{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,7]],"date-time":"2025-10-07T12:06:38Z","timestamp":1759838798108,"version":"3.37.3"},"reference-count":34,"publisher":"Walter de Gruyter GmbH","issue":"4","funder":[{"DOI":"10.13039\/501100001691","name":"Japan Society for the Promotion of Science","doi-asserted-by":"publisher","award":["26730015","24.7985","26310211","15K13460"],"award-info":[{"award-number":["26730015","24.7985","26310211","15K13460"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2016,12,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>Matsumoto, Saito and Matoba recently proposed the Walsh figure of merit (WAFOM),\nwhich is a computable criterion for quasi-Monte Carlo point sets using digital nets.\nSeveral algorithms have been proposed for finding low-WAFOM point sets.\nIn the existing algorithms, the number of points is fixed in advance,\nbut extensible point sets are preferred in some applications.\nIn this paper, we propose a random search algorithm for extensible low-WAFOM point sets.\nFor this, we introduce a method that uses lookup tables to compute WAFOM faster.\nNumerical results show that our extensible low-WAFOM point sets are comparable with Niederreiter\u2013Xing sequences for some low-dimensional and smooth test functions.<\/jats:p>","DOI":"10.1515\/mcma-2016-0119","type":"journal-article","created":{"date-parts":[[2016,11,19]],"date-time":"2016-11-19T10:02:10Z","timestamp":1479549730000},"page":"349-357","source":"Crossref","is-referenced-by-count":2,"title":["A search for extensible low-WAFOM point sets"],"prefix":"10.1515","volume":"22","author":[{"given":"Shin","family":"Harase","sequence":"first","affiliation":[{"name":"College of Science and Engineering, Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu, Shiga, 525-8577, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2016,11,19]]},"reference":[{"key":"2024101615293928780_j_mcma-2016-0119_ref_001_w2aab2b8e1481b1b7b1ab2b1b1Aa","doi-asserted-by":"crossref","unstructured":"Barthelmann V., Novak E. and Ritter K.,\nHigh dimensional polynomial interpolation on sparse grids,\nAdv. Comput. Math. 12 (2000), 273\u2013288.","DOI":"10.1023\/A:1018977404843"},{"key":"2024101615293928780_j_mcma-2016-0119_ref_002_w2aab2b8e1481b1b7b1ab2b1b2Aa","doi-asserted-by":"crossref","unstructured":"Bratley P., Fox B. L. and Niederreiter H.,\nImplementation and tests of low-discrepancy sequences,\nACM Trans. Model. Comput. Simul. 2 (1992), 195\u2013213.","DOI":"10.1145\/146382.146385"},{"key":"2024101615293928780_j_mcma-2016-0119_ref_003_w2aab2b8e1481b1b7b1ab2b1b3Aa","doi-asserted-by":"crossref","unstructured":"Dick J.,\nExplicit constructions of quasi-Monte Carlo rules for the numerical integration of high-dimensional periodic functions,\nSIAM J. Numer. Anal. 45 (2007), 2141\u20132176.","DOI":"10.1137\/060658916"},{"key":"2024101615293928780_j_mcma-2016-0119_ref_004_w2aab2b8e1481b1b7b1ab2b1b4Aa","doi-asserted-by":"crossref","unstructured":"Dick J.,\nWalsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order,\nSIAM J. Numer. Anal. 46 (2008), 1519\u20131553.","DOI":"10.1137\/060666639"},{"key":"2024101615293928780_j_mcma-2016-0119_ref_005_w2aab2b8e1481b1b7b1ab2b1b5Aa","doi-asserted-by":"crossref","unstructured":"Dick J.,\nOn quasi-Monte Carlo rules achieving higher order convergence,\nMonte Carlo and