{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,2]],"date-time":"2026-01-02T17:17:03Z","timestamp":1767374223591,"version":"3.37.3"},"reference-count":32,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2024,2,13]],"date-time":"2024-02-13T00:00:00Z","timestamp":1707782400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,2,13]],"date-time":"2024-02-13T00:00:00Z","timestamp":1707782400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001691","name":"Japan Society for the Promotion of Science","doi-asserted-by":"publisher","award":["20K22301","21K03347"],"award-info":[{"award-number":["20K22301","21K03347"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004721","name":"The University of Tokyo","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100004721","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2024,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Simplices are the fundamental domain when integrating over convex polytopes. The aim of this work is to establish a novel framework of Monte Carlo integration over simplices, throughout from sampling to variance reduction. Namely, we develop a uniform sampling method on the standard simplex consisting of two independent procedures and construct theories on change of measure on each of the two independent elements in the developed sampling technique with a view towards variance reduction by importance sampling. We provide illustrative figures and numerical results to support our theoretical findings and demonstrate the strong potential of the developed framework for effective implementation and acceleration of Monte Carlo integration over simplices.<\/jats:p>","DOI":"10.1007\/s10915-024-02461-0","type":"journal-article","created":{"date-parts":[[2024,2,13]],"date-time":"2024-02-13T18:02:02Z","timestamp":1707847322000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Sampling and Change of Measure for Monte Carlo Integration on Simplices"],"prefix":"10.1007","volume":"98","author":[{"given":"Chenxiao","family":"Song","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3845-015X","authenticated-orcid":false,"given":"Reiichiro","family":"Kawai","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,2,13]]},"reference":[{"key":"2461_CR1","doi-asserted-by":"publisher","DOI":"10.1002\/0471722227","volume-title":"A Primer on Statistical Distributions","author":"N Balakrishnan","year":"2003","unstructured":"Balakrishnan, N., Nevzorov, V.B.: A Primer on Statistical Distributions. John Wiley & Sons Inc, New York (2003)"},{"issue":"273","key":"2461_CR2","doi-asserted-by":"publisher","first-page":"297","DOI":"10.1090\/S0025-5718-2010-02378-6","volume":"80","author":"V Baldoni","year":"2011","unstructured":"Baldoni, V., Berline, N., De Loera, J.A., K\u00f6ppe, M., Vergne, M.: How to integrate a polynomial over a simplex. Math. Comput. 80(273), 297\u2013325 (2011)","journal-title":"Math. Comput."},{"key":"2461_CR3","doi-asserted-by":"publisher","first-page":"817","DOI":"10.1016\/j.jco.2015.06.003","volume":"31","author":"K Basu","year":"2015","unstructured":"Basu, K.: Quasi-Monte Carlo tractability of high dimensional integration over products of simplices. J. Complex. 31, 817\u2013834 (2015)","journal-title":"J. Complex."},{"key":"2461_CR4","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-8643-8","volume-title":"Non-Uniform Random Variate Generation","author":"L Devroye","year":"1986","unstructured":"Devroye, L.: Non-Uniform Random Variate Generation. Springer-Verlag, New York (1986)"},{"issue":"5","key":"2461_CR5","doi-asserted-by":"publisher","first-page":"142","DOI":"10.1007\/s00009-020-01579-3","volume":"17","author":"F Di Tommaso","year":"2020","unstructured":"Di Tommaso, F., Zerroudi, B.: On some numerical integration formulas on the d-dimensional simplex. Mediterr. J. Math. 17(5), 142 (2020)","journal-title":"Mediterr. J. Math."},{"key":"2461_CR6","series-title":"Recent Advances in Optimization and its Applications in Engineering","first-page":"3","volume-title":"Copositive programming\u2014a survey","author":"M D\u00fcr","year":"2010","unstructured":"D\u00fcr, M.: Copositive programming\u2014a survey. Recent Advances in Optimization and its Applications in Engineering, pp. 3\u201320. Springer Berlin Heidelberg, Berlin (2010)"},{"key":"2461_CR7","unstructured":"Fieldsend, J.E.: A short note on the efficient random sampling of the multi-dimensional pyramid between a simplex and the origin lying in the unit hypercube. Technical Report 419, Department of Computer Science, University of Exeter (2005)"},{"issue":"3","key":"2461_CR8","doi-asserted-by":"publisher","first-page":"297","DOI":"10.1145\/838250.838254","volume":"29","author":"A Genz","year":"2003","unstructured":"Genz, A., Cools, R.: An adaptive numerical cubature algorithm for simplices. ACM Trans. Math. Softw. 