Adaptive multi-element polynomial chaos with discrete measure: Algorithms and application to SPDEs
We develop a multi-element probabilistic collocation method (ME-PCM) for arbitrary discrete probability measures with finite moments and apply it to solve partial differential equations with random parameters. The method is based on numerical construction of orthogonal polynomial bases in terms of a...
Saved in:
| Published in | Applied numerical mathematics Vol. 90; pp. 91 - 110 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier B.V
01.04.2015
|
| Subjects | |
| Online Access | Get full text |
| ISSN | 0168-9274 1873-5460 |
| DOI | 10.1016/j.apnum.2014.11.006 |
Cover
| Abstract | We develop a multi-element probabilistic collocation method (ME-PCM) for arbitrary discrete probability measures with finite moments and apply it to solve partial differential equations with random parameters. The method is based on numerical construction of orthogonal polynomial bases in terms of a discrete probability measure. To this end, we compare the accuracy and efficiency of five different constructions. We develop an adaptive procedure for decomposition of the parametric space using the local variance criterion. We then couple the ME-PCM with sparse grids to study the Korteweg–de Vries (KdV) equation subject to random excitation, where the random parameters are associated with either a discrete or a continuous probability measure. Numerical experiments demonstrate that the proposed algorithms lead to high accuracy and efficiency for hybrid (discrete–continuous) random inputs. |
|---|---|
| AbstractList | We develop a multi-element probabilistic collocation method (ME-PCM) for arbitrary discrete probability measures with finite moments and apply it to solve partial differential equations with random parameters. The method is based on numerical construction of orthogonal polynomial bases in terms of a discrete probability measure. To this end, we compare the accuracy and efficiency of five different constructions. We develop an adaptive procedure for decomposition of the parametric space using the local variance criterion. We then couple the ME-PCM with sparse grids to study the Korteweg–de Vries (KdV) equation subject to random excitation, where the random parameters are associated with either a discrete or a continuous probability measure. Numerical experiments demonstrate that the proposed algorithms lead to high accuracy and efficiency for hybrid (discrete–continuous) random inputs. |
| Author | Zheng, Mengdi Karniadakis, George Em Wan, Xiaoliang |
| Author_xml | – sequence: 1 givenname: Mengdi surname: Zheng fullname: Zheng, Mengdi organization: Division of Applied Mathematics, Brown University, United States – sequence: 2 givenname: Xiaoliang surname: Wan fullname: Wan, Xiaoliang organization: Department of Mathematics, Louisiana State University, United States – sequence: 3 givenname: George Em surname: Karniadakis fullname: Karniadakis, George Em email: George_Karniadakis@brown.edu organization: Division of Applied Mathematics, Brown University, United States |
