{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,27]],"date-time":"2026-04-27T23:18:05Z","timestamp":1777331885285,"version":"3.51.4"},"reference-count":34,"publisher":"Walter de Gruyter GmbH","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,9,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We propose the use of randomized (scrambled) quasirandom sequences for the purpose of providing practical error estimates for quasi-Monte Carlo (QMC) applications.\nOne popular quasirandom sequence among practitioners is the Halton sequence.\nHowever, Halton subsequences have correlation problems in their highest dimensions, and so using this sequence for high-dimensional integrals dramatically affects the accuracy of QMC.\nConsequently, QMC studies have previously proposed several scrambling methods; however, to varying degrees, scrambled versions of Halton sequences still suffer from the correlation problem as manifested in two-dimensional projections.\nThis paper proposes a modified Halton sequence (MHalton), created using a linear digital scrambling method, which finds the optimal multiplier for the Halton sequence in the linear scrambling space.\nIn order to generate better uniformity of distributed sequences, we have chosen strong MHalton multipliers up to 360 dimensions.\nThe proposed multipliers have been tested and proved to be stronger than several sets of multipliers used in other known scrambling methods.\nTo compare the quality of our proposed scrambled MHalton sequences with others, we have performed several extensive computational tests that use <jats:inline-formula id=\"j_mcma-2019-2041_ineq_9999_w2aab3b7b2b1b6b1aab1c16b1b1Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msub>\n                              <m:mi>L<\/m:mi>\n                              <m:mn>2<\/m:mn>\n                           <\/m:msub>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_mcma-2019-2041_eq_0117.png\"\/>\n                        <jats:tex-math>{L_{2}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>-discrepancy and high-dimensional integration tests.\nMoreover, we have tested MHalton sequences on Mortgage-backed security (MBS), which is one of the most widely used applications in finance.\nWe have tested our proposed MHalton sequence numerically and empirically, and they show optimal results in QMC applications.\nThese confirm the efficiency and safety of our proposed MHalton over scrambling sequences previously used in QMC applications.<\/jats:p>","DOI":"10.1515\/mcma-2019-2041","type":"journal-article","created":{"date-parts":[[2019,8,14]],"date-time":"2019-08-14T09:04:46Z","timestamp":1565773486000},"page":"187-207","source":"Crossref","is-referenced-by-count":4,"title":["A computational investigation of the optimal Halton sequence in QMC applications"],"prefix":"10.1515","volume":"25","author":[{"given":"Manal","family":"Bayousef","sequence":"first","affiliation":[{"name":"Department of Computer Science , Florida State University , Tallahassee , FL 32306-4530 , USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Michael","family":"Mascagni","sequence":"additional","affiliation":[{"name":"Department of Computer Science , Florida State University , Tallahassee , FL 32306-4530; and National Institute of Standards and Technology, Gaithersburg, MD 20899 , USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2019,8,14]]},"reference":[{"key":"2023040102122324292_j_mcma-2019-2041_ref_001_w2aab3b7b2b1b6b1ab1b9b1Aa","doi-asserted-by":"crossref","unstructured":"E. I.  