29(3), 297\u2013308 (2003)","journal-title":"ACM Trans. Math. Softw."},{"issue":"2","key":"2461_CR9","doi-asserted-by":"publisher","first-page":"282","DOI":"10.1137\/0715019","volume":"15","author":"A Grundmann","year":"1978","unstructured":"Grundmann, A., M\u00f6ller, H.M.: Invariant integration formulas for the n-simplex by combinatorial methods. SIAM J. Numer. Anal. 15(2), 282\u2013290 (1978)","journal-title":"SIAM J. Numer. Anal."},{"issue":"3","key":"2461_CR10","doi-asserted-by":"publisher","first-page":"613","DOI":"10.1007\/s10543-018-0703-3","volume":"58","author":"A Guessab","year":"2018","unstructured":"Guessab, A., Semisalov, B.: Numerical integration using integrals over hyperplane sections of simplices in a triangulation of a polytope. BIT Numer. Math. 58(3), 613\u2013660 (2018)","journal-title":"BIT Numer. Math."},{"issue":"55","key":"2461_CR11","doi-asserted-by":"publisher","first-page":"137","DOI":"10.2307\/2002484","volume":"10","author":"PC Hammer","year":"1956","unstructured":"Hammer, P.C., Stroud, A.H.: Numerical integration over simplexes. Math. Tables and Other Aids Comput. 10(55), 137\u2013139 (1956)","journal-title":"Math. Tables and Other Aids Comput."},{"key":"2461_CR12","doi-asserted-by":"publisher","first-page":"797","DOI":"10.1002\/nme.1620110504","volume":"11","author":"P Hillion","year":"1977","unstructured":"Hillion, P.: Numerical integration on a triangle. Int. J. Numer. Meth. Eng. 11, 797\u2013815 (1977)","journal-title":"Int. J. Numer. Meth. Eng."},{"issue":"1","key":"2461_CR13","doi-asserted-by":"publisher","first-page":"293","DOI":"10.1137\/070680564","volume":"47","author":"R Kawai","year":"2009","unstructured":"Kawai, R.: Optimal importance sampling parameter search for L\u00e9vy processes via stochastic approximation. SIAM J. Numer. Anal. 47(1), 293\u2013307 (2009)","journal-title":"SIAM J. Numer. Anal."},{"issue":"4","key":"2461_CR14","doi-asserted-by":"publisher","first-page":"A1586","DOI":"10.1137\/15M1047192","volume":"39","author":"R Kawai","year":"2017","unstructured":"Kawai, R.: Acceleration on adaptive importance sampling with sample average approximation. SIAM J. Sci. Comput. 39(4), A1586\u2013A1615 (2017)","journal-title":"SIAM J. Sci. Comput."},{"key":"2461_CR15","doi-asserted-by":"publisher","first-page":"440","DOI":"10.1016\/j.cam.2017.01.029","volume":"319","author":"R Kawai","year":"2017","unstructured":"Kawai, R.: Adaptive importance sampling Monte Carlo simulation for general multivariate probability laws. J. Comput. Appl. Math. 319, 440\u2013459 (2017)","journal-title":"J. Comput. Appl. Math."},{"issue":"4","key":"2461_CR16","doi-asserted-by":"publisher","first-page":"A2774","DOI":"10.1137\/18M1173472","volume":"40","author":"R Kawai","year":"2018","unstructured":"Kawai, R.: Optimizing adaptive importance sampling by stochastic approximation. SIAM J. Sci. Comput. 40(4), A2774\u2013A2800 (2018)","journal-title":"SIAM J. Sci. Comput."},{"issue":"1","key":"2461_CR17","doi-asserted-by":"publisher","first-page":"123608","DOI":"10.1016\/j.jmaa.2019.123608","volume":"483","author":"R Kawai","year":"2020","unstructured":"Kawai, R.: Adaptive importance sampling and control variates. J. Math. Anal. Appl. 483(1), 123608 (2020)","journal-title":"J. Math. Anal. Appl."},{"issue":"1","key":"2461_CR18","first-page":"94","volume":"212","author":"M Khosravifard","year":"2009","unstructured":"Khosravifard, M., Esmaeili, M., Saidi, H.: Extension of the Lasserre\u2013Avrachenkov theorem on the integral of multilinear forms over simplices. Appl. Math. Comput. 212(1), 94\u201399 (2009)","journal-title":"Appl. Math. Comput."},{"key":"2461_CR19","doi-asserted-by":"publisher","DOI":"10.1002\/0471722065","volume-title":"Continuous Multivariate Distributions. Models and Applications","author":"S Kotz","year":"2000","unstructured":"Kotz, S., Balakrishnan, N., Johnson, N.L.: Continuous Multivariate Distributions. Models and Applications. Wiley, New York (2000)"},{"key":"2461_CR20","doi-asserted-by":"publisher","first-page":"523","DOI":"10.1007\/s10543-020-00828-x","volume":"61","author":"JB Lasserre","year":"2021","unstructured":"Lasserre, J.B.: Simple formula for integration of polynomials on a simplex. BIT Numer. Math. 61, 523\u2013533 (2021)","journal-title":"BIT Numer. Math."},{"issue":"2","key":"2461_CR21","first-page":"151","volume":"108","author":"JB Lasserre","year":"2001","unstructured":"Lasserre, J.B., Avrachenkov, K.E.: The Multi-Dimensional Version of $$\\int ^b_a x^p dx$$. Am. Math. Mon. 108(2), 151\u2013154 (2001)","journal-title":"Am. Math. Mon."