Quantify uncertainty by estimating the probability density function of the output of interest using MLMC based Bayes method

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Abstract
In uncertainty quantification, the quantity of interest is usually the statistics of the space and/or time integration of system solution. In order to reduce the computational cost, a Bayes estimator based on multilevel Monte Carlo (MLMC) is introduced in this paper. The cumulative distribution function of the output of interest, that is, the expectation of the indicator function, is estimated by MLMC method instead of the classic Monte Carlo simulation. Then, combined with the corresponding probability density function, the quantity of interest is obtained by using some specific quadrature rules. In addition, the smoothing of indicator function and Latin hypercube sampling are used to accelerate the reduction of variance. An elliptic stochastic partial differential equation is used to provide a research context for this model. Numerical experiments are performed to verify the advantage of computational reduction and accuracy improvement of our MLMC-Bayes method.

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