{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:28:18Z","timestamp":1787333298662,"version":"build-2736575974"},"reference-count":19,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>Least squares gradient estimates find application in many fields of computational science, particularly for the purposes of surface fitting and gradient reconstruction in computational fluid dynamics and data visualization. In this paper we derive error bounds for classical and weighted least squares gradient estimates. The bounds reflect how the number of points used in the least squares stencil and the smallest singular value of the least squares matrix impact the accuracy of these estimates. We show how an extrapolation method based on Householder transformations provides substantially tighter bounds. Numerical case studies are presented to elucidate our theory for a data set taken from Franke [Math. Comp., 38 (1982), pp. 181\u2013200].<\/jats:p>","DOI":"10.1137\/080744906","type":"journal-article","created":{"date-parts":[[2010,7,29]],"date-time":"2010-07-29T19:05:29Z","timestamp":1280430329000},"page":"2146-2166","source":"Crossref","is-referenced-by-count":13,"title":["Error Bounds for Least Squares Gradient Estimates"],"prefix":"10.1137","volume":"32","author":[{"given":"Ian W.","family":"Turner","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"John A.","family":"Belward","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Moa'ath N.","family":"Oqielat","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,7,29]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1023\/A:1014299228104"},{"key":"R2","unstructured":"T. J. Barth,\n                      Aspects of unstructured grids and finite-volume solvers for the Euler and Navier\u2013Stokes equations\n                      , in Lecture Notes Presented at the VKI Lecture Series 1994\u201305, Feb. 1994, von Karman Institute, Rhode-St-Genese, Belgium, 1994."},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2007.11.022"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.21914\/anziamj.v50i0.1475"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1016\/S0898-1221(03)80021-4"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1023\/B:ACOM.0000032041.68678.fa"},{"key":"R7","unstructured":"O. Davydov,\n                      On the approximation power of local least squares polynomials\n                      , in Proceedings of Algorithms for Approximation IV, J. Levesley, I. J. Anderson, and J. C. Mason, eds., University of Huddersfield, Huddersfield, UK, 2002, pp. 346\u2013353."},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1982-0637296-4"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1216\/RMJ-1984-14-1-223"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1016\/0096-3003(94)00086-J"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1080\/104077901753243179"},{"key":"R12","unstructured":"P. Lancaster and K. Salkauskas,\n                      CURVE: Curve and Surface Fitting, An Introduction\n                      , Academic Press, San Diego, 1986."},{"key":"R13","doi-asserted-by":"crossref","unstructured":"M. Oqielat, I. Turner, and J. Belward,\n                      A hybrid Clough-Tocher method for surface fitting with application to leaf data\n                      , Appl. Math. Model., 33 (2009) pp. 2582\u20132595.","DOI":"10.1016\/j.apm.2008.07.023"},{"key":"R14","doi-asserted-by":"crossref","unstructured":"C. D. Meyer,\n                      Matrix Analysis and Applied Linear Algebra\n                      , SIAM, Philadelphia, 2000.","DOI":"10.1137\/1.9780898719512"},{"key":"R15","doi-asserted-by":"crossref","unstructured":"C. F. Ollivier-Gooch,\n                      A new class of ENO schemes based on unlimited data-dependent least-squares reconstruction\n                      , in Proceedings of the AIAA 34th Aerospace Sciences Meeting and Exhibit, Reno, NV, 1996, AIAA-96-0887.","DOI":"10.2514\/6.1996-887"},{"key":"R16","unstructured":"H. Pfister, F. Wessels, and A. Kaufman,\n                      Sheared interpolation and gradient estimation for real-time volume rendering\n                      , in Proceedings of the Eurographics Hardware Workshop, Oslo, Norway, 1994, pp. 1\u201310."},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1216\/RMJ-1984-14-1-265"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/21\/2\/013"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1016\/j.apnum.2003.11.001"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/080744906","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:49:02Z","timestamp":1787330942000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/080744906"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":19,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/080744906"],"URL":"https:\/\/doi.org\/10.1137\/080744906","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}