Atanassov and M. K.  Durchova,\nGenerating and testing the modified Halton sequences,\nNumerical Methods and Applications,\nLecture Notes in Comput. Sci. 2542,\nSpringer, Berlin (2003), 91\u201398.","DOI":"10.1007\/3-540-36487-0_9"},{"key":"2023040102122324292_j_mcma-2019-2041_ref_002_w2aab3b7b2b1b6b1ab1b9b2Aa","doi-asserted-by":"crossref","unstructured":"E.  Braaten and G.  Weller,\nImproved low-discrepancy sequence for multidimensional quasi-monte carlo integration,\nJ. Comput. Phys. 33 (1979), no. 2, 249\u2013258.\n10.1016\/0021-9991(79)90019-6","DOI":"10.1016\/0021-9991(79)90019-6"},{"key":"2023040102122324292_j_mcma-2019-2041_ref_003_w2aab3b7b2b1b6b1ab1b9b3Aa","unstructured":"D.  Brunner and A.  Uhl,\nOptimal multipliers for linear congruential pseudo-random number generators with prime moduli: parallel computations and properties,\nBIT 39 (1999), no. 2, 193\u2013209.\n10.1023\/A:1022333627834"},{"key":"2023040102122324292_j_mcma-2019-2041_ref_004_w2aab3b7b2b1b6b1ab1b9b4Aa","doi-asserted-by":"crossref","unstructured":"R. E.  Caflisch,\nMonte Carlo and quasi-Monte Carlo methods,\nActa Numerica 1998,\nActa Numer. 7,\nCambridge University, Cambridge (1998), 1\u201349.","DOI":"10.1017\/S0962492900002804"},{"key":"2023040102122324292_j_mcma-2019-2041_ref_005_w2aab3b7b2b1b6b1ab1b9b5Aa","doi-asserted-by":"crossref","unstructured":"R. E.  Caflisch, W. J.  Morokoff and A. B.  Owen,\nValuation of mortgage backed securities using brownian bridges to reduce effective dimension,\nJ. Comput. Finance 1 (1997), no. 1, 27\u201346.\n10.21314\/JCF.1997.005","DOI":"10.21314\/JCF.1997.005"},{"key":"2023040102122324292_j_mcma-2019-2041_ref_006_w2aab3b7b2b1b6b1ab1b9b6Aa","doi-asserted-by":"crossref","unstructured":"S.  Capstick and B. D.  Keister,\nMultidimensional quadrature algorithms at higher degree and\/or dimension,\nJ. Comput. Phys. 123 (1996), no. 2, 267\u2013273.\n10.1006\/jcph.1996.0023","DOI":"10.1006\/jcph.1996.0023"},{"key":"2023040102122324292_j_mcma-2019-2041_ref_007_w2aab3b7b2b1b6b1ab1b9b7Aa","doi-asserted-by":"crossref","unstructured":"H.  Chi, M.  Mascagni and T.  Warnock,\nOn the optimal Halton sequence,\nMath. Comput. Simulation 70 (2005), no. 1, 9\u201321.\n10.1016\/j.matcom.2005.03.004","DOI":"10.1016\/j.matcom.2005.03.004"},{"key":"2023040102122324292_j_mcma-2019-2041_ref_008_w2aab3b7b2b1b6b1ab1b9b8Aa","doi-asserted-by":"crossref","unstructured":"B.  Fathi Vajargah and A.  Eskandari Chechaglou,\nOptimal Halton sequence via inversive scrambling,\nComm. Statist. Simulation Comput. 42 (2013), no. 2, 476\u2013484.\n10.1080\/03610918.2011.650255","DOI":"10.1080\/03610918.2011.650255"},{"key":"2023040102122324292_j_mcma-2019-2041_ref_009_w2aab3b7b2b1b6b1ab1b9b9Aa","doi-asserted-by":"crossref","unstructured":"H.  