An Exponentially Convergent Nonpolynomial Finite Element Method for Time-Harmonic Scattering from Polygons

In recent years nonpolynomial finite element methods have received increasing attention for the efficient solution of wave problems. As with their close cousin the method of particular solutions, high efficiency comes from using solutions to the Helmholtz equation as basis functions. We present and analyze such a method for the scattering of two-dimensional scalar waves from a polygonal domain that achieves exponential convergence purely by increasing the number of basis functions in each element. Key ingredients are the use of basis functions that capture the singularities at corners and the representation of the scattered field towards infinity by a combination of fundamental solutions. The solution is obtained by minimizing a least-squares functional, which we discretize in such a way that a matrix least-squares problem is obtained. We give computable exponential bounds on the rate of convergence of the least-squares functional that are in very good agreement with the observed numerical convergence. Challenging numerical examples, including a nonconvex polygon with several corner singularities, and a cavity domain, are solved to around 10 digits of accuracy with a few seconds of CPU time. The examples are implemented concisely with MPSpack, a MATLAB toolbox for wave computations with nonpolynomial basis functions, developed by the authors. A code example is included

Designing a high-performance boundary element library with OpenCL and Numba

The Bempp boundary element library is a well known library for the simulation of a range of electrostatic, acoustic and electromagnetic problems in homogeneous bounded and unbounded domains. It originally started as a traditional C++ library with a Python interface. Over the last two years we have completely redesigned Bempp as a native Python library, called Bempp-cl, that provides computational backends for OpenCL (using PyOpenCL) and Numba. The OpenCL backend implements kernels for GPUs and CPUs with SIMD optimization. In this paper, we discuss the design of Bempp-cl, provide performance comparisons on different compute devices, and discuss the advantages and disadvantages of OpenCL as compared to Numba

Overresolving in the Laplace domain for convolution quadrature methods

Convolution quadrature (CQ) methods have enjoyed tremendous interest in recent years as an efficient tool for solving time-domain wave problems in unbounded domains via boundary integral equation techniques. In this paper we consider CQ type formulations for the parallel space-time evaluation of multistep or stiffly accurate Runge--Kutta rules for the wave equation. In particular, we decouple the number of Laplace domain solves from the number of time steps. This allows us to overresolve in the Laplace domain by computing more Laplace domain solutions than there are time steps. We use techniques from complex approximation theory to analyze the error of the CQ approximation of the underlying time-stepping rule when overresolving in the Laplace domain and show that the performance is intimately linked to the location of the poles of the solution operator. Several examples using boundary integral equation formulations in the Laplace domain are presented to illustrate the main results. Read More: http://epubs.siam.org/doi/10.1137/16M106474

Spectral decompositions and nonnormality of boundary integral operators in acoustic scattering - extended version

Understanding the spectral properties of boundary integral operators in acoustic scattering has important practical implications, such as for the analysis of the stability of boundary element discretisations or the convergence of iterative solvers as the wavenumber k grows. Yet little is known about spectral decompo- sitions of the standard boundary integral operators in acoustic scattering. Theoretical results are mainly available on the unit disk, where these operators diagonalise in a simple Fourier basis. In this paper we investigate spectral decompositions for more general smooth domains. Based on the decomposition of the acoustic Green’s function in elliptic coordinates we give spectral decompositions on ellipses. For general smooth domains we show that approximate spectral decompositions can be given in terms of circle Fourier modes transplanted onto the boundary of the domain. An important underlying question is whether or not the operators are normal. Based on previous numerical investigations it appears that the standard boundary integral operators are normal only when the domain is a ball and here we prove that this is indeed the case for the acoustic single layer potential. We show that the acoustic single, double and conjugate double layer potential are normal in a scaled inner product on the ellipse. On more general smooth domains the operators can be split into a normal component plus a smooth perturbation. Numerical computations of pseudospectra are presented to demonstrate the nonnonnormal behaviour on general domains

Product algebras for Galerkin discretisations of boundary integral operators and their applications

