A Multilevel Monte Carlo Method for Computing Failure Probabilities

DOI10.1137/140984294zbMath1343.65004arXiv1408.6856OpenAlexW2153740376MaRDI QIDQ5741181

Fredrik Hellman, Axel Målqvist, Daniel Elfverson

Publication date: 22 July 2016

Published in: SIAM/ASA Journal on Uncertainty Quantification (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1408.6856

zbMATH Keywords

error analysisnumerical experimentquantile estimationfailure probabilityuncertainty quantificationmultilevel Monte Carlo

Mathematics Subject Classification ID

Monte Carlo methods (65C05) Error bounds for boundary value problems involving PDEs (65N15) Stochastic partial differential equations (aspects of stochastic analysis) (60H15) Computational methods for stochastic equations (aspects of stochastic analysis) (60H35) Numerical solutions to stochastic differential and integral equations (65C30)

Related Items (20)

Estimation of distributions via multilevel Monte Carlo with stratified sampling ⋮ A fully adaptive multilevel stochastic collocation strategy for solving elliptic PDEs with random data ⋮ On the optimization of approximate control variates with parametrically defined estimators ⋮ Multilevel Estimation of Rare Events ⋮ Multifidelity Preconditioning of the Cross-Entropy Method for Rare Event Simulation and Failure Probability Estimation ⋮ An Adaptive Multilevel Monte Carlo Method with Stochastic Bounds for Quantities of Interest with Uncertain Data ⋮ An offline-online strategy for multiscale problems with random defects ⋮ Adaptive Multilevel Monte Carlo Methods for Stochastic Variational Inequalities ⋮ Adaptive Multilevel Monte Carlo for Probabilities ⋮ A massively parallel implementation of multilevel Monte Carlo for finite element models ⋮ Deep Importance Sampling Using Tensor Trains with Application to a Priori and a Posteriori Rare Events ⋮ Continuous Level Monte Carlo and Sample-Adaptive Model Hierarchies ⋮ Scheduling Massively Parallel Multigrid for Multilevel Monte Carlo Methods ⋮ Multifidelity probability estimation via fusion of estimators ⋮ Multilevel Sequential Importance Sampling for Rare Event Estimation ⋮ Recycling Samples in the Multigrid Multilevel (Quasi-)Monte Carlo Method ⋮ Quantify uncertainty by estimating the probability density function of the output of interest using MLMC based Bayes method ⋮ Error Analysis for Probabilities of Rare Events with Approximate Models ⋮ Analysis of Nested Multilevel Monte Carlo Using Approximate Normal Random Variables ⋮ Quasi-Monte Carlo and Multilevel Monte Carlo Methods for Computing Posterior Expectations in Elliptic Inverse Problems

Uses Software

• FEniCS
• SciPy

Cites Work

• Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients
• Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients
• Optimization of mesh hierarchies in multilevel Monte Carlo samplers
• Multi-level Monte Carlo finite element method for elliptic PDEs with stochastic coefficients
• Automated solution of differential equations by the finite element method. The FEniCS book
• On irregular functionals of SDEs and the Euler scheme
• A continuation multilevel Monte Carlo algorithm
• Statistical Romberg extrapolation: a new variance reduction method and applications to option pricing
• Uncertainty Quantification for Approximate $p$-Quantiles for Physical Models with Stochastic Inputs
• Multilevel Monte Carlo Path Simulation
• Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality
• Nonparametric Density Estimation for Randomly Perturbed Elliptic Problems I: Computational Methods, A Posteriori Analysis, and Adaptive Error Control
• Finite Element Methods for Navier-Stokes Equations
• Fast and Exact Simulation of Stationary Gaussian Processes through Circulant Embedding of the Covariance Matrix
• Finite Element Error Analysis of Elliptic PDEs with Random Coefficients and Its Application to Multilevel Monte Carlo Methods

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