Multi-order Monte Carlo for uncertainty quantification of multiscale PDEs

Uncertainty quantification (UQ) in multiscale partial differential equations (PDEs) poses significant challenges due to the wide range of scales involved and the high computational cost of traditional numerical methods. This project introduces a novel approach, the Multi-order Monte Carlo (MOMC) method, to efficiently quantify uncertainty in multiscale PDEs. By leveraging different discretization orders based on asymptotic-preserving (AP) implicit-explicit (IMEX) Runge-Kutta methods and combining them with multilevel Monte Carlo (MLMC) techniques, the novel method optimizes computational resources while maintaining accuracy across multiple scales. The potential of MOMC will be investigated on a range of multiscale problems, including the challenging case of the low Mach number limit of the compressible Navier-Stokes equations, with the aim to design a robust approach in handling high dimensional complex systems with varying spatial and temporal scales. This collaboration uniquely combines the complementary expertise of the project participants. Lorenzo Pareschi’s background in IMEX methods and UQ, coupled with Walter Boscheri’s experience in finite volume schemes and HPC, creates a robust framework for tackling the challenges of multiscale PDEs with uncertainty. The fusion of theoretical insights and advanced computational skills ensures that the project will lead to the development of efficient, scalable methods. These innovations will advance the state-of-the-art in uncertainty quantification for multiscale systems, with significant implications for a range of scientific and engineering applications.