{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,16]],"date-time":"2026-06-16T17:16:40Z","timestamp":1781630200136,"version":"3.54.5"},"reference-count":25,"publisher":"American Mathematical Society (AMS)","issue":"270","license":[{"start":{"date-parts":[[2010,11,20]],"date-time":"2010-11-20T00:00:00Z","timestamp":1290211200000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n We present formulas that allow us to decompose a function\u00a0\n \n \n \n f<\/mml:mi>\n f<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of\u00a0\n \n \n \n d<\/mml:mi>\n d<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n variables into a sum of\u00a0\n \n \n \n \n 2<\/mml:mn>\n d<\/mml:mi>\n <\/mml:msup>\n 2^d<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n terms\n \n \n \n \n f<\/mml:mi>\n \n \n u<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:msub>\n f_{\\mathbf {u}}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n indexed by subsets\n \n \n \n \n u<\/mml:mi>\n <\/mml:mrow>\n \\mathbf {u}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of\n \n \n \n \n {<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n \n \u2026\n \n <\/mml:mo>\n ,<\/mml:mo>\n d<\/mml:mi>\n }<\/mml:mo>\n <\/mml:mrow>\n \\{1,\\ldots ,d\\}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , where each term\n \n \n \n \n f<\/mml:mi>\n \n \n u<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:msub>\n f_{\\mathbf {u}}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n depends only on the variables with indices in\n \n \n \n \n u<\/mml:mi>\n <\/mml:mrow>\n \\mathbf {u}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . The decomposition depends on the choice of\n \n \n \n d<\/mml:mi>\n d<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n commuting projections\n \n \n \n \n {<\/mml:mo>\n \n P<\/mml:mi>\n j<\/mml:mi>\n <\/mml:msub>\n \n }<\/mml:mo>\n \n j<\/mml:mi>\n =<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msubsup>\n <\/mml:mrow>\n \\{P_j\\}_{j=1}^d<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , where\n \n \n \n \n \n P<\/mml:mi>\n j<\/mml:mi>\n <\/mml:msub>\n (<\/mml:mo>\n f<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n P_j(f)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n does not depend on the variable\n \n \n \n \n x<\/mml:mi>\n j<\/mml:mi>\n <\/mml:msub>\n x_j<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We present an explicit formula for\n \n \n \n \n f<\/mml:mi>\n \n \n u<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:msub>\n f_{\\mathbf {u}}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , which is new even for the\n anova<\/sc>\n and anchored decompositions; both are special cases of the general decomposition. We show that the decomposition is minimal in the following sense: if\n \n \n \n f<\/mml:mi>\n f<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is expressible as a sum in which there is no term that depends on all of the variables indexed by the subset\n \n \n \n \n z<\/mml:mi>\n <\/mml:mrow>\n \\mathbf {z}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , then, for every choice of\n \n \n \n \n {<\/mml:mo>\n \n P<\/mml:mi>\n j<\/mml:mi>\n <\/mml:msub>\n \n }<\/mml:mo>\n \n j<\/mml:mi>\n =<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msubsup>\n <\/mml:mrow>\n \\{P_j\\}_{j=1}^d<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , the terms\n \n \n \n \n \n f<\/mml:mi>\n \n \n u<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:msub>\n =<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n f_{\\mathbf {u}}=0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for all subsets\n \n \n \n \n u<\/mml:mi>\n <\/mml:mrow>\n \\mathbf {u}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n containing\u00a0\n \n \n \n \n z<\/mml:mi>\n <\/mml:mrow>\n \\mathbf {z}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Furthermore, in a reproducing kernel Hilbert space setting, we give sufficient conditions for the terms\n \n \n \n \n f<\/mml:mi>\n \n \n u<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:msub>\n f_{\\mathbf {u}}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n to be mutually orthogonal.