},{"key":"2461_CR22","first-page":"125140","volume":"376","author":"S Lin","year":"2020","unstructured":"Lin, S., Xie, Z.: A new recursive formula for integration of polynomial over simplex. Appl. Math. Comput. 376, 125140 (2020)","journal-title":"Appl. Math. Comput."},{"key":"2461_CR23","volume-title":"Information Theory, Inference, and Learning Algorithms","author":"DJC MacKay","year":"2003","unstructured":"MacKay, D.J.C.: Information Theory, Inference, and Learning Algorithms. Cambridge University Press, Cambridge (2003)"},{"key":"2461_CR24","doi-asserted-by":"crossref","unstructured":"Ng, K.W., Tian, G.-L., Tang, M.-L.: Dirichlet and Related Distributions: Theory, Methods and Applications. Wiley, Wiley Series in Probability and Statistics (2011)","DOI":"10.1002\/9781119995784"},{"issue":"3\u20134","key":"2461_CR25","first-page":"511","volume":"10","author":"T Pillards","year":"2004","unstructured":"Pillards, T., Cools, R.: A theoretical view on transforming low-discrepancy sequences from a cube to a simplex. Monte Carlo Methods Appl. 10(3\u20134), 511\u2013529 (2004)","journal-title":"Monte Carlo Methods Appl."},{"key":"2461_CR26","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1016\/j.cam.2004.03.019","volume":"174","author":"T Pillards","year":"2005","unstructured":"Pillards, T., Cools, R.: Transforming low-discrepancy sequences from a cube to a simplex. J. Comput. Appl. Math. 174, 29\u201342 (2005)","journal-title":"J. Comput. Appl. Math."},{"key":"2461_CR27","doi-asserted-by":"crossref","unstructured":"Rouigueb, A., Maiza, M., Tkourt, A., Cherchour, I.: Integration of polynomials over n-dimensional simplices. In 2019 Federated Conference on Computer Science and Information Systems (FedCSIS), pp. 157\u2013163 (2019)","DOI":"10.15439\/2019F16"},{"issue":"3","key":"2461_CR28","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1145\/3573386","volume":"33","author":"C Song","year":"2023","unstructured":"Song, C., Kawai, R.: Batching adaptive variance reduction. ACM Trans. Model. Comput. Simul. 33(3), 1\u201324 (2023)","journal-title":"ACM Trans. Model. Comput. Simul."},{"issue":"2","key":"2461_CR29","doi-asserted-by":"publisher","first-page":"A898","DOI":"10.1137\/21M1430996","volume":"45","author":"C Song","year":"2023","unstructured":"Song, C., Kawai, R.: Dynamic finite-budget allocation of stratified sampling with adaptive variance reduction by strata. SIAM J. Sci. Comput. 45(2), A898\u2013A932 (2023)","journal-title":"SIAM J. Sci. Comput."},{"issue":"1","key":"2461_CR30","doi-asserted-by":"publisher","first-page":"90","DOI":"10.1137\/0706009","volume":"6","author":"AH Stroud","year":"1969","unstructured":"Stroud, A.H.: A fifth degree integration formula for the n-simplex. SIAM J. Numer. Anal. 6(1), 90\u201398 (1969)","journal-title":"SIAM J. Numer. Anal."},{"key":"2461_CR31","series-title":"Prentice-Hall series in automatic computation","volume-title":"Approximate Calculation of Multiple Integrals","author":"AH Stroud","year":"1971","unstructured":"Stroud, A.H.: Approximate Calculation of Multiple Integrals. Prentice-Hall series in automatic computation, Englewood Cliffs, Prentice-Hall (1971)"},{"key":"2461_CR32","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1090\/S0025-5718-1970-0258283-6","volume":"24","author":"P Sylvester","year":"1970","unstructured":"Sylvester, P.: Symmetric quadrature formulae for simplexes. Math. Comput. 24, 95\u2013100 (1970)","journal-title":"Math. Comput."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-024-02461-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-024-02461-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-024-02461-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,24]],"date-time":"2024-02-24T11:12:24Z","timestamp":1708773144000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-024-02461-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,2,13]]},"references-count":32,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2024,3]]}},"alternative-id":["2461"],"URL":"https:\/\/doi.org\/10.1007\/s10915-024-02461-0","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"type":"print","value":"0885-7474"},{"type":"electronic","value":"1573-7691"}],"subject":[],"published":{"date-parts":[[2024,2,13]]},"assertion":[{"value":"6 April 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 January 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 January 2024","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 February 2024","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflicts of interest"}}],"article-number":"64"}}