Computational
Mathematics Seminar

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\nUncertainty quantification and inverse prob
lems in many variables are pressingly needed tasks\, yet high-dimensional
functions are notoriously difficult to integrate in order to compute desir
ed quantities of interest.

\nFunctional approximations\, in particular
the low-rank separation of variables into tensor product decompositions\,
have become popular for reducing the computational cost of high-dimensiona
l integration down to linear scaling in the number of variables. However\,
tensor approximations may be inefficient for non-smooth functions. Sampli
ng based Monte Carlo methods are more general\, but they may exhibit a ver
y slow convergence\, overlooking a hidden structure of the function.

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In this talk we review tensor product approximations for the problem of un
certainty quantification and Bayesian inference. This allows efficient int
egration of smooth PDE solutions\, posterior density functions and quantit
ies of interest. Moreover\, we can use the low-rank approximation of the d
ensity function to construct efficient proposals 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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