A reduced basis warm-start iterative solver for the parameterized systems

The reduced basis methods (RBMs) are widely use in fast solution of the parametrized parametrized linear systems. In some problems lacking good order-reduction condition, only the RBMs are not competent to give a high-precision solution with an affordable computational cost of the offline stage. To develop a high-precision solution and balance the offline and online cost, we explore a reasonable and effective framework for accelerating the iterative methods that is based on the RBMs. Firstly, the highly efficient reduced basis (RB) solver is used as the generation tool of accurate initial values. This data-driven initialization method could provide a warm start for the iterative methods. Secondly, we analyze the further acceleration of the RBMs as a preconditioner. For the purpose of high-precision solution, the RBM-preconditioner not only fail to accelerate the convergence but also need to pay more cost for the overuse of the RBMs. Two numerical test on 3D steady-state diffusion equations for two- and six-dimensional parameter space are presented to demonstrate the capability and efficiency of the RBM-initialized pure high-fidelity iterative methods.