{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,5]],"date-time":"2026-03-05T20:50:29Z","timestamp":1772743829964,"version":"3.50.1"},"reference-count":31,"publisher":"Walter de Gruyter GmbH","issue":"3","funder":[{"DOI":"10.13039\/501100004052","name":"King Abdullah University of Science and Technology","doi-asserted-by":"publisher","award":["CRG2020"],"award-info":[{"award-number":["CRG2020"]}],"id":[{"id":"10.13039\/501100004052","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024,7,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We use a Gaussian Process Regression (GPR) strategy to analyze different types of curves that are commonly encountered in parametric eigenvalue problems.\nWe employ an offline-online decomposition method.\nIn the offline phase, we generate the basis of the reduced space by applying the proper orthogonal decomposition (POD) method on a collection of pre-computed, full-order snapshots at a chosen set of parameters.\nThen we generate our GPR model using four different Mat\u00e9rn covariance functions.\nIn the online phase, we use this model to predict both eigenvalues and eigenvectors at new parameters.\nWe then illustrate how the choice of each covariance function influences the performance of GPR.\nFurthermore, we discuss the connection between Gaussian Process Regression and spline methods and compare the performance of the GPR method against linear and cubic spline methods.\nWe show that GPR outperforms other methods for functions with a certain regularity.<\/jats:p>","DOI":"10.1515\/cmam-2023-0086","type":"journal-article","created":{"date-parts":[[2024,6,12]],"date-time":"2024-06-12T11:43:13Z","timestamp":1718192593000},"page":"533-555","source":"Crossref","is-referenced-by-count":5,"title":["A Data-Driven Method for Parametric PDE Eigenvalue Problems Using Gaussian Process with Different Covariance Functions"],"prefix":"10.1515","volume":"24","author":[{"given":"Moataz","family":"Alghamdi","sequence":"first","affiliation":[{"name":"Applied Mathematics and Computational Sciences (AMCS) , 127355 King Abdullah University of Science and Technology , Thuwal , Kingdom of Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fleurianne","family":"Bertrand","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics , TU Chemnitz , Chemnitz , Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Daniele","family":"Boffi","sequence":"additional","affiliation":[{"name":"Applied Mathematics and Computational Sciences (AMCS) , 127355 King Abdullah University of Science and Technology , Thuwal , Kingdom of Saudi Arabia ; and Department of Mathematics \u201cF. Casorati\u201d, University of Pavia, via Ferrata 1, 27100 Pavia, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Abdul","family":"Halim","sequence":"additional","affiliation":[{"name":"Applied Mathematics and Computational Sciences (AMCS) , 127355 King Abdullah University of Science and Technology , Thuwal , Kingdom of Saudi Arabia ; and Department of Mathematics, Memari College, West Bengal, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2024,6,13]]},"reference":[{"key":"2024070116104972723_j_cmam-2023-0086_ref_001","unstructured":"M. M. Alghamdi, D. Boffi and F. Bonizzoni,\nA greedy mor method for the tracking of eigensolutions to parametric elliptic pdes,\nMOX-Report no. 67\/2022, Politecnico di Milano, Milano, 2022."},{"key":"2024070116104972723_j_cmam-2023-0086_ref_002","doi-asserted-by":"crossref","unstructured":"R. Andreev and C. Schwab,\nSparse tensor approximation of parametric eigenvalue problems,\nNumerical Analysis of Multiscale Problems,\nLect. Notes Comput. Sci. Eng. 83,\nSpringer, Heidelberg (2012), 203\u2013241.","DOI":"10.1007\/978-3-642-22061-6_7"},{"key":"2024070116104972723_j_cmam-2023-0086_ref_003","doi-asserted-by":"crossref","unstructured":"F. Bertrand, D. Boffi and A. Halim,\nData-driven reduced order modeling for parametric PDE eigenvalue problems using Gaussian process regression,\nJ. Comput. Phys. 495 (2023), Article ID 112503.","DOI":"10.1016\/j.jcp.2023.112503"},{"key":"2024070116104972723_j_cmam-2023-0086_ref_004","unstructured":"C. M. Bishop,\nPattern Recognition and Machine Learning,\nInform. Sci. Stat.,\nSpringer, New York, 2006."},{"key":"2024070116104972723_j_cmam-2023-0086_ref_005","unstructured":"D. Boffi, A. Halim and G. Priyadarshi,\nReduced basis approximation of parametric eigenvalue problems in presence of clusters and intersections, preprint (2023), https:\/\/arxiv.org\/abs\/2302.00898."},{"key":"2024070116104972723_j_cmam-2023-0086_ref_006","doi-asserted-by":"crossref","unstructured":"A. G. Buchan, C. C. Pain, F. Fang and I. M. Navon,\nA POD reduced-order model for eigenvalue problems with application to reactor physics,\nInternat. J. Numer. Methods Engrg. 95 (2013), no. 12, 1011\u20131032.","DOI":"10.1002\/nme.4533"},{"key":"2024070116104972723_j_cmam-2023-0086_ref_007","doi-asserted-by":"crossref","unstructured":"Z. Chen, J. Fan and K. Wang,\nMultivariate Gaussian processes: Definitions, examples and applications,\nMetron 81 (2023), no. 2, 181\u2013191.","DOI":"10.1007\/s40300-023-00238-3"},{"key":"2024070116104972723_j_cmam-2023-0086_ref_008","doi-asserted-by":"crossref","unstructured":"P. Craven and G. Wahba,\nSmoothing noisy data with spline functions,\nNumer. Math. 31 (1978), 377\u2013403.","DOI":"10.1007\/BF01404567"},{"key":"2024070116104972723_j_cmam-2023-0086_ref_009","unstructured":"D. G. T. Denison, C. C. Holmes, B. K. Mallick and A. F. M. Smith,\nBayesian Methods for Nonlinear Classification and Regression,\nWiley Ser. Probab. Stat.,\nJohn Wiley & Sons, Chichester, 2002."},{"key":"2024070116104972723_j_cmam-2023-0086_ref_010","unstructured":"D. Duvenaud,\nAutomatic model construction with gaussian processes,\nDissertation, Pembroke College, 2014."},{"key":"2024070116104972723_j_cmam-2023-0086_ref_011","doi-asserted-by":"crossref","unstructured":"C. Eckart and G. Young,\nThe approximation of one matrix by another of lower rank,\nPsychometrika 1 (1936), 211\u2013218.","DOI":"10.1007\/BF02288367"},{"key":"2024070116104972723_j_cmam-2023-0086_ref_012","unstructured":"P. I. Frazier,\nA tutorial on Bayesian optimization,\npreprint (2018), https:\/\/arxiv.org\/abs\/1807.02811."},{"key":"2024070116104972723_j_cmam-2023-0086_ref_013","doi-asserted-by":"crossref","unstructured":"I. Fumagalli, A. Manzoni, N. Parolini and M. Verani,\nReduced basis approximation and a posteriori error estimates for parametrized elliptic eigenvalue problems,\nESAIM Math. Model. Numer. Anal. 50 (2016), no. 6, 1857\u20131885.","DOI":"10.1051\/m2an\/2016009"},{"key":"2024070116104972723_j_cmam-2023-0086_ref_014","doi-asserted-by":"crossref","unstructured":"P. German and J. C. Ragusa,\nReduced-order modeling of parameterized multi-group diffusion k-eigenvalue problems,\nAnn. Nuclear Energy 134 (2019), 144\u2013157.","DOI":"10.1016\/j.anucene.2019.05.049"},{"key":"2024070116104972723_j_cmam-2023-0086_ref_015","doi-asserted-by":"crossref","unstructured":"P. J. Green and B. W. Silverman,\nNonparametric Regression and Generalized Linear Models,\nMonogr. Statist. Appl. Probab. 58,\nChapman & Hall, London, 1994.","DOI":"10.1007\/978-1-4899-4473-3"},{"key":"2024070116104972723_j_cmam-2023-0086_ref_016","doi-asserted-by":"crossref","unstructured":"M. Guo and J. S. Hesthaven,\nReduced order modeling for nonlinear structural analysis using Gaussian process regression,\nComput. Methods Appl. Mech. Engrg. 341 (2018), 807\u2013826.","DOI":"10.1016\/j.cma.2018.07.017"},{"key":"2024070116104972723_j_cmam-2023-0086_ref_017","doi-asserted-by":"crossref","unstructured":"M. Guo and J. S. Hesthaven,\nData-driven reduced order modeling for time-dependent problems,\nComput. Methods Appl. Mech. Engrg. 345 (2019), 75\u201399.","DOI":"10.1016\/j.cma.2018.10.029"},{"key":"2024070116104972723_j_cmam-2023-0086_ref_018","doi-asserted-by":"crossref","unstructured":"H. Hakula, V. Kaarnioja and M. Laaksonen,\nApproximate methods for stochastic eigenvalue problems,\nAppl. Math. Comput. 267 (2015), 664\u2013681.","DOI":"10.1016\/j.amc.2014.12.112"},{"key":"2024070116104972723_j_cmam-2023-0086_ref_019","doi-asserted-by":"crossref","unstructured":"T. Hastie and R. Tibshirani,\nGeneralized additive models,\nStatist. Sci. 1 (1986), no. 3, 297\u2013318.","DOI":"10.1214\/ss\/1177013604"},{"key":"2024070116104972723_j_cmam-2023-0086_ref_020","doi-asserted-by":"crossref","unstructured":"T. Horger, B. Wohlmuth and T. Dickopf,\nSimultaneous reduced basis approximation of parameterized elliptic eigenvalue problems,\nESAIM Math. Model. Numer. Anal. 51 (2017), no. 2, 443\u2013465.","DOI":"10.1051\/m2an\/2016025"},{"key":"2024070116104972723_j_cmam-2023-0086_ref_021","doi-asserted-by":"crossref","unstructured":"H. Kano, H. Fujioka and C. F. Martin,\nOptimal smoothing spline with constraints on its derivatives,\n49th IEEE Conference on Decision and Control (CDC),\nIEEE Press, Piscataway (2010), 6785\u20136790.","DOI":"10.1109\/CDC.2010.5717055"},{"key":"2024070116104972723_j_cmam-2023-0086_ref_022","doi-asserted-by":"crossref","unstructured":"G. S. Kimeldorf and G. Wahba,\nA correspondence between Bayesian estimation on stochastic processes and smoothing by splines,\nAnn. Math. Stat. 41 (1970), 495\u2013502.","DOI":"10.1214\/aoms\/1177697089"},{"key":"2024070116104972723_j_cmam-2023-0086_ref_023","doi-asserted-by":"crossref","unstructured":"L. Machiels, Y. Maday, I. B. Oliveira, A. T. Patera and D. V. Rovas,\nOutput bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems,\nC. R. Acad. Sci. Paris S\u00e9r. I Math. 331 (2000), no. 2, 153\u2013158.","DOI":"10.1016\/S0764-4442(00)00270-6"},{"key":"2024070116104972723_j_cmam-2023-0086_ref_024","doi-asserted-by":"crossref","unstructured":"G. S. H. Pau,\nReduced-basis method for band structure calculations,\nPhys. Rev. E 76 (2007), Article ID 046704.","DOI":"10.1103\/PhysRevE.76.046704"},{"key":"2024070116104972723_j_cmam-2023-0086_ref_025","unstructured":"G. S. H. Pau,\nReduced basis method for quantum models of crystalline solids,\nPh.D. thesis, Massachusetts Institute of Technology, 2008."},{"key":"2024070116104972723_j_cmam-2023-0086_ref_026","doi-asserted-by":"crossref","unstructured":"A. Quarteroni, A. Manzoni and F. Negri,\nReduced Basis Methods for Partial Differential Equations,\nUnitext 92,\nSpringer, Cham, 2016.","DOI":"10.1007\/978-3-319-15431-2"},{"key":"2024070116104972723_j_cmam-2023-0086_ref_027","doi-asserted-by":"crossref","unstructured":"A. Quarteroni and G. Rozza,\nNumerical solution of parametrized Navier\u2013Stokes equations by reduced basis methods,\nNumer. Methods Partial Differential Equations 23 (2007), no. 4, 923\u2013948.","DOI":"10.1002\/num.20249"},{"key":"2024070116104972723_j_cmam-2023-0086_ref_028","doi-asserted-by":"crossref","unstructured":"C. E. Rasmussen and C. K. I. Williams,\nGaussian Processes for Machine Learning,\nAdapt. Comput. Mach. Learn.,\nMIT Press, Cambridge, 2005.","DOI":"10.7551\/mitpress\/3206.001.0001"},{"key":"2024070116104972723_j_cmam-2023-0086_ref_029","doi-asserted-by":"crossref","unstructured":"S. Vallagh\u00e9, P. Huynh, D. J. Knezevic, L. Nguyen and A. T. Patera,\nComponent-based reduced basis for parametrized symmetric eigenproblems,\nAdv. Model. Simulat. Eng. Sci. 2 (2015), no. 1, 1\u201330.","DOI":"10.1186\/s40323-015-0021-0"},{"key":"2024070116104972723_j_cmam-2023-0086_ref_030","doi-asserted-by":"crossref","unstructured":"G. Wahba,\nSpline Models for Observational Data,\nCBMS-NSF Regional Conf. Ser. in Appl. Math. 59,\nSociety for Industrial and Applied Mathematics, Philadelphia, 1990.","DOI":"10.1137\/1.9781611970128"},{"key":"2024070116104972723_j_cmam-2023-0086_ref_031","doi-asserted-by":"crossref","unstructured":"B. Wang and T. Chen,\nGaussian process regression with multiple response variables,\nChemometrics Intell. Lab. Syst. 142 (2015), 159\u2013165.","DOI":"10.1016\/j.chemolab.2015.01.016"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2023-0086\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2023-0086\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,1]],"date-time":"2024-07-01T16:14:09Z","timestamp":1719850449000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2023-0086\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,13]]},"references-count":31,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2024,6,26]]},"published-print":{"date-parts":[[2024,7,1]]}},"alternative-id":["10.1515\/cmam-2023-0086"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2023-0086","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,6,13]]}}}