Faure and C.  Lemieux,\nGeneralized halton sequences in 2008: A comparative study,\nACM Trans. Model. Comput. Simul. (TOMACS) 19 (2009), no. 4, Article ID 15.","DOI":"10.1145\/1596519.1596520"},{"key":"2023040102122324292_j_mcma-2019-2041_ref_010_w2aab3b7b2b1b6b1ab1b9c10Aa","doi-asserted-by":"crossref","unstructured":"B. L.  Fox,\nAlgorithm 647: Implementation and relative efficiency of quasirandom sequence generators,\nACM Trans. Math. Software (TOMS) 12 (1986), no. 4, 362\u2013376.","DOI":"10.1145\/22721.356187"},{"key":"2023040102122324292_j_mcma-2019-2041_ref_011_w2aab3b7b2b1b6b1ab1b9c11Aa","doi-asserted-by":"crossref","unstructured":"S.  Heinrich,\nEfficient algorithms for computing the L2L_{2}-discrepancy,\nMath. Comp. 65 (1996), no. 216, 1621\u20131633.","DOI":"10.1090\/S0025-5718-96-00756-9"},{"key":"2023040102122324292_j_mcma-2019-2041_ref_012_w2aab3b7b2b1b6b1ab1b9c12Aa","doi-asserted-by":"crossref","unstructured":"R.  Hofer,\nHalton-type sequences in rational bases in the ring of rational integers and in the ring of polynomials over a finite field,\nMath. Comput. Simulation 143 (2018), 78\u201388.\n10.1016\/j.matcom.2016.07.005","DOI":"10.1016\/j.matcom.2016.07.005"},{"key":"2023040102122324292_j_mcma-2019-2041_ref_013_w2aab3b7b2b1b6b1ab1b9c13Aa","doi-asserted-by":"crossref","unstructured":"S.  Joe and F. Y.  Kuo,\nRemark on Algorithm 659: implementing Sobol\u2019s quasirandom sequence generator,\nACM Trans. Math. Software 29 (2003), no. 1, 49\u201357.\n10.1145\/641876.641879","DOI":"10.1145\/641876.641879"},{"key":"2023040102122324292_j_mcma-2019-2041_ref_014_w2aab3b7b2b1b6b1ab1b9c14Aa","doi-asserted-by":"crossref","unstructured":"B. D.  Keister,\nMultidimensional quadrature algorithms,\nComput. Phys. 10 (1996), no. 2, 119\u2013122.\n10.1063\/1.168565","DOI":"10.1063\/1.168565"},{"key":"2023040102122324292_j_mcma-2019-2041_ref_015_w2aab3b7b2b1b6b1ab1b9c15Aa","doi-asserted-by":"crossref","unstructured":"L.  Kocis and W. J.  Whiten,\nComputational investigations of low-discrepancy sequences,\nACM Trans. Math. Software (TOMS) 23 (1997), no. 2, 266\u2013294.\n10.1145\/264029.264064","DOI":"10.1145\/264029.264064"},{"key":"2023040102122324292_j_mcma-2019-2041_ref_016_w2aab3b7b2b1b6b1ab1b9c16Aa","doi-asserted-by":"crossref","unstructured":"J.  Matou\u0161ek,\nOn the L2L_{2}-discrepancy for anchored boxes,\nJ. Complexity 14 (1998), no. 4, 527\u2013556.","DOI":"10.1006\/jcom.1998.0489"},{"key":"2023040102122324292_j_mcma-2019-2041_ref_017_w2aab3b7b2b1b6b1ab1b9c17Aa","doi-asserted-by":"crossref","unstructured":"W. J.  Morokoff and R. E.  Caflisch,\nQuasi-random sequences and their discrepancies,\nSIAM J. Sci. Comput. 15 (1994), no. 6, 1251\u20131279.\n10.1137\/0915077","DOI":"10.1137\/0915077"},{"key":"2023040102122324292_j_mcma-2019-2041_ref_018_w2aab3b7b2b1b6b1ab1b9c18Aa","doi-asserted-by":"crossref","unstructured":"H.  Niederreiter,\nRandom Number Generation and Quasi-Monte Carlo Methods,\nCBMS-NSF Regional Conf. Ser. in Appl. Math. 63,\nSIAM, Philadelphia, 1992.","DOI":"10.1137\/1.9781611970081"},{"key":"2023040102122324292_j_mcma-2019-2041_ref_019_w2aab3b7b2b1b6b1ab1b9c19Aa","doi-asserted-by":"crossref","unstructured":"S.  