| BookMark | eNqFkE1LxDAQQIMouH78Ai85emlNtmnaCh4Wv0FQUM9hNp1qljSpSbriv7e6njzoaQ7z3jC8PbLtvENCjjjLOePyZJXD4MY-nzMucs5zxuQWmfG6KrJSSLZNZhNVZ828ErtkL8YVY6wsBZuR5aKFIZk10n60yWRosUeX6ODth_O9AUv1K_hI3016pa2JOmCaYIQ4BjylC_viw7TqIwXXUhgGazQk4x1Nnj4-XFzGA7LTgY14-DP3yfPV5dP5TXZ3f317vrjLdFHIlLVY45yjZCWvim65ZEJA3WClUXQAAspKtrxqBGt0g6WUGmRXIQKUGnkBTbFPjjd3h-DfRoxJ9dO7aC049GNUXMqmrgQXbEKbDaqDjzFgp7RJ31-nAMYqztRXV7VS313VV1fFuZq6Tm7xyx2C6SF8_GOdbSycCqwNBhW1QaexNQF1Uq03f_qfmZaXQw |
| CitedBy_id | crossref_primary_10_1016_j_jsv_2015_06_039 crossref_primary_10_1155_2024_9945531 crossref_primary_10_1016_j_probengmech_2022_103366 crossref_primary_10_3390_aerospace10120990 crossref_primary_10_1016_j_jcp_2019_07_048 crossref_primary_10_1016_j_jcp_2018_12_020 crossref_primary_10_1016_j_ress_2017_08_010 crossref_primary_10_1063_5_0096285 crossref_primary_10_1002_nla_2314 crossref_primary_10_1063_5_0209701 crossref_primary_10_1016_j_cma_2019_03_014 crossref_primary_10_1016_j_jcp_2023_112008 crossref_primary_10_1137_140975930 crossref_primary_10_1109_TMAG_2017_2741519 crossref_primary_10_1016_j_jcp_2022_111425 crossref_primary_10_1137_140966083 crossref_primary_10_1016_j_jcp_2022_111313 |
| Cites_doi | 10.4171/ZAA/815 10.1090/memo/0319 10.1016/j.jcp.2005.03.023 10.1137/040615201 10.1016/j.jcp.2008.07.009 10.1137/0903018 10.1016/j.ress.2012.05.002 10.1088/1478-3975/6/4/046001 10.4171/ZAA/696 10.1016/j.jcp.2005.08.029 10.1090/S0025-5718-69-99647-1 10.1137/S1064827501387826 10.1088/0266-5611/3/4/010 10.1007/s003659900119 10.1137/1037125 10.1007/s002110050231 10.1137/0105007 10.1081/STM-200046459 10.1103/PhysRevLett.68.580 |
| ContentType | Journal Article |
| Copyright | 2014 IMACS |
| Copyright_xml | – notice: 2014 IMACS |
| DBID | AAYXX CITATION 7SC 8FD JQ2 L7M L~C L~D |
| DOI | 10.1016/j.apnum.2014.11.006 |
| DatabaseName | CrossRef Computer and Information Systems Abstracts Technology Research Database ProQuest Computer Science Collection Advanced Technologies Database with Aerospace Computer and Information Systems Abstracts Academic Computer and Information Systems Abstracts Professional |
| DatabaseTitle | CrossRef Computer and Information Systems Abstracts Technology Research Database Computer and Information Systems Abstracts – Academic Advanced Technologies Database with Aerospace ProQuest Computer Science Collection Computer and Information Systems Abstracts Professional |
| DatabaseTitleList | Computer and Information Systems Abstracts |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Applied Sciences |
| EISSN | 1873-5460 |
| EndPage | 110 |
| ExternalDocumentID | 10_1016_j_apnum_2014_11_006 S0168927414001895 |
| GroupedDBID | --K --M -~X .DC .~1 0R~ 1B1 1RT 1~. 1~5 23M 4.4 457 4G. 5GY 5VS 7-5 71M 8P~ 9JN AACTN AAEDT AAEDW AAIAV AAIKJ AAKOC AALRI AAOAW AAQFI AAQXK AAXUO ABAOU ABEFU ABJNI ABMAC ABXDB ABYKQ ACAZW ACDAQ ACGFS ACRLP ADBBV ADEZE ADGUI ADIYS ADMUD AEBSH AEKER AENEX AFFNX AFKWA AFTJW AGHFR AGUBO AGYEJ AHHHB AI. AIEXJ AIGVJ AIKHN AITUG AJBFU AJOXV ALMA_UNASSIGNED_HOLDINGS AMFUW AMRAJ ARUGR ASPBG AVWKF AXJTR AZFZN BKOJK BLXMC CS3 EBS EFJIC EFLBG EJD EO8 EO9 EP2 EP3 F5P FDB FEDTE FGOYB FIRID FNPLU FYGXN G-2 G-Q GBLVA HMJ HVGLF HZ~ IHE J1W KOM M26 M41 MHUIS MO0 N9A O-L O9- OAUVE OZT P-8 P-9 P2P PC. Q38 R2- RIG RNS ROL RPZ SDF SDG SDP SES SEW SME SPC SPCBC SSW SSZ T5K TN5 VH1 VOH WH7 WUQ XPP ZMT ~02 ~G- AATTM AAXKI AAYWO AAYXX ABWVN ACLOT ACRPL ACVFH ADCNI ADNMO AEIPS AEUPX AFJKZ AFPUW AGQPQ AIGII AIIUN AKBMS AKRWK AKYEP ANKPU APXCP CITATION EFKBS ~HD 7SC 8FD JQ2 L7M L~C L~D |
