Preconditioning Techniques for Reduced Basis Methods for Parameterized Partial Differential Equations
Abstract
The reduced basis methodology is an efficient approach to solve
parameterized discrete partial differential equations when the solution
is needed at many parameter values. An offline step approximates the
solution space and an online step utilizes this approximation, the
reduced basis, to solve a smaller reduced problem, which provides an
accurate estimate of the solution. Traditionally, the reduced problem is
solved using direct methods. However, the size of the reduced system
needed to produce solutions of a given accuracy depends on the
characteristics of the problem, and it may happen that the size is
significantly smaller than that of the original discrete problem but
large enough to make direct solution costly. In this scenario, it may be
more effective to use iterative methods to solve the reduced problem. We
construct preconditioners for reduced iterative methods which are
derived from preconditioners for the full problem. This approach permits
reduced basis methods to be practical for larger bases than direct
methods allow. We illustrate the effectiveness of iterative methods for
solving reduced problems by considering two examples, the steady-state
diffusion and convection-diffusion-reaction equations.