A stochastic collocation approach for parabolic PDEs with random domain deformations
It provides a rigorous numerical framework for uncertainty quantification in parabolic PDEs with random geometry, which is important for engineering applications but represents an incremental extension of existing methods.
The paper develops a stochastic collocation method for parabolic PDEs with random domain deformations, achieving convergence rates for statistical moments that match numerical experiments.
This work considers the problem of numerically approximating statistical moments of a Quantity of Interest (QoI) that depends on the solution of a linear parabolic partial differential equation. The geometry is assumed to be random and is parameterized by $N$ random variables. The parabolic problem is remapped to a fixed deterministic domain with random coefficients and shown to admit an extension on a well defined region embedded in the complex hyperplane. A Stochastic collocation method with an isotropic Smolyak sparse grid is used to compute the statistical moments of the QoI. In addition, convergence rates for the stochastic moments are derived and compared to numerical experiments.