\n <\/p>","DOI":"10.1090\/s0025-5718-09-02319-9","type":"journal-article","created":{"date-parts":[[2010,2,4]],"date-time":"2010-02-04T11:19:48Z","timestamp":1265282388000},"page":"953-966","source":"Crossref","is-referenced-by-count":114,"title":["On decompositions of multivariate functions"],"prefix":"10.1090","volume":"79","author":[{"given":"F.","family":"Kuo","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"I.","family":"Sloan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"G.","family":"Wasilkowski","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"H.","family":"Wo\u017aniakowski","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2009,11,20]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"337","DOI":"10.2307\/1990404","article-title":"Theory of reproducing kernels","volume":"68","author":"Aronszajn, N.","year":"1950","journal-title":"Trans. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9947","issn-type":"print"},{"key":"2","doi-asserted-by":"crossref","unstructured":"R. E. Caflisch, W. Morokoff, and A. Owen, Valuation of mortgage backed securities using Brownian bridges to reduce effective dimension, J. Comput. Finance 1, 27\u201346 (1997).","DOI":"10.21314\/JCF.1997.005"},{"issue":"4-6","key":"3","doi-asserted-by":"publisher","first-page":"436","DOI":"10.1016\/j.jco.2007.02.001","article-title":"Strong tractability of multivariate integration of arbitrary high order using digitally shifted polynomial lattice rules","volume":"23","author":"Dick, Josef","year":"2007","journal-title":"J. Complexity","ISSN":"https:\/\/id.crossref.org\/issn\/0885-064X","issn-type":"print"},{"issue":"5","key":"4","doi-asserted-by":"publisher","first-page":"593","DOI":"10.1016\/j.jco.2003.06.002","article-title":"Liberating the weights","volume":"20","author":"Dick, Josef","year":"2004","journal-title":"J. Complexity","ISSN":"https:\/\/id.crossref.org\/issn\/0885-064X","issn-type":"print"},{"key":"5","isbn-type":"print","doi-asserted-by":"publisher","first-page":"106","DOI":"10.1017\/CBO9780511721571.004","article-title":"Sparse grids and related approximation schemes for higher dimensional problems","author":"Griebel, Michael","year":"2006","ISBN":"https:\/\/id.crossref.org\/isbn\/9780521681612"},{"key":"6","unstructured":"M. Griebel, F. Y. Kuo and I. H. Sloan, The smoothing effect of the anova decomposition, submitted."},{"issue":"248","key":"7","doi-asserted-by":"publisher","first-page":"1885","DOI":"10.1090\/S0025-5718-04-01624-2","article-title":"On tractability of weighted integration over bounded and unbounded regions in \u211d^{\ud835\udd64}","volume":"73","author":"Hickernell, Fred J.","year":"2004","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"8","isbn-type":"print","first-page":"259","article-title":"The strong tractability of multivariate integration using lattice rules","author":"Hickernell, Fred J.","year":"2004","ISBN":"https:\/\/id.crossref.org\/isbn\/3540204660"},{"issue":"11","key":"9","first-page":"1320","article-title":"Lifting the curse of dimensionality","volume":"52","author":"Kuo, Frances Y.","year":"2005","journal-title":"Notices Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9920","issn-type":"print"},{"key":"10","unstructured":"F. Y. Kuo, I. H. Sloan, G. W. Wasilkowski, and H. Wo\u017aniakowski, Liberating the dimension, submitted."},{"key":"11","doi-asserted-by":"crossref","unstructured":"G. Li, J. Schoendorf, T.-S. Ho, and H. Rabitz, Multicut-HDMR with an application to an ionospheric model, J. Comput. Chem. 25, 1149\u20131156 (2004).","DOI":"10.1002\/jcc.20040"},{"key":"12","doi-asserted-by":"crossref","unstructured":"G. Li, J. Hu, S.-W. Wang, P. G. Georgopoulos, J. Schoendorf, and H. Rabitz, Random sampling-high dimensional model representation (RS-HDMR) and orthogonality of its different order component functions, J. Phys. Chem. A 110, 2474\u20132485 (2006).","DOI":"10.1021\/jp054148m"},{"key":"13","doi-asserted-by":"crossref","unstructured":"G. Li and H. Rabitz, Ratio control variate method for efficiently determining high-dimensional model representations, J. Comput. Chem. 27, 1112\u20131118 (2006).","DOI":"10.1002\/jcc.20435"},{"issue":"474","key":"14","doi-asserted-by":"publisher","first-page":"712","DOI":"10.1198\/016214505000001410","article-title":"Estimating mean dimensionality of analysis of variance decompositions","volume":"101","author":"Liu, Ruixue","year":"2006","journal-title":"J. Amer. Statist. Assoc.","ISSN":"https:\/\/id.crossref.org\/issn\/0162-1459","issn-type":"print"},{"key":"15","doi-asserted-by":"crossref","unstructured":"H. Rabitz, O. F. Alis, J. Shorter and K. Shim, Efficient input-output model representation, Comput. Phys. Commun. 