A Multi-Level Reduced Basis Method for Surrogate Modeling

Surrogate modeling is the most common approach for reducing the computational burden of performing statistical analysis of complex models.
We consider parametric partial differential equations (PDE) with operators that have affine dependence on the input parameters, in which case, the reduced basis (RB) method offers the theoretically fastest convergence rate in terms of accuracy per number of samples (model evaluations). However, theory assumes that the samples are drawn from an infinite dimensional space, while in practice, a finite element discretization is use with a fixed dense mesh. We present a method that incorporates multiple meshes with different density, which allow for further reduction of total computational cost in a manner similar to Multi-Level Monte Carlo (MLMC) and Multi-Level Stochastic Collocation (MLSC) methods. Unlike MLMC and MLSC methods, RB works with the PDE operators in addition to the model outputs which presents a set of unique challenges. Specifically, we derive a different PDE for each mesh density where we have to give considerationt o boundary conditions. Furthermor, MLMC and MLSC rely on sharp apriori error estimates to compute optimal distribution of the computation work, which is not available in our context. Thus, we employ a greedy seach similar to the knapsack multi-index problem. We present several examples where we demonstrate $60$\%-$90$\% reduction in cost of the multi-level RB approach compared to the standard RB method.