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Abstract
General multivariate distributions are notoriously expensive to sample from, particularly the highdimensional posterior distributions in PDEconstrained inverse problems. This paper develops a sampler for arbitrary continuous multivariate distributions that is based on lowrank surrogates in the tensor train format, a methodology that has been exploited for many years for scalable, highdimensional density function approximation in quantum physics and chemistry. We build upon recent developments of the cross approximation algorithms in linear algebra to construct a tensor train approximation to the target probability density function using a small number of function evaluations. For sufficiently smooth distributions, the storage required for accurate tensor train approximations is moderate, scaling linearly with dimension. In turn, the structure of the tensor train surrogate allows sampling by an efficient conditional distribution method since marginal distributions are computable with linear complexity in dimension. Expected values of nonsmooth quantities of interest, with respect to the surrogate distribution, can be estimated using transformed independent uniformlyrandom seeds that provide Monte Carlo quadrature or transformed points from a quasiMonte Carlo lattice to give more efficient quasiMonte Carlo quadrature. Unbiased estimates may be calculated by correcting the transformed random seeds using a Metropolis–Hastings accept/reject step, while the quasiMonte Carlo quadrature may be corrected either by a controlvariate strategy or by importance weighting. We show that the error in the tensor train approximation propagates linearly into the Metropolis–Hastings rejection rate and the integrated autocorrelation time of the resulting Markov chain; thus, the integrated autocorrelation time may be made arbitrarily close to 1, implying that, asymptotic in sample size, the cost per effectively independent sample is one target density evaluation plus the cheap tensor train surrogate proposal that has linear cost with dimension. These methods are demonstrated in three computed examples: fitting failure time of shock absorbers; a PDEconstrained inverse diffusion problem; and sampling from the Rosenbrock distribution. The delayed rejection adaptive Metropolis (DRAM) algorithm is used as a benchmark. In all computed examples, the importance weightcorrected quasiMonte Carlo quadrature performs best and is more efficient than DRAM by orders of magnitude across a wide range of approximation accuracies and sample sizes. Indeed, all the methods developed here significantly outperform DRAM in all computed examples.
Original language  English 

Pages (fromto)  603625 
Number of pages  23 
Journal  Statistics and Computing 
Volume  30 
Issue number  3 
Early online date  2 Nov 2019 
DOIs  
Publication status  Published  31 Dec 2019 
Keywords
 Importance weights
 MCMC
 Multivariate distributions
 Surrogate models
 Tensor decomposition
ASJC Scopus subject areas
 Theoretical Computer Science
 Statistics and Probability
 Statistics, Probability and Uncertainty
 Computational Theory and Mathematics
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Projects
 1 Finished

Tensor product numerical methods for highdimensional problems in probability and quantum calculations
1/01/16 → 31/12/18
Project: Research council