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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Low-rank tensor approximation for sampling high di
mensional distributions - Robert Scheichl (Univers
ity of Bath)
DTSTART;TZID=Europe/London:20180409T150000
DTEND;TZID=Europe/London:20180409T153000
UID:TALK103546AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/103546
DESCRIPTION:High-dimensional distributions are notoriously dif
ficult to sample from\, particularly in the contex
t of PDE-constrained inverse problems. In this tal
k\, we will present general purpose samplers based
on low-rank tensor surrogates in the tensor-train
(TT) format\, a methodology that has been exploit
ed already for many years for scalable\, high-dime
nsional function approximations in quantum chemist
ry. In the Bayesian context\, the TT surrogate is
built in a two stage process. First we build a sur
rogate of the entire PDE solution in the TT format
\, using a novel combination of alternating least
squares and the TT cross algorithm. It exploits an
d preserves the block diagonal structure of the di
scretised operator in stochastic collocation schem
es\, requiring only independent PDE solutions at a
few parameter values\, thus allowing the use of e
xisting high performance PDE 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 metho
d (or Rosenblatt transform) that only requires a s
ampling algorithm for one-dimensional conditionals
. This conditional distribution method can also be
used for other high-dimensional distributions\, n
ot necessarily coming from a PDE-constrained inver
se problem. The overall computational cost and sto
rage requirements of the sampler grow linearly wit
h the dimension. For sufficiently smooth distribut
ions\, the ranks required for accurate TT approxim
ations are moderate\, leading to significant compu
tational gains. We compare our new sampling method
with established methods\, such as the delayed re
jection adaptive Metropolis (DRAM) algorithm\, as
well as with multilevel quasi-Monte Carlo ratio es
timators. This is joint work with Sergey Dolgov (
Bath)\, Colin Fox (Otago) and Karim Anaya-Izquierd
o (Bath).
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:info@newton.ac.uk
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