Certified Reduced Basis Method for Parametrized Partial Differential Equations: a Combination with ANOVA Expansion
In many fields, the computation of an output depending on a field variable is of great interest. If the field variable depends on a high-dimensional parameter, the computational cost involved can be huge. Hence, it is necessary to find efficient and reliable methods to solve such a problem. In this report, we describe a method to solve in an efficient and reliable way elliptic coercive parametric partial differential equations which depend on high-dimensional parameters. The idea is to combine two methods already known, the Reduced Basis (RB) method and the ANOVA expansion. Since also the use of the Reduced Basis method for the approximation of high-dimensional parametric partial differential equations can be computationally expensive, it is important to find a method to approximate the solution in an efficient way. The method is divided in three steps, RB-ANOVA-RB. First, we use the Reduced Basis method with a big tolerance to have a coarse approximation of the output of interest. This allows us to use the ANOVA expansion in order to determine if there exists some less important parameters. If it is the case, we freeze them and then reapply the Reduced Basis method with a low tolerance to get a fine approximation of the output.