Quasi-Monte Carlo Methods 2008 (Montr\u00e9al 2009),\nSpringer, Berlin (2009), 73\u201396.","DOI":"10.1007\/978-3-642-04107-5_5"},{"key":"2024101615293928780_j_mcma-2016-0119_ref_006_w2aab2b8e1481b1b7b1ab2b1b6Aa","doi-asserted-by":"crossref","unstructured":"Dick J. and Matsumoto M.,\nOn the fast computation of the weight enumerator polynomial and the t value of digital nets over finite abelian groups,\nSIAM J. Discrete Math. 27 (2013), 1335\u20131359.","DOI":"10.1137\/120893677"},{"key":"2024101615293928780_j_mcma-2016-0119_ref_007_w2aab2b8e1481b1b7b1ab2b1b7Aa","doi-asserted-by":"crossref","unstructured":"Dick J. and Pillichshammer F.,\nDigital Nets and Sequences. Discrepancy Theory and Quasi-Monte Carlo Integration,\nCambridge University Press, Cambridge, 2010.","DOI":"10.1017\/CBO9780511761188"},{"key":"2024101615293928780_j_mcma-2016-0119_ref_008_w2aab2b8e1481b1b7b1ab2b1b8Aa","unstructured":"Genz A.,\nTesting multidimensional integration routines,\nTools, Methods, and Languages for Scientific and Engineering Computation,\nNorth-Holland, Amsterdam (1984), 81\u201394."},{"key":"2024101615293928780_j_mcma-2016-0119_ref_009_w2aab2b8e1481b1b7b1ab2b1b9Aa","doi-asserted-by":"crossref","unstructured":"Genz A.,\nA package for testing multiple integration subroutines,\nNumerical Integration: Recent Developments, Software and Applications,\nSpringer, Berlin (1987), 337\u2013340.","DOI":"10.1007\/978-94-009-3889-2_33"},{"key":"2024101615293928780_j_mcma-2016-0119_ref_010_w2aab2b8e1481b1b7b1ab2b1c10Aa","doi-asserted-by":"crossref","unstructured":"Goda T., Ohori R., Suzuki K. and Yoshiki T.,\nThe mean square quasi-Monte Carlo error for digitally shifted digital nets,\nMonte Carlo and Quasi-Monte Carlo Methods,\nSpringer Proc. Math. Stat. 163,\nSpringer, Berlin (2016), 331\u2013350.","DOI":"10.1007\/978-3-319-33507-0_16"},{"key":"2024101615293928780_j_mcma-2016-0119_ref_011_w2aab2b8e1481b1b7b1ab2b1c11Aa","doi-asserted-by":"crossref","unstructured":"Harase S.,\nQuasi-Monte Carlo point sets with small t-values and WAFOM,\nAppl. Math. Comput. 254 (2015), 318\u2013326.","DOI":"10.1016\/j.amc.2014.12.144"},{"key":"2024101615293928780_j_mcma-2016-0119_ref_012_w2aab2b8e1481b1b7b1ab2b1c12Aa","doi-asserted-by":"crossref","unstructured":"Hellekalek P. and Leeb H.,\nDyadic diaphony,\nActa Arith. 80 (1997), 187\u2013196.","DOI":"10.4064\/aa-80-2-187-196"},{"key":"2024101615293928780_j_mcma-2016-0119_ref_013_w2aab2b8e1481b1b7b1ab2b1c13Aa","doi-asserted-by":"crossref","unstructured":"Joe S. and Kuo F. Y.,\nConstructing Sobol\u2019 sequences with better two-dimensional projections,\nSIAM J. Sci. Comput. 30 (2008), 2635\u20132654.","DOI":"10.1137\/070709359"},{"key":"2024101615293928780_j_mcma-2016-0119_ref_014_w2aab2b8e1481b1b7b1ab2b1c14Aa","doi-asserted-by":"crossref","unstructured":"Matou\u0161ek J.,\nOn the L2${L_{2}}$-discrepancy for anchored boxes,\nJ. Complexity 14 (1998), 527\u2013556.","DOI":"10.1006\/jcom.1998.0489"},{"key":"2024101615293928780_j_mcma-2016-0119_ref_015_w2aab2b8e1481b1b7b1ab2b1c15Aa","doi-asserted-by":"crossref","unstructured":"Matsumoto M. and Ohori R.,\nWalsh figure of merit for digital nets: An easy measure for higher order convergent QMC,\nMonte Carlo and Quasi-Monte Carlo Methods,\nSpringer Proc. Math. Stat. 163,\nSpringer, Berlin (2016), 