FEM-based discretization-invariant MCMC methods for PDE-constrained Bayesian inverse problems

DOI10.3934/ipi.2016028zbMath1348.65013OpenAlexW2539071726MaRDI QIDQ338594

Tan Bui-Thanh, Quoc P. Nguyen

Publication date: 7 November 2016

Published in: Inverse Problems and Imaging (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.3934/ipi.2016028

zbMATH Keywords

Markov chain Monte CarloGalerkin finite element methodhybrid Monte CarloBayesian inferenceinfinite-dimensional inverse problemsmatrix transfer techniqueMetropolis-adjusted Langevin algorithm

Mathematics Subject Classification ID

Bayesian inference (62F15) Inverse problems for PDEs (35R30) Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs (65N30) Numerical analysis or methods applied to Markov chains (65C40) PDEs in connection with statistics (35Q62)

Related Items

On the implementation of a quasi-Newton interior-point method for PDE-constrained optimization using finite element discretizations ⋮ Stein Variational Gradient Descent on Infinite-Dimensional Space and Applications to Statistical Inverse Problems ⋮ A Hadamard fractional total variation-Gaussian (HFTG) prior for Bayesian inverse problems ⋮ On the accept-reject mechanism for Metropolis-Hastings algorithms ⋮ On unifying randomized methods for inverse problems ⋮ An optimization problem based on a Bayesian approach for the 2D Helmholtz equation ⋮ A Bayesian level set method for an inverse medium scattering problem in acoustics ⋮ Equivalence of weak and strong modes of measures on topological vector spaces ⋮ Posterior contraction for empirical Bayesian approach to inverse problems under non-diagonal assumption ⋮ Generalized parallel tempering on Bayesian inverse problems ⋮ Stabilities of shape identification inverse problems in a Bayesian framework ⋮ Optimal experimental design for infinite-dimensional Bayesian inverse problems governed by PDEs: a review ⋮ Mixing rates for Hamiltonian Monte Carlo algorithms in finite and infinite dimensions ⋮ Variational Bayes' Method for Functions with Applications to Some Inverse Problems ⋮ A Randomized Maximum A Posteriori Method for Posterior Sampling of High Dimensional Nonlinear Bayesian Inverse Problems

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• An Explicit Link between Gaussian Fields and Gaussian Markov Random Fields: The Stochastic Partial Differential Equation Approach
• Hybrid Monte Carlo on Hilbert spaces
• A note on Metropolis-Hastings kernels for general state spaces
• Weak convergence and optimal scaling of random walk Metropolis algorithms
• Optimal tuning of the hybrid Monte Carlo algorithm
• Analysis of the Hessian for inverse scattering problems. III: Inverse medium scattering of electromagnetic waves in three dimensions.
• Analysis of SPDEs arising in path sampling. II: The nonlinear case
• Well-posed stochastic extensions of ill-posed linear problems
• Conditional path sampling of SDEs and the Langevin MCMC method
• Analysis of the Hessian for inverse scattering problems: I. Inverse shape scattering of acoustic waves
• Analysis of the Hessian for inverse scattering problems: II. Inverse medium scattering of acoustic waves
• Inverse problems: A Bayesian perspective
• A Stochastic Newton MCMC Method for Large-Scale Statistical Inverse Problems with Application to Seismic Inversion
• Solving large-scale PDE-constrained Bayesian inverse problems with Riemann manifold Hamiltonian Monte Carlo
• An Analysis of Infinite Dimensional Bayesian Inverse Shape Acoustic Scattering and Its Numerical Approximation
• Computational Optimization of Systems Governed by Partial Differential Equations
• Stochastic Equations in Infinite Dimensions
• Optimal Scaling of Discrete Approximations to Langevin Diffusions
• Riemann Manifold Langevin and Hamiltonian Monte Carlo Methods
• Linear inverse problems for generalised random variables
• Computational Methods for Inverse Problems
• Perspectives in Flow Control and Optimization
• Adaptive Finite Element Methods for Parameter Identification Problems
• Equation of State Calculations by Fast Computing Machines
• A Computational Framework for Infinite-Dimensional Bayesian Inverse Problems Part I: The Linearized Case, with Application to Global Seismic Inversion
• Monte Carlo sampling methods using Markov chains and their applications
• MAP estimators and their consistency in Bayesian nonparametric inverse problems
• A function space HMC algorithm with second order Langevin diffusion limit
• MCMC methods for functions: modifying old algorithms to make them faster