117, 11\u201320 (1999).","DOI":"10.1016\/S0010-4655(98)00152-0"},{"issue":"1","key":"16","doi-asserted-by":"publisher","first-page":"46","DOI":"10.1016\/j.jco.2003.11.003","article-title":"Finite-order weights imply tractability of multivariate integration","volume":"20","author":"Sloan, Ian H.","year":"2004","journal-title":"J. Complexity","ISSN":"https:\/\/id.crossref.org\/issn\/0885-064X","issn-type":"print"},{"key":"17","volume-title":"{\\cyr Mnogomernye kvadraturnye formuly i funktsii Khaara}","author":"Tsobol\u2032, I. M.","year":"1969"},{"issue":"4","key":"18","first-page":"407","article-title":"Sensitivity estimates for nonlinear mathematical models","volume":"1","author":"Sobol\u2032, I. M.","year":"1993","journal-title":"Math. Modeling Comput. Experiment","ISSN":"https:\/\/id.crossref.org\/issn\/1061-7590","issn-type":"print"},{"key":"19","doi-asserted-by":"crossref","unstructured":"I. M. Sobol\u2019, Theorems and examples on high dimensional model representation, Reliability Engrg. and System Safety 79, 187\u2013193 (2003).","DOI":"10.1016\/S0951-8320(02)00229-6"},{"key":"20","series-title":"CBMS-NSF Regional Conference Series in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970128","volume-title":"Spline models for observational data","volume":"59","author":"Wahba, Grace","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/0898712440"},{"issue":"2","key":"21","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1016\/S0885-064X(03)00003-7","article-title":"The effective dimension and quasi-Monte Carlo integration","volume":"19","author":"Wang, Xiaoqun","year":"2003","journal-title":"J. Complexity","ISSN":"https:\/\/id.crossref.org\/issn\/0885-064X","issn-type":"print"},{"issue":"4","key":"22","doi-asserted-by":"publisher","first-page":"631","DOI":"10.1093\/imanum\/drl044","article-title":"Brownian bridge and principal component analysis: towards removing the curse of dimensionality","volume":"27","author":"Wang, Xiaoqun","year":"2007","journal-title":"IMA J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"},{"key":"23","isbn-type":"print","first-page":"497","article-title":"Tractability of approximation and integration for weighted tensor product problems over unbounded domains","author":"Wasilkowski, G. W.","year":"2002","ISBN":"https:\/\/id.crossref.org\/isbn\/354042718X"},{"issue":"1","key":"24","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1016\/j.jat.2004.06.011","article-title":"Finite-order weights imply tractability of linear multivariate problems","volume":"130","author":"Wasilkowski, G. W.","year":"2004","journal-title":"J. Approx. Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9045","issn-type":"print"},{"issue":"4","key":"25","doi-asserted-by":"publisher","first-page":"451","DOI":"10.1007\/s10208-005-0171-4","article-title":"Polynomial-time algorithms for multivariate linear problems with finite-order weights: worst case setting","volume":"5","author":"Wasilkowski, G. W.","year":"2005","journal-title":"Found. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1615-3375","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2010-79-270\/S0025-5718-09-02319-9\/S0025-5718-09-02319-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2010-79-270\/S0025-5718-09-02319-9\/S0025-5718-09-02319-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:23:57Z","timestamp":1776788637000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2010-79-270\/S0025-5718-09-02319-9\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,11,20]]},"references-count":25,"journal-issue":{"issue":"270","published-print":{"date-parts":[[2010,4]]}},"alternative-id":["S0025-5718-09-02319-9"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-09-02319-9","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2009,11,20]]}}}