143\u2013160.","DOI":"10.1007\/978-3-319-33507-0_5"},{"key":"2024101615293928780_j_mcma-2016-0119_ref_016_w2aab2b8e1481b1b7b1ab2b1c16Aa","doi-asserted-by":"crossref","unstructured":"Matsumoto M., Saito M. and Matoba K.,\nA computable figure of merit for quasi-Monte Carlo point sets,\nMath. Comp. 83 (2014), 1233\u20131250.","DOI":"10.1090\/S0025-5718-2013-02774-3"},{"key":"2024101615293928780_j_mcma-2016-0119_ref_017_w2aab2b8e1481b1b7b1ab2b1c17Aa","doi-asserted-by":"crossref","unstructured":"Matsumoto M. and Yoshiki T.,\nExistence of higher order convergent quasi-Monte Carlo rules via Walsh figure of merit,\nMonte Carlo and Quasi-Monte Carlo Methods 2012,\nSpringer Proc. Math. Stat. 65,\nSpringer, Berlin (2013), 569\u2013579.","DOI":"10.1007\/978-3-642-41095-6_29"},{"key":"2024101615293928780_j_mcma-2016-0119_ref_018_w2aab2b8e1481b1b7b1ab2b1c18Aa","doi-asserted-by":"crossref","unstructured":"Niederreiter H.,\nPoint sets and sequences with small discrepancy,\nMonatsh. Math. 104 (1987), 273\u2013337.","DOI":"10.1007\/BF01294651"},{"key":"2024101615293928780_j_mcma-2016-0119_ref_019_w2aab2b8e1481b1b7b1ab2b1c19Aa","doi-asserted-by":"crossref","unstructured":"Niederreiter H.,\nRandom Number Generation and Quasi-Monte Carlo Methods,\nCBMS-NSF Regional Conf. Ser. in Appl. Math. 63,\nSociety for Industrial and Applied Mathematics (SIAM), Philadelphia, 1992.","DOI":"10.1137\/1.9781611970081"},{"key":"2024101615293928780_j_mcma-2016-0119_ref_020_w2aab2b8e1481b1b7b1ab2b1c20Aa","doi-asserted-by":"crossref","unstructured":"Novak E. and Ritter K.,\nHigh-dimensional integration of smooth functions over cubes,\nNumer. Math. 75 (1996), 79\u201397.","DOI":"10.1007\/s002110050231"},{"key":"2024101615293928780_j_mcma-2016-0119_ref_021_w2aab2b8e1481b1b7b1ab2b1c21Aa","unstructured":"Ohori R.,\nE,fficient quasi-Monte Carlo integration by adjusting the derivation-sensitivity parameter of Walsh figure of merit\nMaster\u2019s thesis, Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo, 2015."},{"key":"2024101615293928780_j_mcma-2016-0119_ref_022_w2aab2b8e1481b1b7b1ab2b1c22Aa","unstructured":"Ohori R. and Yoshiki T.,\nWalsh figure of merit is efficiently approximable,\nin preparation."},{"key":"2024101615293928780_j_mcma-2016-0119_ref_023_w2aab2b8e1481b1b7b1ab2b1c23Aa","unstructured":"Owen A. B.,\nThe dimension distribution and quadrature test functions,\nStatist. Sinica 13 (2003), 1\u201317."},{"key":"2024101615293928780_j_mcma-2016-0119_ref_024_w2aab2b8e1481b1b7b1ab2b1c24Aa","unstructured":"Patterson D. A. and Hennessy J. L.,\nComputer Organization and Design, Fifth Edition: The Hardware\/Software Interface, 5th ed.,\nMorgan Kaufmann Publishers, San Francisco, 2013."},{"key":"2024101615293928780_j_mcma-2016-0119_ref_025_w2aab2b8e1481b1b7b1ab2b1c25Aa","doi-asserted-by":"crossref","unstructured":"Pirsic G.,\nA software implementation of Niederreiter\u2013Xing sequences,\nMonte Carlo and Quasi-Monte Carlo Methods 2000 (Hong Kong 2000),\nSpringer, Berlin (2002), 434\u2013445.","DOI":"10.1007\/978-3-642-56046-0_30"},{"key":"2024101615293928780_j_mcma-2016-0119_ref_026_w2aab2b8e1481b1b7b1ab2b1c26Aa","doi-asserted-by":"crossref","unstructured":"Pirsic G. and Schmid W. C.,\nCalculation of the quality parameter of digital nets and application to their construction,\nJ. Complexity 17 (2001), 