  This paper proposes and tests the first-ever reduced basis warm-start
iterative method for the parametrized linear systems, exemplified by those
discretizing the parametric partial differential equations. Traditional
iterative methods are usually used to obtain the high-fidelity solutions of
these linear systems. However, they typically come with a significant
computational cost which becomes challenging if not entirely untenable when the
parametrized systems need to be solved a large number of times (e.g.
corresponding to different parameter values or time steps). Classical
techniques for mitigating this cost mainly include acceleration approaches such
as preconditioning. This paper advocates for the generation of an initial
prediction with controllable fidelity as an alternative approach to achieve the
same goal. The proposed reduced basis warm-start iterative method leverages the
mathematically rigorous and efficient reduced basis method to generate a
high-quality initial guess thereby decreasing the number of iterative steps.
Via comparison with the iterative method initialized with a zero solution and
the RBM preconditioned and initialized iterative method tested on two 3D
steady-state diffusion equations, we establish the efficacy of the proposed
reduced basis warm-start approach.