| ID | FETCH-LOGICAL-c336t-de8e21e605173fbb044a89e7ce4faa4a576d179409c9e566ca6f7eeaa5ce13a93 |
| IEDL.DBID | .~1 |
| ISSN | 0168-9274 |
| IngestDate | Thu Oct 02 10:58:53 EDT 2025 Wed Oct 01 01:58:22 EDT 2025 Thu Apr 24 23:11:24 EDT 2025 Fri Feb 23 02:32:57 EST 2024 |
| IsPeerReviewed | true |
| IsScholarly | true |
| Keywords | KdV equation Discrete–continuous inputs Stochastic collocation Data-driven model Charlier polynomials |
| Language | English |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c336t-de8e21e605173fbb044a89e7ce4faa4a576d179409c9e566ca6f7eeaa5ce13a93 |
| Notes | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 23 |
| PQID | 1669874140 |
| PQPubID | 23500 |
| PageCount | 20 |
| ParticipantIDs | proquest_miscellaneous_1669874140 crossref_citationtrail_10_1016_j_apnum_2014_11_006 crossref_primary_10_1016_j_apnum_2014_11_006 elsevier_sciencedirect_doi_10_1016_j_apnum_2014_11_006 |
| ProviderPackageCode | CITATION AAYXX |
| PublicationCentury | 2000 |
| PublicationDate | 2015-04-01 |
| PublicationDateYYYYMMDD | 2015-04-01 |
| PublicationDate_xml | – month: 04 year: 2015 text: 2015-04-01 day: 01 |
| PublicationDecade | 2010 |
| PublicationTitle | Applied numerical mathematics |
| PublicationYear | 2015 |
| Publisher | Elsevier B.V |
| Publisher_xml | – name: Elsevier B.V |
| References | Genz (br0170) 1987 Askey, Wilson (br0020) 1985 Atkinson (br0030) 1989 Chihara (br0060) 1978; vol. 13 Erban, Chapman (br0100) 2009; 6 Dalibard, Castin (br0080) 1992; 68 Lin, Grinberg, Karniadakis (br0210) 2006; 213 Evans (br0090) 1998 Tzenova, Adan, Kulkarni (br0280) 2005; 21 Gautschi (br0160) 1982; 3 Novak, Ritter (br0240) 1999; 15 Novak, Ritter (br0230) 1996; 75 Oladyshkin, Nowak (br0250) 2012; 106 Boley, Golub (br0040) 1987; 3 Fischer (br0120) 1996; 15 Golub, Van Loan (br0180) 1983 Fischer (br0130) 1998; 17 Xiu, Hesthaven (br0310) 2005; 27 Karniadakis, Sherwin (br0200) 2005 Yan, Hanson (br0320) 2006 Ypma (br0330) 1995; 37 Smolyak (br0260) 1963; 4 Golub, Welsch (br0190) 1969; 23 Xiu, Karniadakis (br0300) 2002; 24 Trench (br0270) 1964; 12 Forsythe (br0150) 1957; 5 Wan, Karniadakis (br0290) 2005; 209 Adams (br0010) 1975 Favard (br0110) 1935; 200 Foo, Wan, Karniadakis (br0140) 2008; 227 Foo (10.1016/j.apnum.2014.11.006_br0140) 2008; 227 Golub (10.1016/j.apnum.2014.11.006_br0180) 1983 Erban (10.1016/j.apnum.2014.11.006_br0100) 2009; 6 Yan (10.1016/j.apnum.2014.11.006_br0320) 2006 Forsythe (10.1016/j.apnum.2014.11.006_br0150) 1957; 5 Askey (10.1016/j.apnum.2014.11.006_br0020) 1985 Fischer (10.1016/j.apnum.2014.11.006_br0120) 1996; 15 Lin (10.1016/j.apnum.2014.11.006_br0210) 2006; 213 Atkinson (10.1016/j.apnum.2014.11.006_br0030) 1989 Xiu (10.1016/j.apnum.2014.11.006_br0300) 2002; 24 Smolyak (10.1016/j.apnum.2014.11.006_br0260) 1963; 4 Wan (10.1016/j.apnum.2014.11.006_br0290) 2005; 209 Fischer (10.1016/j.apnum.2014.11.006_br0130) 1998; 17 Karniadakis (10.1016/j.apnum.2014.11.006_br0200) 2005 Evans (10.1016/j.apnum.2014.11.006_br0090) 1998 Golub (10.1016/j.apnum.2014.11.006_br0190) 1969; 23 Tzenova (10.1016/j.apnum.2014.11.006_br0280) 2005; 21 Adams (10.1016/j.apnum.2014.11.006_br0010) 1975 Trench (10.1016/j.apnum.2014.11.006_br0270) 1964; 12 Xiu (10.1016/j.apnum.2014.11.006_br0310) 2005; 27 Novak (10.1016/j.apnum.2014.11.006_br0240) 1999; 15 Genz (10.1016/j.apnum.2014.11.006_br0170) 1987 Gautschi (10.1016/j.apnum.2014.11.006_br0160) 1982; 3 Novak (10.1016/j.apnum.2014.11.006_br0230) 1996; 75 Favard (10.1016/j.apnum.2014.11.006_br0110) 1935; 200 Ypma (10.1016/j.apnum.2014.11.006_br0330) 1995; 37 Oladyshkin (10.1016/j.apnum.2014.11.006_br0250) 2012; 106 Chihara (10.1016/j.apnum.2014.11.006_br0060) 1978; vol. 13 Dalibard (10.1016/j.apnum.2014.11.006_br0080) 1992; 68 Boley (10.1016/j.apnum.2014.11.006_br0040) 1987; 3 |
| References_xml | – volume: 27 start-page: 1118 year: 2005 end-page: 1139 ident: br0310 article-title: High-order collocation methods for differential equations with random inputs publication-title: SIAM J. Sci. Comput. – volume: 3 start-page: 595 year: 1987 end-page: 622 ident: br0040 article-title: A survey of matrix inverse eigenvalue problems publication-title: Inverse Problems – year: 1998 ident: br0090 article-title: Partial Differential Equations – year: 1989 ident: br0030 article-title: An Introduction to Numerical Analysis – volume: 15 start-page: 1 year: 1996 end-page: 18 ident: br0120 article-title: On the condition of orthogonal polynomials via modified moments publication-title: Z. Anal. Anwend. – volume: 106 start-page: 179 year: 2012 end-page: 190 ident: br0250 article-title: Data-driven uncertainty quantification using the arbitrary polynomial chaos expansion publication-title: Reliab. Eng. Syst. Saf. – volume: 227 start-page: 9572 year: 2008 end-page: 9595 ident: br0140 article-title: A multi-element probabilistic collocation method for PDEs with parametric uncertainty: error analysis and applications publication-title: J. Comput. Phys. – volume: 213 start-page: 676 year: 2006 end-page: 703 ident: br0210 article-title: Numerical studies of the stochastic Korteweg–de Vries equation publication-title: J. Comput. Phys. – start-page: 2989 year: 2006 end-page: 2994 ident: br0320 article-title: Option pricing for a stochastic-volatility jump-diffusion model with log uniform jump-amplitudes publication-title: Proceedings of the 2006 American Control Conference – volume: 4 start-page: 240 year: 1963 end-page: 243 ident: br0260 article-title: Quadrature and interpolation formulas for tensor products of certain classes of functions publication-title: Sov. Math. Dokl. – year: 1983 ident: br0180 article-title: Matrix Computations – volume: 6 start-page: 046001 year: 2009 ident: br0100 article-title: Stochastic modeling of reaction diffusion processes: algorithms for bimolecular reactions publication-title: Phys. Biol. – volume: 37 start-page: 531 year: 1995 end-page: 551 ident: br0330 article-title: Historical development of the Newton–Raphson method publication-title: SIAM Rev. – volume: 5 start-page: 74 year: 1957 end-page: 88 ident: br0150 article-title: Generation and use of orthogonal polynomials for data-fitting with a digital computer publication-title: J. Soc. Ind. Appl. Math. – volume: 75 