Ninomiya and S.  Tezuka,\nToward real-time pricing of complex financial derivatives,\nAppl. Math. Finance 3 (1996), no. 1, 1\u201320.\n10.1080\/13504869600000001","DOI":"10.1080\/13504869600000001"},{"key":"2023040102122324292_j_mcma-2019-2041_ref_020_w2aab3b7b2b1b6b1ab1b9c20Aa","unstructured":"A. B.  Owen,\nThe dimension distribution and quadrature test functions,\nStatist. Sinica 13 (2003), no. 1, 1\u201317."},{"key":"2023040102122324292_j_mcma-2019-2041_ref_021_w2aab3b7b2b1b6b1ab1b9c21Aa","doi-asserted-by":"crossref","unstructured":"A.  Papageorgiou and J. F.  Traub,\nFaster evaluation of multidimensional integrals,\nComput. Phys. 11 (1997), no. 6, 574\u2013578.\n10.1063\/1.168616","DOI":"10.1063\/1.168616"},{"key":"2023040102122324292_j_mcma-2019-2041_ref_022_w2aab3b7b2b1b6b1ab1b9c22Aa","doi-asserted-by":"crossref","unstructured":"S. H.  Paskov and J. F.  Traub,\nFaster valuation of financial derivatives,\nJ. Portfolio Manag. 22 (1995), no. 1, 113\u2013123.\n10.3905\/jpm.1995.409541","DOI":"10.3905\/jpm.1995.409541"},{"key":"2023040102122324292_j_mcma-2019-2041_ref_023_w2aab3b7b2b1b6b1ab1b9c23Aa","doi-asserted-by":"crossref","unstructured":"F.  Pausinger,\nWeak multipliers for generalized van der Corput sequences,\nJ. Th\u00e9or. Nombres Bordeaux 24 (2012), no. 3, 729\u2013749.\n10.5802\/jtnb.819","DOI":"10.5802\/jtnb.819"},{"key":"2023040102122324292_j_mcma-2019-2041_ref_024_w2aab3b7b2b1b6b1ab1b9c24Aa","doi-asserted-by":"crossref","unstructured":"T.  Pillards and R.  Cools,\nA note on E. Thi\u00e9mard\u2019s algorithm to compute bounds for the star discrepancy,\nJ. Complexity 21 (2005), no. 3, 320\u2013323.\n10.1016\/j.jco.2004.05.004","DOI":"10.1016\/j.jco.2004.05.004"},{"key":"2023040102122324292_j_mcma-2019-2041_ref_025_w2aab3b7b2b1b6b1ab1b9c25Aa","doi-asserted-by":"crossref","unstructured":"I.  Radovi\u0107, I. M.  Sobol\u2019 and R. F.  Tichy,\nQuasi-Monte Carlo methods for numerical integration: Comparison of different low discrepancy sequences,\nMonte Carlo Methods Appl. 2 (1996), no. 1, 1\u201314.\n10.1515\/mcma.1996.2.1.1","DOI":"10.1515\/mcma.1996.2.1.1"},{"key":"2023040102122324292_j_mcma-2019-2041_ref_026_w2aab3b7b2b1b6b1ab1b9c26Aa","doi-asserted-by":"crossref","unstructured":"I. M.  Sobol,\nGlobal sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates,\nMath. Comput. Simulation 55 (2001), no. 1\u20133, 271\u2013280.\n10.1016\/S0378-4754(00)00270-6","DOI":"10.1016\/S0378-4754(00)00270-6"},{"key":"2023040102122324292_j_mcma-2019-2041_ref_027_w2aab3b7b2b1b6b1ab1b9c27Aa","doi-asserted-by":"crossref","unstructured":"I. M.  Sobol and D. I.  Asotsky,\nOne more experiment on estimating high-dimensional integrals by quasi-Monte Carlo methods,\nMath. Comput. Simulation 62 (2003), no. 3\u20136, 255\u2013263.\n10.1016\/S0378-4754(02)00228-8","DOI":"10.1016\/S0378-4754(02)00228-8"},{"key":"2023040102122324292_j_mcma-2019-2041_ref_028_w2aab3b7b2b1b6b1ab1b9c28Aa","doi-asserted-by":"crossref","unstructured":"S.  