Unbiased Sampling of Multidimensional Partial Differential Equations with Random Coefficients

We construct an unbiased estimator for function value evaluated at the solution of a partial differential equation with random coefficients. We show that the variance and expected computational cost of our estimator are finite and our estimator is unsensitive to the problem dimension so that it can be applied in problems across various disciplines. For the error analysis, we connect the parabolic partial differential equations with the expectations of stochastic differential equations and perform rough path estimation.