start-page: 79 year: 1996 end-page: 97 ident: br0230 article-title: High dimensional integration of smooth functions over cubes publication-title: Numer. Math. – start-page: 597 year: 2005 end-page: 598 ident: br0200 article-title: Spectral/hp Element Methods for Computational Fluid Dynamics – year: 1975 ident: br0010 article-title: Sobolev Spaces – volume: 17 start-page: 183 year: 1998 end-page: 205 ident: br0130 article-title: On generating orthogonal polynomials for discrete measures publication-title: Z. Anal. Anwend. – start-page: 337 year: 1987 end-page: 340 ident: br0170 article-title: A package for testing multiple integration subroutines, numerical integration: recent developments publication-title: Softw. Appl. – volume: 209 start-page: 617 year: 2005 end-page: 642 ident: br0290 article-title: An adaptive multi-element generalized polynomial chaos method for stochastic differential equations publication-title: J. Comput. Phys. – volume: vol. 13 year: 1978 ident: br0060 article-title: An Introduction to Orthogonal Polynomials publication-title: Mathematics and Its Applications – volume: 23 start-page: 221 year: 1969 end-page: 230 ident: br0190 article-title: Calculation of Gauss quadrature rules publication-title: Math. Comput. – volume: 24 start-page: 619 year: 2002 end-page: 644 ident: br0300 article-title: The Wiener–Askey polynomial chaos for stochastic differential equations publication-title: SIAM J. Sci. Comput. – volume: 21 start-page: 37 year: 2005 end-page: 55 ident: br0280 article-title: Fluid models with jumps publication-title: Stoch. Models – volume: 3 start-page: 289 year: 1982 end-page: 317 ident: br0160 article-title: On generating orthogonal polynomials publication-title: SIAM J. Sci. Stat. Comput. – volume: 12 start-page: 512 year: 1964 end-page: 522 ident: br0270 article-title: An algorithm for the inversion of finite Toeplitz matrices publication-title: SIAM J. Control Optim. – volume: 200 start-page: 2052 year: 1935 end-page: 2053 ident: br0110 article-title: Sur les polynomes de Tchebicheff publication-title: C. R. Acad. Sci., Paris – volume: 15 start-page: 49 year: 1999 end-page: 522 ident: br0240 article-title: Simple cubature formulas with high polynomial exactness publication-title: Constr. Approx. – year: 1985 ident: br0020 article-title: Some Basic Hypergeometric Polynomials That Generalize Jacobi Polynomials publication-title: Mem. Am. Math. Soc. – volume: 68 start-page: 580 year: 1992 end-page: 583 ident: br0080 article-title: Wave-function approach to dissipative processes in quantum optics publication-title: Phys. Rev. Lett. – year: 1989 ident: 10.1016/j.apnum.2014.11.006_br0030 – year: 1998 ident: 10.1016/j.apnum.2014.11.006_br0090 – volume: 17 start-page: 183 year: 1998 ident: 10.1016/j.apnum.2014.11.006_br0130 article-title: On generating orthogonal polynomials for discrete measures publication-title: Z. Anal. Anwend. doi: 10.4171/ZAA/815 – year: 1985 ident: 10.1016/j.apnum.2014.11.006_br0020 article-title: Some Basic Hypergeometric Polynomials That Generalize Jacobi Polynomials doi: 10.1090/memo/0319 – start-page: 597 year: 2005 ident: 10.1016/j.apnum.2014.11.006_br0200 – volume: 209 start-page: 617 issue: 2 year: 2005 ident: 10.1016/j.apnum.2014.11.006_br0290 article-title: An adaptive multi-element generalized polynomial chaos method for stochastic differential equations publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2005.03.023 – volume: 27 start-page: 1118 issue: 3 year: 2005 ident: 10.1016/j.apnum.2014.11.006_br0310 article-title: High-order collocation methods for differential equations with random inputs publication-title: SIAM J. Sci. Comput. doi: 10.1137/040615201 – volume: 227 start-page: 9572 year: 2008 ident: 10.1016/j.apnum.2014.11.006_br0140 article-title: A multi-element probabilistic collocation method for PDEs with parametric uncertainty: error analysis and applications publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2008.07.009 – volume: 3 start-page: 289 issue: 3 year: 1982 ident: 10.1016/j.apnum.2014.11.006_br0160 article-title: On generating orthogonal polynomials publication-title: SIAM J. Sci. Stat. Comput. doi: 10.1137/0903018 – volume: 106 start-page: 179 year: 2012 ident: 10.1016/j.apnum.2014.11.006_br0250 article-title: Data-driven uncertainty quantification using the arbitrary polynomial chaos expansion publication-title: Reliab. Eng. Syst. Saf. doi: 10.1016/j.ress.2012.05.002 – volume: 6 start-page: 046001 issue: 4 year: 2009 ident: 10.1016/j.apnum.2014.11.006_br0100 article-title: Stochastic modeling of reaction diffusion processes: algorithms for bimolecular reactions publication-title: Phys. Biol. doi: 10.1088/1478-3975/6/4/046001 – volume: 15 start-page: 1 year: 1996 ident: 10.1016/j.apnum.2014.11.006_br0120 article-title: On the condition of orthogonal polynomials via modified moments publication-title: Z. Anal. Anwend. doi: 10.4171/ZAA/696 – volume: 213 start-page: 676 issue: 2 year: 2006 ident: 10.1016/j.apnum.2014.11.006_br0210 article-title: Numerical studies of the stochastic Korteweg–de Vries equation publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2005.08.029 – volume: 200 start-page: 2052 year: 1935 ident: 10.1016/j.apnum.2014.11.006_br0110 article-title: Sur les polynomes de Tchebicheff publication-title: C. R. Acad. Sci., Paris – volume: 23 start-page: 221 issue: 106 year: 1969 ident: 10.1016/j.apnum.2014.11.006_br0190 article-title: Calculation of Gauss quadrature rules publication-title: Math. Comput. doi: 10.1090/S0025-5718-69-99647-1 – volume: 4 start-page: 240 year: 1963 ident: 10.1016/j.apnum.2014.11.006_br0260 article-title: Quadrature and interpolation formulas for tensor products of certain classes of functions publication-title: Sov. Math. Dokl. – year: 1975 ident: 10.1016/j.apnum.2014.11.006_br0010 – volume: 12 start-page: 512 year: 1964 ident: 10.1016/j.apnum.2014.11.006_br0270 article-title: An algorithm for the inversion of finite Toeplitz matrices publication-title: SIAM J. Control Optim. – volume: 24 start-page: 619 year: 2002 ident: 10.1016/j.apnum.2014.11.006_br0300 article-title: The Wiener–Askey polynomial chaos for stochastic differential equations publication-title: SIAM J. Sci. Comput. doi: 10.1137/S1064827501387826 – volume: 3 start-page: 595 year: 1987 ident: 10.1016/j.apnum.2014.11.006_br0040 article-title: A survey of matrix inverse eigenvalue problems publication-title: Inverse