| Abstract: |
| The approximation of very high-dimensional integrals is a common challenge in
many uncertainty quantification problems. These integrals pop up as
expectations of functionals of the solution of a PDE with random coefficients,
or they appear when calculating the Bayesian posterior mean of a quantity of
interest in an inverse problem. Sometimes the uncertainties can be modelled by
very smooth random fields and then polynomial chaos expansions (PCE) methods
can be used to approximate the uncertainty with very few terms (say 10),
however, when the needed dimensionality is really high (and we are talking
thousands or millions), e.g., for a rough random field with possibly high
variance, the PCE method will have problems. In such a context quasi-Monte
Carlo (QMC) methods can deliver a solution. Sometimes giving
dimension-independent convergence of $1/N$ or even higher algebraic rates. |
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