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General multivariate distributions are notoriously expensive to sample from,particularly the high-dimensional posterior distributions in PDE-constrainedinverse problems. This paper develops a sampler for arbitrary continuousmultivariate distributions that is based on low-rank surrogates in thetensor-train format. We construct a tensor-train approximation to the targetprobability density function using the cross interpolation, which requires asmall number of function evaluations. For sufficiently smooth distributions thestorage required for the TT approximation is moderate, scaling linearly withdimension. The structure of the tensor-train surrogate allows efficientsampling by the conditional distribution method. Unbiased estimates may becalculated by correcting the transformed random seeds using aMetropolis--Hastings accept/reject step. Moreover, one can use a more efficientquasi-Monte Carlo quadrature that may be corrected either by a control-variatestrategy, or by importance weighting. We show that the error in thetensor-train approximation propagates linearly into the Metropolis--Hastingsrejection rate and the integrated autocorrelation time of the resulting Markovchain. These methods are demonstrated in three computed examples: fittingfailure time of shock absorbers; a PDE-constrained inverse diffusion problem;and sampling from the Rosenbrock distribution. The delayed rejection adaptiveMetropolis (DRAM) algorithm is used as a benchmark. We find that theimportance-weight corrected quasi-Monte Carlo quadrature performs best in allcomputed examples, and is orders-of-magnitude more efficient than DRAM across awide range of approximation accuracies and sample sizes. Indeed, all themethods developed here significantly outperform DRAM in all computed examples.

Approximation and sampling of multivariate probability distributions in the tensor train decomposition