Proposals which speed-up function-space MCMC
Kody J. H. Law

TL;DR
This paper proposes new methods to accelerate function-space MCMC algorithms by leveraging spectral properties and Hessian information to improve mixing times in Bayesian inverse problems.
Contribution
It introduces two novel rescaling techniques, characteristic function truncation and Hessian-based interpolation, to enhance the efficiency of MCMC in high-dimensional Bayesian inverse problems.
Findings
Improved mixing times demonstrated through theoretical analysis.
Spectral rescaling methods outperform standard MCMC schemes.
Enhanced convergence in high-dimensional inverse problems.
Abstract
Inverse problems lend themselves naturally to a Bayesian formulation, in which the quantity of interest is a posterior distribution of state and/or parameters given some uncertain observations. For the common case in which the forward operator is smoothing, then the inverse problem is ill-posed. Well-posedness is imposed via regularisation in the form of a prior, which is often Gaussian. Under quite general conditions, it can be shown that the posterior is absolutely continuous with respect to the prior and it may be well-defined on function space in terms of its density with respect to the prior. In this case, by constructing a proposal for which the prior is invariant, one can define Metropolis-Hastings schemes for MCMC which are well-defined on function space, and hence do not degenerate as the dimension of the underlying quantity of interest increases to infinity, e.g. under mesh…
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