# Accelerating MCMC with active subspaces

@inproceedings{Constantine2015AcceleratingMW, title={Accelerating MCMC with active subspaces}, author={Paul G. Constantine and Carson Kent and Tan Bui-Thanh}, year={2015} }

The Markov chain Monte Carlo (MCMC) method is the computational workhorse for Bayesian inverse problems. However, MCMC struggles in high-dimensional parameter spaces, since its iterates must sequentially explore the high-dimensional space. This struggle is compounded in physical applications when the nonlinear forward model is computationally expensive. One approach to accelerate MCMC is to reduce the dimension of the state space. Active subspaces are part of an emerging set of tools for… Expand

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#### References

SHOWING 1-10 OF 32 REFERENCES

Computing active subspaces with Monte Carlo

- Mathematics
- 2014

Active subspaces can effectively reduce the dimension of high-dimensional parameter studies enabling otherwise infeasible experiments with expensive simulations. The key components of active subspace… Expand

Dimension-independent likelihood-informed MCMC

- Mathematics, Computer Science
- J. Comput. Phys.
- 2016

This work introduces a family of Markov chain Monte Carlo samplers that can adapt to the particular structure of a posterior distribution over functions that may be useful for a large class of high-dimensional problems where the target probability measure has a density with respect to a Gaussian reference measure. Expand

Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems

- Mathematics, Computer Science
- J. Comput. Phys.
- 2009

This work considers a Bayesian approach to nonlinear inverse problems in which the unknown quantity is a spatial or temporal field, endowed with a hierarchical Gaussian process prior, and introduces truncated Karhunen-Loeve expansions, based on the prior distribution, to efficiently parameterize the unknown field. Expand

Accelerating Markov Chain Monte Carlo Simulation by Differential Evolution with Self-Adaptive Randomized Subspace Sampling

- Mathematics
- 2009

Markov chain Monte Carlo (MCMC) methods have found widespread use in many fields of study to estimate the average properties of complex systems, and for posterior inference in a Bayesian framework.… Expand

A Stochastic Newton MCMC Method for Large-Scale Statistical Inverse Problems with Application to Seismic Inversion

- Computer Science, Mathematics
- SIAM J. Sci. Comput.
- 2012

This work addresses the solution of large-scale statistical inverse problems in the framework of Bayesian inference with a so-called Stochastic Monte Carlo method. Expand

Riemann manifold Langevin and Hamiltonian Monte Carlo methods

- Mathematics
- 2011

The paper proposes Metropolis adjusted Langevin and Hamiltonian Monte Carlo sampling methods defined on the Riemann manifold to resolve the shortcomings of existing Monte Carlo algorithms when… Expand

Sampling the posterior: An approach to non-Gaussian data assimilation

- Mathematics
- 2007

The viewpoint taken in this paper is that data assimilation is fundamentally a statistical problem and that this problem should be cast in a Bayesian framework. In the absence of model error, the… Expand

Optimal Low-rank Approximations of Bayesian Linear Inverse Problems

- Mathematics, Computer Science
- SIAM J. Sci. Comput.
- 2015

Two fast approximations of the posterior mean are proposed and proved optimality with respect to a weighted Bayes risk under squared-error loss and the Hessian of the negative log-likelihood and the prior precision are proved. Expand

Extreme-scale UQ for Bayesian inverse problems governed by PDEs

- Computer Science
- 2012 International Conference for High Performance Computing, Networking, Storage and Analysis
- 2012

This work addresses uncertainty quantification for large-scale inverse problems in a Bayesian inference framework by exploiting the fact that the data are typically informative about low-dimensional manifolds of parameter space to construct low rank approximations of the covariance matrix of the posterior pdf via a matrix-free randomized method. Expand

Active Subspaces - Emerging Ideas for Dimension Reduction in Parameter Studies

- Mathematics, Computer Science
- SIAM spotlights
- 2015

Scientists and engineers use computer simulations to study relationships between a model's input parameters and its outputs. However, thorough parameter studies are challenging, if not impossible,… Expand