations
, 2014
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MULTI INDEX MONTE CARLO: WHEN SPARSITY MEETS SAMPLING
"... Abstract. 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 combinati ..."
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Abstract. 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, instead of using firstorder differences as in MLMC, we use in MIMC highorder mixed differences 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 problem parameters for which we achieve the optimal convergence, O(TOL−2). Moreover, we motivate the systematic construction of optimal sets of indices for MIMC based on properly defined profits that in turn depend
Comput Visual Sci (2011) 14:3–15 DOI 10.1007/s007910110160x Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients
"... Abstract We consider the numerical solution of elliptic partial differential equations with random coefficients. Such problems arise, for example, in uncertainty quantification for groundwater flow. We describe a novel variance reduction technique for the standard Monte Carlo method, called the mu ..."
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Abstract We consider the numerical solution of elliptic partial differential equations with random coefficients. Such problems arise, for example, in uncertainty quantification for groundwater flow. We describe a novel variance reduction technique for the standard Monte Carlo method, called the multilevel Monte Carlo method, and demonstrate numerically its superiority. The asymptotic cost of solving the stochastic problem with the multilevel method is always significantly lower than that of the standard method and grows only proportionally to the cost of solving the deterministic problem in certain circumstances. Numerical calculations demonstrating the effectiveness of the method for one and twodimensional model problems arising in groundwater flow are presented.
Multiscale Methods and Uncertainty Quantification
, 2015
"... of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: ..."
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of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: