{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,16]],"date-time":"2025-05-16T16:27:31Z","timestamp":1747412851885},"reference-count":31,"publisher":"Walter de Gruyter GmbH","issue":"3","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CMMI-1265511","CMMI-1639669"],"award-info":[{"award-number":["CMMI-1265511","CMMI-1639669"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2017,9,1]]},"abstract":"Abstract<\/jats:title>An algorithm is developed for generating samples of vector-valued Gaussian processes and fields. The algorithm is based on Karhunen\u2013Lo\u00e8ve (KL) representations of vector-valued random functionsZ<\/m:mi>\u2062<\/m:mo>(<\/m:mo>x<\/m:mi>)<\/m:mo><\/m:mrow><\/m:mrow><\/m:math>{Z(x)}<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>with finite variances and their construction involves two steps. First, truncation levels{<\/m:mo>m<\/m:mi>i<\/m:mi><\/m:msub>}<\/m:mo><\/m:mrow><\/m:math>{\\{m_{i}\\}}<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>are selected for the KL representations of the components{<\/m:mo>Z<\/m:mi>i<\/m:mi><\/m:msub>\u2062<\/m:mo>(<\/m:mo>x<\/m:mi>)<\/m:mo><\/m:mrow><\/m:mrow>}<\/m:mo><\/m:mrow><\/m:math>{\\{Z_{i}(x)\\}}<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>ofZ<\/m:mi>\u2062<\/m:mo>(<\/m:mo>x<\/m:mi>)<\/m:mo><\/m:mrow><\/m:mrow><\/m:math>{Z(x)}<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>such that they meet imposed accuracies. Second, the truncation levels{<\/m:mo>m<\/m:mi>i<\/m:mi><\/m:msub>}<\/m:mo><\/m:mrow><\/m:math>{\\{m_{i}\\}}<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>are accepted or increased if the accuracies of resulting cross correlation functions ofZ<\/m:mi>\u2062<\/m:mo>(<\/m:mo>x<\/m:mi>)<\/m:mo><\/m:mrow><\/m:mrow><\/m:math>{Z(x)}<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>satisfy or violate preset constraints. Theoretical arguments are used to prove the validity of the proposed KL-based models ofZ<\/m:mi>\u2062<\/m:mo>(<\/m:mo>x<\/m:mi>)<\/m:mo><\/m:mrow><\/m:mrow><\/m:math>{Z(x)}<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. The models are applied to develop an efficient Monte Carlo algorithm for generating samples of vector-valued Gaussian functions. Numerical examples illustrate the implementation of the proposed Monte Carlo algorithm and demonstrate its performance.<\/jats:p>","DOI":"10.1515\/mcma-2017-0112","type":"journal-article","created":{"date-parts":[[2017,7,20]],"date-time":"2017-07-20T10:01:39Z","timestamp":1500544899000},"page":"165-188","source":"Crossref","is-referenced-by-count":4,"title":["Monte Carlo algorithm for vector-valued Gaussian functions with preset component accuracies"],"prefix":"10.1515","volume":"23","author":[{"given":"Mircea","family":"Grigoriu","sequence":"first","affiliation":[{"name":"School of Civil & Environmental Engineering , Cornell University , Ithaca , NY 14853-3501 , USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2017,7,20]]},"reference":[{"key":"2023040101530559604_j_mcma-2017-0112_ref_001_w2aab2b8d761b1b7b1ab2ab1Aa","unstructured":"R. J. 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