Solving helmholtz equation with compressed sensing enhanced finite element methods

Abstract

The generalized finite element method (FEM) approach towards solving the Helmholtz equation involves a very high computational complexity of the order of O(n3) where n is the number of nodes of the FEM formulation. Prior research involved exploiting the special properties of FEM matrices for reducing the computation time and memory involved in solving the large FEM problems. These included both direct and iterative solvers. In more recent times, graphical processing units (GPUs) are being used to accelerate the solvers. In this work, we propose an alternative different approach for solving the Helmholtz equation with reduced memory requirements by incorporating compressed sensing (CS) techniques into the original FEM formulation. Our approach is based on the fundamental assumption that electromagnetic fields are continuous except at source locations and can be represented with sparse coefficients in alternate transform domains such as wavelets or DCT. We present different practical aspects of this approach with respect to one-dimensional FEM problems and conclude by pointing out some open-ended questions with respect to this area of research