Approximating distribution functions in uncertainty quantification using quasi-Monte Carlo methods

Abstract

As high-dimensional problems become increasingly prevalent in many applications, the effective evaluation of these problems within the limits of our current technology poses a great hurdle due to the exponential increase in computational cost as dimensionality increases. One class of strategies for evaluating such problems efficiently  are quasi-Monte Carlo (QMC) methods. Recently the application of quasi-Monte Carlo methods to approximate expected values associated with solutions to elliptic partial differential equations with random coefficients in uncertainty quantification has been of great interest.

In this talk, we look into extending this from the computation of expected values of functionals of the PDE solution to the approximation of distribution functions. This done by reformulating these functions as expectations of an indicator function. However due to the discontinuous nature of the indicator functions, we do not have an integrand that is conducive to obtaining the optimal rate of error convergence. We seek to alleviate this issue using preintegration, whereby we integrate out a single variable of the discontinuous function in order to obtain a function of one dimension less with a sufficient level of smoothness to apply QMC methods. Some theoretical results regarding the error bounds associated with such approximations and the results of numerical experiments will be presented.