Journal ArticleDOI
Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients
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TLDR
A novel variance reduction technique for the standard Monte Carlo method, called the multilevel Monte Carlo Method, is described, and numerically its superiority is demonstrated.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 two-dimensional model problems arising in groundwater flow are presentedread more
Citations
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Journal ArticleDOI
Survey of Multifidelity Methods in Uncertainty Propagation, Inference, and Optimization
TL;DR: In many situations across computational science and engineering, multiple computational models are available that describe a system of interest as discussed by the authors, and these different models have varying evaluation costs, i.e.
Journal ArticleDOI
Multilevel Monte Carlo methods
TL;DR: A review of the progress in multilevel Monte Carlo path simulation can be found in this article, where the authors highlight the simplicity, flexibility and generality of the multi-level Monte Carlo approach.
Journal ArticleDOI
High-dimensional integration: The quasi-Monte Carlo way
TL;DR: A survey of recent developments in lattice methods, digital nets, and related themes can be found in this paper, where the authors present a contemporary review of QMC (quasi-Monte Carlo) methods, that is, equalweight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0, 1] s, w heres may be large, or even infinite.
Acta Numerica: High dimensional integration - the Quasi-Monte Carlo way
TL;DR: This paper is a contemporary review of QMC (‘quasi-Monte Carlo’) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0,1]s, where s may be large, or even infinite.
References
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Book
Stochastic Finite Elements: A Spectral Approach
Roger Ghanem,Pol D. Spanos +1 more
TL;DR: In this article, a representation of stochastic processes and response statistics are represented by finite element method and response representation, respectively, and numerical examples are provided for each of them.
Book
Stochastic Equations in Infinite Dimensions
Giuseppe Da Prato,Jerzy Zabczyk +1 more
TL;DR: In this paper, the existence and uniqueness of nonlinear equations with additive and multiplicative noise was investigated. But the authors focused on the uniqueness of solutions and not on the properties of solutions.
Journal ArticleDOI
Multilevel Monte Carlo Path Simulation
TL;DR: It is shown that multigrid ideas can be used to reduce the computational complexity of estimating an expected value arising from a stochastic differential equation using Monte Carlo path simulations.
Book
Numerical Methods for Stochastic Computations: A Spectral Method Approach
TL;DR: This book describes the class of numerical methods based on generalized polynomial chaos (gPC), an extension of the classical spectral methods of high-dimensional random spaces designed to simulate complex systems subject to random inputs.