Operator products occur naturally in a range of regularized boundary integral equation formulations. However, while a Galerkin discretisation only depends on the domain space and the test (or dual) space of the operator, products require a notion of the range. In the boundary element software package Bempp we have implemented a complete operator algebra that depends on knowledge of the domain, range and test space. The aim was to develop a way of working with Galerkin operators in boundary element software that is as close to working with the strong form on paper as possible while hiding the complexities of Galerkin discretisations. In this paper, we demonstrate the implementation of this operator algebra and show, using various Laplace and Helmholtz example problems, how it significantly simplifies the definition and solution of a wide range of typical boundary integral equation problems

Product algebras for Galerkin discretisations of boundary integral operators and their applications

Operator products occur naturally in a range of regularised boundary integral equation formulations. However, while a Galerkin discretisation only depends on the domain space and the test (or dual) space of the operator, products require a notion of the range. In the boundary element software package Bempp, we have implemented a complete operator algebra that depends on knowledge of the domain, range, and test space. The aim was to develop a way of working with Galerkin operators in boundary element software that is as close to working with the strong form on paper as possible, while hiding the complexities of Galerkin discretisations. In this article, we demonstrate the implementation of this operator algebra and show, using various Laplace and Helmholtz example problems, how it significantly simplifies the definition and solution of a wide range of typical boundary integral equation problems

On the Adjoint Operator in Photoacoustic Tomography

Photoacoustic Tomography (PAT) is an emerging biomedical "imaging from coupled physics" technique, in which the image contrast is due to optical absorption, but the information is carried to the surface of the tissue as ultrasound pulses. Many algorithms and formulae for PAT image reconstruction have been proposed for the case when a complete data set is available. In many practical imaging scenarios, however, it is not possible to obtain the full data, or the data may be sub-sampled for faster data acquisition. In such cases, image reconstruction algorithms that can incorporate prior knowledge to ameliorate the loss of data are required. Hence, recently there has been an increased interest in using variational image reconstruction. A crucial ingredient for the application of these techniques is the adjoint of the PAT forward operator, which is described in this article from physical, theoretical and numerical perspectives. First, a simple mathematical derivation of the adjoint of the PAT forward operator in the continuous framework is presented. Then, an efficient numerical implementation of the adjoint using a k-space time domain wave propagation model is described and illustrated in the context of variational PAT image reconstruction, on both 2D and 3D examples including inhomogeneous sound speed. The principal advantage of this analytical adjoint over an algebraic adjoint (obtained by taking the direct adjoint of the particular numerical forward scheme used) is that it can be implemented using currently available fast wave propagation solvers.Comment: submitted to "Inverse Problems

Simple and efficient GPU parallelization of existing H-Matrix accelerated BEM code

In this paper, we demonstrate how GPU-accelerated BEM routines can be used in a simple black-box fashion to accelerate fast boundary element formulations based on Hierarchical Matrices (H-Matrices) with ACA (Adaptive Cross Approximation). In particular, we focus on the expensive evaluation of the discrete weak form of boundary operators associated with the Laplace and the Helmholtz equation in three space dimensions. The method is based on offloading the CPU assembly of elements during the ACA assembly onto a GPU device and to use threading strategies across ACA blocks to create sufficient workload for the GPU. The proposed GPU strategy is designed such that it can be implemented in existing code with minimal changes to the surrounding application structure. This is in particular interesting for existing legacy code that is not from the ground-up designed with GPU computing in mind. Our benchmark study gives realistic impressions of the benefits of GPU-accelerated BEM simulations by using state-of-the-art multi-threaded computations on modern high-performance CPUs as a reference, rather than drawing synthetic comparisons with single-threaded codes. Speed-up plots illustrate that performance gains up to a factor of 5.5 could be realized with GPU computing under these conditions. This refers to a boundary element model with about 4 million unknowns, whose H-Matrix weak form associated with a real-valued (Laplace) boundary operator is set up in only 100 minutes harnessing the two GPUs instead of 9 hours when using the 20 CPU cores at disposal only. The benchmark study is followed by a particularly demanding real-life application, where we compute the scattered high-frequency sound field of a submarine to demonstrate the increase in overall application performance from moving to a GPU-based ACA assembly