Hybrid collocation perturbation for PDEs with random domains
This work addresses the computational challenge of high-dimensional stochastic PDEs for researchers in uncertainty quantification, offering a more efficient approach than standard methods.
The paper develops a hybrid collocation-perturbation method for solving PDEs with random domains, achieving significant dimensionality reduction and computational cost that scales at most quadratically with the number of small-variation dimensions, and linearly when variations are independent.
In this work we consider the problem of approximating the statistics of a given Quantity of Interest (QoI) that depends on the solution of a linear elliptic PDE defined over a random domain parameterized by $N$ random variables. The random domain is split into large and small variations contributions. The large variations are approximated by applying a sparse grid stochastic collocation method. The small variations are approximated with a stochastic collocation-perturbation method. Convergence rates for the variance of the QoI are derived and compared to those obtained in numerical experiments. Our approach significantly reduces the dimensionality of the stochastic problem. The computational cost of this method increases at most quadratically with respect to the number of dimensions of the small variations. Moreover, for the case that the small and large variations are independent the cost increases linearly.