A practical determination strategy of optimal threshold parameter for matrix compression in wavelet BEM

Abstract

SUMMARYA practical strategy is developed to determine the optimal threshold parameter for wavelet-based boundary element (BE) analysis. The optimal parameter is determined so that the amount of storage (and computational work) is minimized without reducing the accuracy of the BE solution. In the present study, the Beylkin-type truncation scheme is used in the matrix assembly. To avoid unnecessary integration concerning the truncated entries of a coe cient matrix, a priori estimation of the matrix entries is introduced and thus the truncated entries are determined twice: before and after matrix assembly. The optimal threshold parameter is set based on the equilibrium of the truncation and discretization errors. These errors are estimated in the residual sense. For Laplace problems the discretization error is, in particular, indicated with the potential's contribution c to the residual norm R used in error estimation for mesh adaptation. Since the normalized residual norm c = u (u: the potential components of BE solution) cannot be computed without main BE analysis, the discretization error is estimated by the approximate expression constructed through subsidiary BE calculation with smaller degree of freedom (DOF). The matrix compression using the proposed optimal threshold parameter enables us to generate a sparse matrix with O(N 1+ ) (06 ¡1) non-zero entries. Although the quasi-optimal memory requirements and complexity are not attained, the compression rate of a few per cent can be achieved for N ∼ 1000.

Wavelet compression of parabolic diffusion integral kernels

“…We used time steps Δt = 10 −2 , Δt = 10 −3 and Δt = 10 −4 for calculation of integral kernels in equation (11). We used wavelet transform for matrices of arbitrary size, developed by Ravnik et al [14], to compress the resulting integral matrices.…”

“…Ravnik et al [16] compared wavelet and fast multipole data sparse approximations for boundary-domain integral equations of Poisson type. Koro and Abe worked on developing a practical determination strategy of optimal threshold parameter for matrix compression in wavelet BEM [11]. Xiao et al [21] developed an a-posteriori technique for BEM compression.…”

Matrix-Compression Property of Beylkin-Type Truncation Scheme for Wavelet Bem

“…This is because we did not have an appropriate way for determining the threshold. We can now overcome this difficulty by using the semi-analytical technique proposed by the authors 12) . The applicability of our determination strategy is independent of e.g., the kind of Fredholm integral equations and the kind of wavelets.…”

“…The difference between the performance of the main two schemes in actual BE analysis has thus never been discussed. Moreover, the estimation on the number of non-zero entries for Beylkin-type truncation is limited to 2-D problems 12) . In our previous paper, we have never discussed the efficiency of the Beylkin-type compression in 3-D problems.…”

“…The truncation of small matrix entries is based on either Schneider's level-dependent or the Beylkintype schemes. The truncation threshold for the Beylkin-type scheme is determined with our semianalytical technique 12) . The theoretical estimate claims that the performance of this truncation may be inferior to that of the level-dependent scheme.…”