Application of the inverse fast multipole method as a preconditioner in a 3D Helmholtz boundary element method

Abstract

We investigate an efficient preconditioning of iterative methods (such as GMRES) for solving dense linear systems Ax = b that follow from a boundary element method (BEM) for the 3D Helmholtz equation, focusing on the low-frequency regime. While matrix-vector products in GMRES can be accelerated through the low-frequency fast multipole method (LFFMM), the BEM often remains computationally expensive due to the large number of GMRES iterations. We propose the application of the inverse fast multipole method (IFMM) as a preconditioner to accelerate the convergence of GMRES. The IFMM is in essence an approximate direct solver that uses a multilevel hierarchical decomposition and low-rank approximations. The proposed IFMM-based preconditioning has a tunable parameter ε that balances the cost to construct a preconditioner M, which is an approximation of A −1 , and the cost to perform the iterative process by means of M. Namely, using a small (respectively, large) value of ε takes a long (respectively, short) time to construct M, while the number of iterations can be small (respectively, large). A comprehensive set of numerical examples involving various boundary value problems with complicated geometries and mixed boundary conditions is presented to validate the efficiency of the proposed method. We show that the IFMM preconditioner (with a nearly optimal ε of 10 −2) clearly outperforms some common preconditioners for the BEM, achieving 1.2-10.8 times speed-up of the computations, in particular when the scale of the underlying scatterer is about five wavelengths or more. In addition, the IFMM preconditioner is capable of solving complicated problems (in a reasonable amount of time) that BD preconditioner can not.

Are you using the right tools in computational electromagnetics?

“…It is worth mentioning a few words about the recent developments in the integral equation solvers. These advanced solvers use the state‐of‐the‐art fast multipole methods for dealing with the large matrices resulting from integral equation methods …”

“…These advanced solvers use the state-of-the-art fast multipole methods for dealing with the large matrices resulting from integral equation methods. [59][60][61][62] Though integral equation solvers come with the aforementioned advantages, they too have some limitations. For instance, they are not ideal for modeling inhomogeneous complex (anisotropic) materials.…”

Parallelization of the inverse fast multipole method with an application to boundary element method

“…In particular, every leaf node stores all six FMM operators associated its partition, and every internal tree node stores M2M, M2L, and L2P associated with its partition (the other three operators are initially empty and will be computed in the algorithm; see lines 41 -42 in 1). In the context of BEM, the expressions of the FMM translation operators can be found in [15]. Note this initialization step is embarrassingly parallel with respect to all tree nodes, and this was introduced in our previous work [15].…”

“…The first step -hierarchical domain decomposition -can be done using existing parallel partitioning packages, such as Zoltan [25]. The second step -initialization of tree data structure -is embarrassingly parallel with respect to all tree nodes, and this has been introduced in our previous work [15], so we skip the details here. The last two steps -factorization and solve -are the main challenges and the focus of this section.…”

“…26 in [6]), and the overhead of serialization is paid back by the tremendous improvement of parallel efficiency from large group sizes and a small number of groups/colors. In our previous study [15], a naive parallel IFMM algorithm was developed by simply replacing all matrix computations such as matrix-matrix multiplication, LU decomposition, and the SVD with their parallel counterparts in the sequential IFMM algorithm. This approach can be realized easily by linking the sequential IFMM code with a multi-threaded linear algebra library such as the Intel MKL library [20] and the OpenBLAS library [21].…”