Computational Mathematics Seminar

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\nUncertainty qu
antification and inverse problems in many variables are pressingly needed
tasks\, yet high-dimensional functions are notoriously difficult to integr
ate in order to compute desired quantities of interest.

\nFunctional ap
proximations\, in particular the low-rank separation of variables into ten
sor product decompositions\, have become popular for reducing the computat
ional cost of high-dimensional integration down to linear scaling in the n
umber of variables. However\, tensor approximations may be inefficient for
non-smooth functions. Sampling based Monte Carlo methods are more general
\, but they may exhibit a very slow convergence\, overlooking a hidden str
ucture of the function.

\nIn this talk we review tensor product approxi
mations for the problem of uncertainty quantification and Bayesian inferen
ce. This allows efficient integration of smooth PDE solutions\, posterior
density functions and quantities of interest. Moreover\, we can use the lo
w-rank approximation of the density function to construct efficient propos
als in the MCMC algorithm for inverse problems. This combined MCMC method
is more accurate also if the quantity of interest is not smooth\, such as
the indicator function of an event.

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