Generative modeling of Sparse Approximate Inverse Preconditioners

We present a new deep learning paradigm for the generation of sparse approximate inverse (SPAI) preconditioners for matrix systems arising from the mesh-based discretization of elliptic differential operators. Our approach is based upon the observation that matrices generated in this manner are not arbitrary, but inherit properties from differential operators that they discretize. Consequently, we seek to represent a learnable distribution of high-performance preconditioners from a low-dimensional subspace through a carefully-designed autoencoder, which is able to generate SPAI preconditioners for these systems. The concept has been implemented on a variety of finite element discretizations of second- and fourth-order elliptic partial differential equations with highly promising results.