{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,24]],"date-time":"2026-08-24T03:17:37Z","timestamp":1787541457144,"version":"build-2736575974"},"reference-count":18,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>The fast Gauss transform allows for the calculation of the sum of N Gaussians at M points in $\\mathcal{O}(N+M)$ time. Here, we extend the algorithm to a wider class of kernels, motivated by quadrature issues that arise in using integral equation methods for solving the heat equation on moving domains. In particular, robust high-order product integration methods require convolution with $\\mathcal{O}(q)$ distinct Gaussian-type kernels in order to obtain qth-order accuracy in time. The generalized Gauss transform permits the computation of each of these kernels and, thus, the construction of fast solvers with optimal computational complexity. We also develop plane-wave representations of these Gaussian-type fields, permitting the \u201cdiagonal translation\u201d version of the Gauss transform to be applied. When the sources and targets lie on a uniform grid, or a hierarchy of uniform grids, we show that the curse of dimensionality (the exponential growth of complexity constants with dimension) can be mitigated. Under these conditions, the algorithm has a lower operation count than the fast Fourier transform even for modest values of N and M.<\/jats:p>","DOI":"10.1137\/100790744","type":"journal-article","created":{"date-parts":[[2010,10,12]],"date-time":"2010-10-12T19:07:47Z","timestamp":1286910467000},"page":"3092-3107","source":"Crossref","is-referenced-by-count":28,"title":["The Fast Generalized Gauss Transform"],"prefix":"10.1137","volume":"32","author":[{"given":"Marina","family":"Spivak","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shravan K.","family":"Veerapaneni","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Leslie","family":"Greengard","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,10,12]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/0152075"},{"key":"R2","doi-asserted-by":"crossref","unstructured":"C. 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Math."},{"key":"R11","unstructured":"R. B. Guenther and J. W. Lee,\n                      Partial Differential Equations of Mathematical Physics and Integral Equations\n                      , Prentice\u2013Hall, Englewood Cliffs, NJ, 1988."},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2005.854442"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1137\/080732389"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1137\/0912059"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2006.11.001"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2008.05.003"},{"key":"R17","unstructured":"S. K. Veerapaneni and G. Biros,\n                      Arbitrary-Order Accurate Schemes for Computing Boundary Heat Potentials\n                      , preprint, 2009."},{"key":"R18","doi-asserted-by":"crossref","unstructured":"C. Yang, R. Duraiswami, N. Gumerov, and L. S. Davis,\n                      Improved fast Gauss transform and efficient kernel density estimation\n                      , in Proceedings of the Ninth IEEE International Conference on Computer Vision, 2003, pp. 664\u2013671.","DOI":"10.1109\/ICCV.2003.1238383"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/100790744","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:45:49Z","timestamp":1787330749000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/100790744"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":18,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/100790744"],"URL":"https:\/\/doi.org\/10.1137\/100790744","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}