Tezuka,\nFinancial applications of quasi-Monte Carlo methods,\nA Mathematical Approach to Research Problems of Science and Technology,\nSpringer, Tokyo (2014), 379\u2013392.","DOI":"10.1007\/978-4-431-55060-0_28"},{"key":"2023040102122324292_j_mcma-2019-2041_ref_029_w2aab3b7b2b1b6b1ab1b9c29Aa","doi-asserted-by":"crossref","unstructured":"E.  Thi\u00e9mard,\nAn algorithm to compute bounds for the star discrepancy,\nJ. Complexity 17 (2001), no. 4, 850\u2013880.\n10.1006\/jcom.2001.0600","DOI":"10.1006\/jcom.2001.0600"},{"key":"2023040102122324292_j_mcma-2019-2041_ref_030_w2aab3b7b2b1b6b1ab1b9c30Aa","doi-asserted-by":"crossref","unstructured":"B.  Vandewoestyne and R.  Cools,\nGood permutations for deterministic scrambled Halton sequences in terms of L2L_{2}-discrepancy,\nJ. Comput. Appl. Math. 189 (2006), no. 1\u20132, 341\u2013361.","DOI":"10.1016\/j.cam.2005.05.022"},{"key":"2023040102122324292_j_mcma-2019-2041_ref_031_w2aab3b7b2b1b6b1ab1b9c31Aa","doi-asserted-by":"crossref","unstructured":"X.  Wang and K.-T.  Fang,\nThe effective dimension and quasi-Monte Carlo integration,\nJ. Complexity 19 (2003), no. 2, 101\u2013124.\n10.1016\/S0885-064X(03)00003-7","DOI":"10.1016\/S0885-064X(03)00003-7"},{"key":"2023040102122324292_j_mcma-2019-2041_ref_032_w2aab3b7b2b1b6b1ab1b9c32Aa","doi-asserted-by":"crossref","unstructured":"X.  Wang and I. H.  Sloan,\nWhy are high-dimensional finance problems often of low effective dimension?,\nSIAM J. Sci. Comput. 27 (2005), no. 1, 159\u2013183.\n10.1137\/S1064827503429429","DOI":"10.1137\/S1064827503429429"},{"key":"2023040102122324292_j_mcma-2019-2041_ref_033_w2aab3b7b2b1b6b1ab1b9c33Aa","doi-asserted-by":"crossref","unstructured":"X.  Wang and K. S.  Tan,\nHow do path generation methods affect the accuracy of quasi-Monte Carlo methods for problems in finance?,\nJ. Complexity 28 (2012), no. 2, 250\u2013277.\n10.1016\/j.jco.2011.10.011","DOI":"10.1016\/j.jco.2011.10.011"},{"key":"2023040102122324292_j_mcma-2019-2041_ref_034_w2aab3b7b2b1b6b1ab1b9c34Aa","doi-asserted-by":"crossref","unstructured":"T. T.  Warnock,\nComputational investigations of low-discrepancy point sets\nApplications of Number Theory to Numerical Analysis,\nAcademic Press, New York (1972), 319\u2013343.","DOI":"10.1016\/B978-0-12-775950-0.50015-7"}],"container-title":["Monte Carlo Methods and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.degruyter.com\/view\/j\/mcma.2019.25.issue-3\/mcma-2019-2041\/mcma-2019-2041.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/mcma-2019-2041\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/mcma-2019-2041\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,2]],"date-time":"2023-04-02T00:43:52Z","timestamp":1680396232000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/mcma-2019-2041\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,8,14]]},"references-count":34,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2019,8,17]]},"published-print":{"date-parts":[[2019,9,1]]}},"alternative-id":["10.1515\/mcma-2019-2041"],"URL":"https:\/\/doi.org\/10.1515\/mcma-2019-2041","relation":{},"ISSN":["1569-3961","0929-9629"],"issn-type":[{"value":"1569-3961","type":"electronic"},{"value":"0929-9629","type":"print"}],"subject":[],"published":{"date-parts":[[2019,8,14]]}}}