A bi-fidelity stochastic collocation method for transport equations with diffusive scaling and multi-dimensional random inputs

Abstract

In this paper, we consider the development of efficient numerical methods for linear transport equations with random parameters and under the diffusive scaling. We extend to the present case the bi-fidelity stochastic collocation method introduced in [33,50,51]. For the high-fidelity transport model, the asymptotic-preserving scheme [29] is used for each stochastic sample. We employ the simple two-velocity Goldstein-Taylor equation as low-fidelity model to accelerate the convergence of the uncertainty quantification process. The choice is motivated by the fact that both models, high fidelity and low fidelity, share the same diffusion limit. Speed-up is achieved by proper selection of the collocation points and relative approximation of the high-fidelity solution. Extensive numerical experiments are conducted to show the efficiency and accuracy of the proposed method, even in non diffusive regimes, with empirical error bound estimations as studied in [16].

Bi-fidelity stochastic collocation methods for epidemic transport models with uncertainties

“…In this context, multi-fidelity methods have shown to be able to efficiently alleviate such limitations through control variate techniques based on an appropriate use of low-fidelity surrogate models able to accelerate the convergence of stochastic sampling [19,20,29,34,48]. Specifically, general multi-fidelity approaches for kinetic equations have been developed in [19,20], while bi-fidelity techniques with greedy sample selection in [29,30]. We refer also to the recent survey in [16].…”

“…The present paper is devoted to extend the bi-fidelity method for transport equations in the diffusive limit developed in [29] to the case of compartmental systems of multiscale equations designed to model mobility dynamics in an epidemic setting with uncertainty [6,10]. For this purpose, the corresponding hyperbolic system recently introduced in [8,9] will be used as a reduced low-fidelity model.…”