Problems doi: 10.1088/0266-5611/3/4/010 – volume: vol. 13 year: 1978 ident: 10.1016/j.apnum.2014.11.006_br0060 article-title: An Introduction to Orthogonal Polynomials – year: 1983 ident: 10.1016/j.apnum.2014.11.006_br0180 – start-page: 2989 year: 2006 ident: 10.1016/j.apnum.2014.11.006_br0320 article-title: Option pricing for a stochastic-volatility jump-diffusion model with log uniform jump-amplitudes – volume: 15 start-page: 49 year: 1999 ident: 10.1016/j.apnum.2014.11.006_br0240 article-title: Simple cubature formulas with high polynomial exactness publication-title: Constr. Approx. doi: 10.1007/s003659900119 – start-page: 337 year: 1987 ident: 10.1016/j.apnum.2014.11.006_br0170 article-title: A package for testing multiple integration subroutines, numerical integration: recent developments publication-title: Softw. Appl. – volume: 37 start-page: 531 issue: 4 year: 1995 ident: 10.1016/j.apnum.2014.11.006_br0330 article-title: Historical development of the Newton–Raphson method publication-title: SIAM Rev. doi: 10.1137/1037125 – volume: 75 start-page: 79 year: 1996 ident: 10.1016/j.apnum.2014.11.006_br0230 article-title: High dimensional integration of smooth functions over cubes publication-title: Numer. Math. doi: 10.1007/s002110050231 – volume: 5 start-page: 74 year: 1957 ident: 10.1016/j.apnum.2014.11.006_br0150 article-title: Generation and use of orthogonal polynomials for data-fitting with a digital computer publication-title: J. Soc. Ind. Appl. Math. doi: 10.1137/0105007 – volume: 21 start-page: 37 issue: 1 year: 2005 ident: 10.1016/j.apnum.2014.11.006_br0280 article-title: Fluid models with jumps publication-title: Stoch. Models doi: 10.1081/STM-200046459 – volume: 68 start-page: 580 year: 1992 ident: 10.1016/j.apnum.2014.11.006_br0080 article-title: Wave-function approach to dissipative processes in quantum optics publication-title: Phys. Rev. Lett. doi: 10.1103/PhysRevLett.68.580 |
| SSID | ssj0005540 |
| Score | 2.1719143 |
| Snippet | We develop a multi-element probabilistic collocation method (ME-PCM) for arbitrary discrete probability measures with finite moments and apply it to solve... |
| SourceID | proquest crossref elsevier |
| SourceType | Aggregation Database Enrichment Source Index Database Publisher |
| StartPage | 91 |
| SubjectTerms | Accuracy Adaptive algorithms Algorithms Charlier polynomials Construction Criteria Data-driven model Discrete–continuous inputs KdV equation Mathematical analysis Mathematical models Polynomials Stochastic collocation |
| Title | Adaptive multi-element polynomial chaos with discrete measure: Algorithms and application to SPDEs |
| URI | https://dx.doi.org/10.1016/j.apnum.2014.11.006 https://www.proquest.com/docview/1669874140 |
| Volume | 90 |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| journalDatabaseRights | – providerCode: PRVESC databaseName: Baden-Württemberg Complete Freedom Collection (Elsevier) customDbUrl: eissn: 1873-5460 dateEnd: 99991231 omitProxy: true ssIdentifier: ssj0005540 issn: 0168-9274 databaseCode: GBLVA dateStart: 20110101 isFulltext: true titleUrlDefault: https://www.sciencedirect.com providerName: Elsevier – providerCode: PRVESC databaseName: Elsevier SD Complete Freedom