Efficient Reduced Basis Methods for Saddle Point Problems With Applications in Groundwater Flow

Reduced basis methods (RBMs) are recommended to reduce the computational cost of solving parameter-dependent PDEs in scenarios where many choices of parameters need to be considered, for example in uncertainty quantification (UQ). A reduced basis is constructed during a computationally demanding offline (or set-up) stage that allows the user to obtain cheap approximations for parameters choices of interest, online. In this paper we consider RBMs for parameter-dependent saddle point problems, in particular the one that arises in the mixed formulation of the Darcy flow problem in groundwater flow modelling. We apply a discrete empirical interpolation method (DEIM) to approximate the inverse of the diffusion coefficient, which depends non-affinely on the system parameters. We develop an efficient RBM that exploits the DEIM approximation and combine it with a sparse grid stochastic collocation mixed finite element method (SCMFEM) to construct a surrogate solution, which then allows for efficient forward UQ. Through numerical experiments we demonstrate that significant computational savings can be made when we use the RB-DEIM-SCMFEM scheme over standard high fidelity methods. For groundwater flow problems, we provide a thorough cost assessment of the new method and show how the size of the reduced basis, and hence, the extent of the savings, depends on the statistical properties of the input parameters.