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SUMMARY:Low-rank tensor approximation for sampling high dimensional distri
 butions - Robert Scheichl (University of Bath)
DTSTART:20180409T140000Z
DTEND:20180409T143000Z
UID:TALK103549@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:High-dimensional distributions are notoriously difficult to sa
 mple from\, particularly in the context of PDE-constrained inverse problem
 s. In this talk\, we will present general purpose samplers based on low-ra
 nk tensor surrogates in the tensor-train (TT) format\, a methodology that 
 has been exploited already for many years for scalable\, high-dimensional 
 function approximations in quantum chemistry. In the Bayesian context\, th
 e TT surrogate is built in a two stage process. First we build a surrogate
  of the entire PDE solution in the TT format\, using a novel combination o
 f alternating least squares and the TT cross algorithm. It exploits and pr
 eserves the block diagonal structure of the discretised operator in stocha
 stic collocation schemes\, requiring only independent PDE solutions at a f
 ew parameter values\, thus allowing the use of existing high performance P
 DE solvers. In a second stage\, we approximate the high-dimensional poster
 ior density function also in TT format. Due to the particular structure of
  the TT surrogate\, we can build an efficient conditional distribution met
 hod (or Rosenblatt transform) that only requires a sampling algorithm for 
 one-dimensional conditionals. This conditional distribution method can als
 o be used for other high-dimensional distributions\, not necessarily comin
 g from a PDE-constrained inverse problem. The overall computational cost a
 nd storage requirements of the sampler grow linearly with the dimension. F
 or sufficiently smooth distributions\, the ranks required for accurate TT 
 approximations are moderate\, leading to significant computational gains. 
 We compare our new sampling method with established methods\, such as the 
 delayed rejection adaptive Metropolis (DRAM) algorithm\, as well as with m
 ultilevel quasi-Monte Carlo ratio estimators.  This is joint work with Ser
 gey Dolgov (Bath)\, Colin Fox (Otago) and Karim Anaya-Izquierdo (Bath).
LOCATION:Seminar Room 1\, Newton Institute
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