Volume 32, Issue 5
The Effect of Global Smoothness on the Accuracy of Treecodes

Commun. Comput. Phys., 32 (2022), pp. 1332-1360.

Published online: 2023-01

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• Abstract

Treecode algorithms are widely used in evaluation of $N$-body pairwise interactions in $\mathcal{O}(N)$ or $\mathcal{O}(NlogN)$ operations. While they can provide high accuracy approximations, a criticism leveled at the methods is that they lack global smoothness. In this work, we study the effect of smoothness on the accuracy of treecodes by comparing three tricubic interpolation based treecodes with differing smoothness properties: a global $\mathcal{C}^1$ continuous tricubic, and two new tricubic interpolants, one that is globally $\mathcal{C}^0$ continuous and one that is discontinuous. We present numerical results which show that higher smoothness leads to higher accuracy for properties dependent on the derivatives of the kernel, nevertheless the global $\mathcal{C}^0$ continuous and discontinuous treecodes are competitive with the $\mathcal{C}^1$ continuous treecode. One advantage of the discontinuous treecode over the $\mathcal{C}^1$ continuous is that, in addition to function evaluations, the discontinuous treecode only requires evaluations of the first derivatives of the kernel while the $\mathcal{C}^1$ continuous treecode requires evaluations up to third order derivatives. When the first derivatives are computed using finite differences, the discontinuous version can be viewed as kernel independent and of utility for a wider array of kernels with minimal effort.

70-08, 70-10, 76-04, 85-04, 65D99

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@Article{CiCP-32-1332, author = {Boateng , Henry A. and Tlupova , Svetlana}, title = {The Effect of Global Smoothness on the Accuracy of Treecodes}, journal = {Communications in Computational Physics}, year = {2023}, volume = {32}, number = {5}, pages = {1332--1360}, abstract = {

Treecode algorithms are widely used in evaluation of $N$-body pairwise interactions in $\mathcal{O}(N)$ or $\mathcal{O}(NlogN)$ operations. While they can provide high accuracy approximations, a criticism leveled at the methods is that they lack global smoothness. In this work, we study the effect of smoothness on the accuracy of treecodes by comparing three tricubic interpolation based treecodes with differing smoothness properties: a global $\mathcal{C}^1$ continuous tricubic, and two new tricubic interpolants, one that is globally $\mathcal{C}^0$ continuous and one that is discontinuous. We present numerical results which show that higher smoothness leads to higher accuracy for properties dependent on the derivatives of the kernel, nevertheless the global $\mathcal{C}^0$ continuous and discontinuous treecodes are competitive with the $\mathcal{C}^1$ continuous treecode. One advantage of the discontinuous treecode over the $\mathcal{C}^1$ continuous is that, in addition to function evaluations, the discontinuous treecode only requires evaluations of the first derivatives of the kernel while the $\mathcal{C}^1$ continuous treecode requires evaluations up to third order derivatives. When the first derivatives are computed using finite differences, the discontinuous version can be viewed as kernel independent and of utility for a wider array of kernels with minimal effort.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2022-0153}, url = {http://global-sci.org/intro/article_detail/cicp/21366.html} }
TY - JOUR T1 - The Effect of Global Smoothness on the Accuracy of Treecodes AU - Boateng , Henry A. AU - Tlupova , Svetlana JO - Communications in Computational Physics VL - 5 SP - 1332 EP - 1360 PY - 2023 DA - 2023/01 SN - 32 DO - http://doi.org/10.4208/cicp.OA-2022-0153 UR - https://global-sci.org/intro/article_detail/cicp/21366.html KW - Fast summation, treecode, tricubic interpolation. AB -

Treecode algorithms are widely used in evaluation of $N$-body pairwise interactions in $\mathcal{O}(N)$ or $\mathcal{O}(NlogN)$ operations. While they can provide high accuracy approximations, a criticism leveled at the methods is that they lack global smoothness. In this work, we study the effect of smoothness on the accuracy of treecodes by comparing three tricubic interpolation based treecodes with differing smoothness properties: a global $\mathcal{C}^1$ continuous tricubic, and two new tricubic interpolants, one that is globally $\mathcal{C}^0$ continuous and one that is discontinuous. We present numerical results which show that higher smoothness leads to higher accuracy for properties dependent on the derivatives of the kernel, nevertheless the global $\mathcal{C}^0$ continuous and discontinuous treecodes are competitive with the $\mathcal{C}^1$ continuous treecode. One advantage of the discontinuous treecode over the $\mathcal{C}^1$ continuous is that, in addition to function evaluations, the discontinuous treecode only requires evaluations of the first derivatives of the kernel while the $\mathcal{C}^1$ continuous treecode requires evaluations up to third order derivatives. When the first derivatives are computed using finite differences, the discontinuous version can be viewed as kernel independent and of utility for a wider array of kernels with minimal effort.

Henry A. Boateng & Svetlana Tlupova. (2023). The Effect of Global Smoothness on the Accuracy of Treecodes. Communications in Computational Physics. 32 (5). 1332-1360. doi:10.4208/cicp.OA-2022-0153
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