Performance of Low-Rank QR Approximation of the Finite Element Biot–Savart Law

In this paper, we present a low-rank QR method for evaluating the discrete Biot–Savart law. Our goal is to develop an algorithm that is easily implemented on parallel computers. It is assumed that the known current density and the unknown magnetic field are both expressed in a finite-element expansion, and we wish to compute the degrees-of-freedom (DOF) in the basis function expansion of the magnetic field. The matrix that maps the current DOF to the field DOF is full, but if the spatial domain is properly partitioned the matrix can be written as a block matrix, with blocks representing distant interactions being low rank and having a compressed QR representation. While an octree partitioning of the matrix may be ideal, for ease of parallel implementation, we employ a partitioning based on number of processors. The rank of each block (i.e., the compression) is determined by the specific geometry and is computed dynamically. In this paper, we provide the algorithmic details and present computational results for large-scale computations.