Reduced multiscale finite element basis methods for elliptic PDEs with parameterized inputs

In this paper, we present some reduced basis methods for elliptic PDEs with parameterized inputs. In the framework of Galerkin projection, dimension reduction techniques are used to construct a reduced order model. If the PDEs have multiscale structures, multiscale finite element method (MsFEM) is one of the efficient approaches to numerically solve the equations. When the inputs of the PDEs are parameterized by a few parameters, the MsFE basis functions usually depend on the parameters. This impacts on the computation efficiency. In order to get the multiscale basis functions independent of parameters, we can build multiscale basis functions based on a set of samples in the parameter space. This will result in a high dimensional MsFE space for approximation and bring great challenge for simulation. To treat this difficulty, we use some optimal strategies to identify a set of optimal reduced basis functions from the high dimensional MsFE space and obtain a reduced order multiscale model. We consider three optimal strategies for model reduction: cross-validation method, greedy algorithm and proper orthogonal decomposition. The dimension of the space spanned by the set of reduced basis functions is much smaller than the dimension of the original full order model. An offline–online computational decomposition is achieved in the reduced multiscale basis methods to significantly improve computation efficiency. Careful comparison is addressed for the reduced basis methods using different optimization strategies. A few numerical results are presented to illustrate the efficacy of the reduced basis methods.