{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,26]],"date-time":"2026-08-26T04:54:23Z","timestamp":1787720063794,"version":"build-2784847793"},"reference-count":34,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2006,1]]},"abstract":"<jats:p>We develop a multi-element generalized polynomial chaos (ME-gPC) method for arbitrary probability measures and apply it to solve ordinary and partial differential equations with stochastic inputs. Given a stochastic input with an arbitrary probability measure, its random space is decomposed into smaller elements. Subsequently, in each element a new random variable with respect to a conditional probability density function (PDF) is defined, and a set of orthogonal polynomials in terms of this random variable is constructed numerically. Then, the generalized polynomial chaos (gPC) method is implemented element-by-element. Numerical experiments show that the cost for the construction of orthogonal polynomials is negligible compared to the total time cost. Efficiency and convergence of ME-gPC are studied numerically by considering some commonly used random variables. ME-gPC provides an efficient and flexible approach to solving differential equations with random inputs, especially for problems related to long-term integration, large perturbation, and stochastic discontinuities.<\/jats:p>","DOI":"10.1137\/050627630","type":"journal-article","created":{"date-parts":[[2006,8,24]],"date-time":"2006-08-24T15:29:50Z","timestamp":1156433390000},"page":"901-928","source":"Crossref","is-referenced-by-count":404,"title":["Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures"],"prefix":"10.1137","volume":"28","author":[{"given":"Xiaoliang","family":"Wan","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"George Em","family":"Karniadakis","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,8,4]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142902418680"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/3\/4\/010"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.2307\/1969178"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1016\/S0045-7825(01)00237-7"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.4171\/ZAA\/696"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.4171\/ZAA\/815"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2004.04.008"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1137\/0903018"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1145\/174603.174605"},{"key":"R10","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1080\/10407788608913478","volume":"9","author":"Ghaddar N. 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