Forward and Inverse Uncertainty Quantification using Multilevel Monte Carlo Algorithms for an Elliptic Nonlocal Equation

摘要

This paper considers uncertainty quantification for an elliptic nonlocalequation. In particular, it is assumed that the parameters which define thekernel in the nonlocal operator are uncertain and a priori distributedaccording to a probability measure. It is shown that the induced probabilitymeasure on some quantities of interest arising from functionals of the solutionto the equation with random inputs is well-defined; as is the posteriordistribution on parameters given observations. As the elliptic nonlocalequation cannot be solved approximate posteriors are constructed. Themultilevel Monte Carlo (MLMC) and multilevel sequential Monte Carlo (MLSMC)sampling algorithms are used for a priori and a posteriori estimation,respectively, of quantities of interest. These algorithms reduce the amount ofwork to estimate posterior expectations, for a given level of error, relativeto Monte Carlo and i.i.d. sampling from the posterior at a given level ofapproximation of the solution of the elliptic nonlocal equation.