Model Reduction with the Reduced Basis Method and Sparse Grids

The reduced basis (RB) method has become increasingly popular for problems where PDEs have to be solved for varying parameters \(\mu \in \mathcal{D}\) in order to evaluate a parameter-dependent output function \(s : \mathcal{D}\rightarrow \mathrm{IR}\). The idea of the RB method is to compute the solution of the PDE for varying parameters in a problem-specific low-dimensional subspace X N of the high-dimensional finite element space \({X}^{\mathcal{N}}\). We will discuss how sparse grids can be employed within the RB method or to circumvent the RB method altogether. One drawback of the RB method is that the solvers of the governing equations have to be modified and tailored to the reduced basis. This is a severe limitation of the RB method. Our approach interpolates the output function s on a sparse grid. Thus, we compute the respond to a new parameter \(\mu \in \mathcal{D}\) with a simple function evaluation. No modification or in-depth knowledge of the governing equations and its solver are necessary. We present numerical examples to show that we obtain not only competitive results with the interpolation on sparse grids but that we can even be better than the RB approximation if we are only interested in a rough but very fast approximation.