The panel-clustering method for the wave equation in two spatial dimensions

Abstract

We consider the numerical solution of the wave equation in a two-dimensional domain and start from a boundary integral formulation for its discretization. We employ the convolution quadrature (CQ) for the temporal and a Galerkin boundary element method (BEM) for the spatial discretization. Our main focus is the sparse approximation of the arising sequence of boundary integral operators by panel clustering. This requires the definition of an appropriate admissibility condition such that the arising kernel functions can be efficiently approximated on admissible blocks.The resulting method has log-linear complexity O N (N + M ) q 4+s , s ∈ {0, 1}, where N is the number of time points, M denotes the dimension of the boundary element space, and q = O (log N + log M ) is the order of the panelclustering expansion.Numerical experiments will illustrate the efficiency and accuracy of the proposed CQ-BEM method with panel clustering.

Exact non-reflecting boundary condition for 3D time-dependent multiple scattering–multiple source problems

“…It is easy to check that the volume term of the NRBC is given by I f = I f 1 + I f 2 , where I f i , i = 1, 2 are given by (36).…”

“…This includes the use of discrete convolution quadratures alternative to those of Lubich (see [31,26,32]), which should allow the construction of highly sparse K m and V m matrices, with the position of the non zero elements known a priori, but at the cost of losing the FFT benefits; the use of higher order Lubich convolution rules (see for example [21,33]) or of time integration formulas which do not require to fix a priori the final time instant T and to proceed with constant time step-size (see [34,35]). Finally, we recall that very recently data-sparse techniques, such as panel-clustering, H-matrices and highfrequency fast multipole methods, have been used to reduce the overall computational cost of a 3D space-time BIE Galerkin discretization (see [36,37]). However, the use of these strategies for reducing the computational cost of our NRBC is still at an early stage and needs further investigation.…”

An Improved Grey Clustering Model with Multiattribute Spatial-Temporal Feature for Panel Data and Its Application

“…Considering that the sample is affected by other viewpoints or clustering behavior and the panel data are nonlinear, George et al [23] proposed a nonlinear panel data model which can produce endogenous "strong" and "weak" cross-sectional dependence and used the relevant approximation theory to estimate and infer the model, clustering the objects. Falletta and Sauer [29] started from boundary integral to discretization, considered numerical solution of wave equation in two-dimensional space, and generated loosely approximated sequences by panel clustering and boundary integral operators to improve the validity and accuracy of panel clustering; Li et al [24] constructed the cumulative generating sequence of time series with different targets. e dynamic trend of the original sequence was characterized by the average generating rate of the generating sequence, and then a mean-AGRA grey incidence correlation clustering algorithm is proposed under panel data.…”

On the discretization and application of two space–time boundary integral equations for 3D wave propagation problems in unbounded domains

“…In the last decades, space-time Boundary Integral Equations (BIE) have been successfully applied to wave propagation problems defined in the exterior of a bounded domain (see, for example, [9], [24], [3], [19], [26], [15], [10], [2], [27], [16], [5], [4], [25], [18], [17], [22], [12]).…”

“…We note that in some papers (see [20], [19], [5], [17]), to reduce the method computational cost and storage, a rule for approximating the matrices generated by the chosen Lubich discrete convolution, when this is combined with a Galerkin space discretization, by corresponding sparse ones have been proposed and examined when c = 1, d ≈ 1. However, this is valid only for the single-layer BIE representation, not for the single-double BIE case we consider in this paper.…”