827\u2013839.","DOI":"10.1006\/jcom.2001.0597"},{"key":"2024101615293928780_j_mcma-2016-0119_ref_027_w2aab2b8e1481b1b7b1ab2b1c27Aa","doi-asserted-by":"crossref","unstructured":"Sloan I. H. and Joe S.,\nLattice Methods for Multiple Integration,\nClarendon Press, New York, 1994.","DOI":"10.1093\/oso\/9780198534723.001.0001"},{"key":"2024101615293928780_j_mcma-2016-0119_ref_028_w2aab2b8e1481b1b7b1ab2b1c28Aa","doi-asserted-by":"crossref","unstructured":"Sobol\u2019 I. M.,\nDistribution of points in a cube and approximate evaluation of integrals,\nZ\u0306. Vy\u010disl. Mat. i Mat. Fiz. 7 (1967), 784\u2013802.","DOI":"10.1016\/0041-5553(67)90144-9"},{"key":"2024101615293928780_j_mcma-2016-0119_ref_029_w2aab2b8e1481b1b7b1ab2b1c29Aa","doi-asserted-by":"crossref","unstructured":"Suzuki K.,\nAn explicit construction of point sets with large minimum Dick weight,\nJ. Complexity 30 (2014), 347\u2013354.","DOI":"10.1016\/j.jco.2013.12.002"},{"key":"2024101615293928780_j_mcma-2016-0119_ref_030_w2aab2b8e1481b1b7b1ab2b1c30Aa","doi-asserted-by":"crossref","unstructured":"Suzuki K.,\nWAFOM over abelian groups for quasi-Monte Carlo point sets,\nHiroshima Math. J. 45 (2015), 341\u2013364.","DOI":"10.32917\/hmj\/1448323769"},{"key":"2024101615293928780_j_mcma-2016-0119_ref_031_w2aab2b8e1481b1b7b1ab2b1c31Aa","doi-asserted-by":"crossref","unstructured":"Suzuki K.,\nSuper-polynomial convergence and tractability of multivariate integration for infinitely times differentiable functions,\nJ. Complexity (2016), 10.1016\/j.jco.2016.10.002.","DOI":"10.1016\/j.jco.2016.10.002"},{"key":"2024101615293928780_j_mcma-2016-0119_ref_032_w2aab2b8e1481b1b7b1ab2b1c32Aa","doi-asserted-by":"crossref","unstructured":"Xing C. P. and Niederreiter H.,\nA construction of low-discrepancy sequences using global function fields,\nActa Arith. 73 (1995), 87\u2013102.","DOI":"10.4064\/aa-73-1-87-102"},{"key":"2024101615293928780_j_mcma-2016-0119_ref_033_w2aab2b8e1481b1b7b1ab2b1c33Aa","doi-asserted-by":"crossref","unstructured":"Yoshiki T.,\nA lower bound on WAFOM,\nHiroshima Math. J. 44 (2014), 261\u2013266.","DOI":"10.32917\/hmj\/1419619746"},{"key":"2024101615293928780_j_mcma-2016-0119_ref_034_w2aab2b8e1481b1b7b1ab2b1c34Aa","unstructured":"Yoshiki T.,\nBounds on Walsh coefficients by dyadic difference and a new Koksma\u2013Hlawka type inequality for quasi-Monte Carlo integration,\npreprint 2015, https:\/\/arxiv.org\/abs\/1504.03175."}],"container-title":["Monte Carlo Methods and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.degruyter.com\/view\/j\/mcma.2016.22.issue-4\/mcma-2016-0119\/mcma-2016-0119.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/mcma-2016-0119\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/mcma-2016-0119\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,10,16]],"date-time":"2024-10-16T15:29:56Z","timestamp":1729092596000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/mcma-2016-0119\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,11,19]]},"references-count":34,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2016,11,19]]},"published-print":{"date-parts":[[2016,12,1]]}},"alternative-id":["10.1515\/mcma-2016-0119"],"URL":"https:\/\/doi.org\/10.1515\/mcma-2016-0119","relation":{},"ISSN":["0929-9629","1569-3961"],"issn-type":[{"type":"print","value":"0929-9629"},{"type":"electronic","value":"1569-3961"}],"subject":[],"published":{"date-parts":[[2016,11,19]]}}}