Collection (subscription) customDbUrl: eissn: 1873-5460 dateEnd: 99991231 omitProxy: true ssIdentifier: ssj0005540 issn: 0168-9274 databaseCode: ACRLP dateStart: 19950101 isFulltext: true titleUrlDefault: https://www.sciencedirect.com providerName: Elsevier – providerCode: PRVESC databaseName: Elsevier SD Freedom Collection Journals [SCFCJ] customDbUrl: eissn: 1873-5460 dateEnd: 99991231 omitProxy: true ssIdentifier: ssj0005540 issn: 0168-9274 databaseCode: AIKHN dateStart: 19950101 isFulltext: true titleUrlDefault: https://www.sciencedirect.com providerName: Elsevier – providerCode: PRVESC databaseName: ScienceDirect (Elsevier) customDbUrl: eissn: 1873-5460 dateEnd: 99991231 omitProxy: true ssIdentifier: ssj0005540 issn: 0168-9274 databaseCode: .~1 dateStart: 19950101 isFulltext: true titleUrlDefault: https://www.sciencedirect.com providerName: Elsevier – providerCode: PRVLSH databaseName: Elsevier Journals customDbUrl: mediaType: online eissn: 1873-5460 dateEnd: 99991231 omitProxy: true ssIdentifier: ssj0005540 issn: 0168-9274 databaseCode: AKRWK dateStart: 19930101 isFulltext: true providerName: Library Specific Holdings |
| link | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1LS8QwEA6LXrz4Fp9LBI_W7SNpG2-LD1YFEVTwFpJ06oO1Ldvdgxd_u5m0FRXx4LHtJJRJMvkm-WaGkIMwjw1wbr0ToSOPJdr3dG5yL1BCh8CNBe0u2-d1PLpnlw_8oUdOulgYpFW2tr-x6c5at28GrTYH1fPz4NaClVRg9hWGleUEBpozlmAVg6P3LzQP7oIiUdhD6S7zkON4qaqYYTh6wI4wlSeWPfp9d_php93mc75MFlvUSIfNj62QHhSrZKlFkLRdn_Ua0cNMVWjAqCMKetCQw2lVjt8w_tj2YZ5UWVM8fqUYkTuxoJm-NgeFx3Q4fiwn9tNrTVWR0S-323Ra0tub07N6ndyfn92djLy2ioJnoiieehmkEAYQYzKuKNfaZ0ylAhIDLFeKKetwZLgqfWEEWHBnVJwnAEpxA0GkRLRB5oqygE1CTRbkLPGRGQqMZaEGHqZBnlivK_U197dI2GlPmjbFOFa6GMuOS_Yincolqtw6H9KqfIscfjaqmgwbf4vH3bDIbxNF2j3g74b73SBKu4TwXkQVUM5qGcSxSN082v5v5ztkwT7xhtKzS-amkxnsWbQy1X03HftkfnhxNbr-AD3p6vo |
| linkProvider | Elsevier |
| linkToHtml | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1Lj9MwELa65cBeeCPK8vBKeyRtHnYSc6vKVgVKhbS7Um-W7UygqE2ipj3shd-Ox0lWXYR64Bo_FI3H42_sb2YIuQjz2ADn1jsROvJYon1P5yb3AiV0CNxY0O6yfS7i2Q37suTLHpl0sTBIq2xtf2PTnbVuv4xaaY6q1Wp0ZcFKKjD7CsPKcoKfkAeMhwl6YMPfBzwP7qIisbeH3bvUQ47kpapij_HoARtiLk-se_Tv4-kvQ-1On-kT8qiFjXTc_NlT0oPiGXncQkjabtD6OdHjTFVowahjCnrQsMNpVa5vMQDZzmF-qrKmeP9KMSR3a1Ez3TQ3hR_peP2j3NqmTU1VkdGD5226K-nV90-X9QtyM728nsy8toyCZ6Io3nkZpBAGEGM2rijX2mdMpQISAyxXiinrcWS4LX1hBFh0Z1ScJwBKcQNBpET0kvSLsoBXhJosyFniIzUUGMtCDTxMgzyxblfqa-4PSNhJT5o2xziWuljLjkz2SzqRSxS59T6kFfmAfLgbVDUpNo53j7tlkfc0RdpD4PjA824Rpd1D-DCiCij3tQziWKROkV7_7-TvycPZ9be5nH9efD0jp7aFN_yeN6S_2-7hrYUuO_3OqeYfudDsjw |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Adaptive+multi-element+polynomial+chaos+with+discrete+measure%3A+Algorithms+and+application+to+SPDEs&rft.jtitle=Applied+numerical+mathematics&rft.au=Zheng%2C+Mengdi&rft.au=Wan%2C+Xiaoliang&rft.au=Karniadakis%2C+George+Em&rft.date=2015-04-01&rft.pub=Elsevier+B.V&rft.issn=0168-9274&rft.eissn=1873-5460&rft.volume=90&rft.spage=91&rft.epage=110&rft_id=info:doi/10.1016%2Fj.apnum.2014.11.006&rft.externalDocID=S0168927414001895 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0168-9274&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0168-9274&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0168-9274&client=summon |