Quasi-Monte Carlo and Multilevel Monte Carlo Methods for Computing Posterior Expectations in Elliptic Inverse Problems

DOI10.1137/16M1061692MaRDI QIDQ5269873

A. L. Teckentrup, Robert Scheichl, Andrew M. Stuart

Publication date: 28 June 2017

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

Full work available at URL: https://arxiv.org/abs/1602.04704

zbMATH Keywords

inverse problemfinite element methodsBayesianhigh-dimensional approximationquasi-Monte Carlouncertainty quantificationmultilevel Monte Carlo

Mathematics Subject Classification ID

Random fields; image analysis (62M40) Bayesian inference (62F15) Monte Carlo methods (65C05) Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs (65N30) PDEs with randomness, stochastic partial differential equations (35R60) Numerical methods for inverse problems for boundary value problems involving PDEs (65N21)

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Cites Work

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• 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
• Quasi-Monte Carlo finite element methods for elliptic PDEs with lognormal random coefficients
• Optimization of mesh hierarchies in multilevel Monte Carlo samplers
• Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications
• Convergence rates of best \(N\)-term Galerkin approximations for a class of elliptic SPDEs
• Evaluation for moments of a ratio with application to regression estimation
• Multi-level Monte Carlo finite element method for elliptic PDEs with stochastic coefficients
• Simple upper bounds for partition functions
• Circulant embedding with QMC: analysis for elliptic PDE with lognormal coefficients
• Importance sampling: intrinsic dimension and computational cost
• A continuation multilevel Monte Carlo algorithm
• Multi-level quasi-Monte Carlo finite element methods for a class of elliptic PDEs with random coefficients
• Analysis of quasi-optimal polynomial approximations for parameterized PDEs with deterministic and stochastic coefficients
• MAP estimators for piecewise continuous inversion
• Sparse, adaptive Smolyak quadratures for Bayesian inverse problems
• Complexity analysis of accelerated MCMC methods for Bayesian inversion
• Strong and Weak Error Estimates for Elliptic Partial Differential Equations with Random Coefficients
• Sparse deterministic approximation of Bayesian inverse problems
• Inverse problems: A Bayesian perspective
• Scaling limits in computational Bayesian inversion
• Multilevel higher-order quasi-Monte Carlo Bayesian estimation
• Higher-Order Quasi-Monte Carlo for Bayesian Shape Inversion
• Multilevel Monte Carlo Path Simulation
• A Hierarchical Multilevel Markov Chain Monte Carlo Algorithm with Applications to Uncertainty Quantification in Subsurface Flow
• THE DISTRIBUTION OF THE INDEX IN A NORMAL BIVARIATE POPULATION
• Fast and Exact Simulation of Stationary Gaussian Processes through Circulant Embedding of the Covariance Matrix
• The quotient of two correlated normal variables with applications
• A Multivariate Faa di Bruno Formula with Applications
• Quasi-Monte Carlo Finite Element Methods for a Class of Elliptic Partial Differential Equations with Random Coefficients
• Finite Element Error Analysis of Elliptic PDEs with Random Coefficients and Its Application to Multilevel Monte Carlo Methods
• Sparsity in Bayesian inversion of parametric operator equations
• Multilevel Quasi-Monte Carlo methods for lognormal diffusion problems
• High-dimensional integration: The quasi-Monte Carlo way
• On the ratio of two correlated normal random variables
• A Multilevel Monte Carlo Method for Computing Failure Probabilities
• MCMC methods for functions: modifying old algorithms to make them faster