{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T09:44:31Z","timestamp":1777455871105,"version":"3.51.4"},"reference-count":35,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2020,5,8]],"date-time":"2020-05-08T00:00:00Z","timestamp":1588896000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,5,8]],"date-time":"2020-05-08T00:00:00Z","timestamp":1588896000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["F 65"],"award-info":[{"award-number":["F 65"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["P 29197-N32"],"award-info":[{"award-number":["P 29197-N32"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["F 65"],"award-info":[{"award-number":["F 65"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["T 998"],"award-info":[{"award-number":["T 998"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001821","name":"Vienna Science and Technology Fund","doi-asserted-by":"publisher","award":["MA14-006"],"award-info":[{"award-number":["MA14-006"]}],"id":[{"id":"10.13039\/501100001821","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001711","name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung","doi-asserted-by":"publisher","award":["182236"],"award-info":[{"award-number":["182236"]}],"id":[{"id":"10.13039\/501100001711","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv Comput Math"],"published-print":{"date-parts":[[2020,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The present work deals with rational model order reduction methods based on the single-point Least-Square (LS) Pad\u00e9 approximation techniques introduced in Bonizzoni et al. (ESAIM Math. Model. Numer. Anal., <jats:bold>52<\/jats:bold>(4), 1261\u20131284 2018, Math. Comput. <jats:bold>89<\/jats:bold>, 1229\u20131257 2020). Algorithmical aspects concerning the construction of rational LS-Pad\u00e9 approximants are described. In particular, we show that the computation of the Pad\u00e9 denominator can be carried out efficiently by solving an eigenvalue-eigenvector problem involving a Gramian matrix. The LS-Pad\u00e9 techniques are employed to approximate the frequency response map associated with two parametric time-harmonic acoustic wave problems, namely a transmission-reflection problem and a scattering problem. In both cases, we establish the meromorphy of the frequency response map. The Helmholtz equation with stochastic wavenumber is also considered. In particular, for Lipschitz functionals of the solution and their corresponding probability measures, we establish weak convergence of the measure derived from the LS-Pad\u00e9 approximant to the true one. 2D numerical tests are performed, which confirm the effectiveness of the approximation methods.<\/jats:p>","DOI":"10.1007\/s10444-020-09749-3","type":"journal-article","created":{"date-parts":[[2020,5,8]],"date-time":"2020-05-08T12:02:45Z","timestamp":1588939365000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Least-Squares Pad\u00e9 approximation of parametric and stochastic Helmholtz maps"],"prefix":"10.1007","volume":"46","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6222-3352","authenticated-orcid":false,"given":"Francesca","family":"Bonizzoni","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fabio","family":"Nobile","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ilaria","family":"Perugia","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Davide","family":"Pradovera","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,5,8]]},"reference":[{"key":"9749_CR1","unstructured":"Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, vol. 140. Academic Press (2003)"},{"issue":"6","key":"9749_CR2","doi-asserted-by":"publisher","first-page":"2392","DOI":"10.1137\/S0036142994269186","volume":"34","author":"IM Babu\u0161ka","year":"1997","unstructured":"Babu\u0161ka, I.M., Sauter, S.A.: Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? SIAM J. Numer. Anal. 34 (6), 2392\u20132423 (1997)","journal-title":"SIAM J. Numer. Anal."},{"issue":"4","key":"9749_CR3","doi-asserted-by":"publisher","first-page":"1261","DOI":"10.1051\/m2an\/2017050","volume":"52","author":"F Bonizzoni","year":"2018","unstructured":"Bonizzoni, F., Nobile, F., Perugia, I.: Convergence analysis of Pad\u00e9, approximations for Helmholtz frequency response problems. ESAIM Math. Model. Numer Anal. 52(4), 1261\u20131284 (2018)","journal-title":"ESAIM Math. Model. Numer Anal."},{"key":"9749_CR4","doi-asserted-by":"publisher","first-page":"1229","DOI":"10.1090\/mcom\/3511","volume":"89","author":"F Bonizzoni","year":"2020","unstructured":"Bonizzoni, F., Nobile, F, Perugia, I., Pradovera, D.: Fast Least-Squares Pad\u00e9 approximation of problems with normal operators and meromorphic structure. Math. Comput. 89, 1229\u20131257 (2020)","journal-title":"Math. Comput."},{"key":"9749_CR5","doi-asserted-by":"publisher","unstructured":"Bonizzoni, F., Pradovera, D.: Distributed sampling for rational approximation of the acoustic scattering of an airfoil. Proc. Appl. Math. Mech. https:\/\/doi.org\/10.1002\/pamm.201900422 (2019)","DOI":"10.1002\/pamm.201900422"},{"issue":"2","key":"9749_CR6","doi-asserted-by":"publisher","first-page":"970","DOI":"10.1137\/09075250X","volume":"32","author":"Y Chen","year":"2010","unstructured":"Chen, Y., Hesthaven, J.S., Maday, Y., Rodr\u00edguez, J.: Certified reduced basis methods and output bounds for the harmonic Maxwell\u2019s equations. SIAM J. Sci Comput. 32(2), 970\u2013996 (2010)","journal-title":"SIAM J. Sci Comput."},{"issue":"5","key":"9749_CR7","doi-asserted-by":"publisher","first-page":"678","DOI":"10.1109\/TCAD.2004.826583","volume":"23","author":"L Daniel","year":"2004","unstructured":"Daniel, L., Siong, O.C., Chay, L.S., Lee, K.H., White, J.: A multiparameter moment-matching model-reduction approach for generating geometrically parameterized interconnect performance models. IEEE Trans Comput.-Aided Design Integr. Circuits Syst. 23(5), 678\u2013693 (2004)","journal-title":"IEEE Trans Comput.-Aided Design Integr. Circuits Syst."},{"issue":"2","key":"9749_CR8","doi-asserted-by":"publisher","first-page":"219","DOI":"10.1007\/s00466-016-1348-1","volume":"59","author":"O Ezvan","year":"2017","unstructured":"Ezvan, O., Batou, A., Soize, C., Gagliardini, L.: Multilevel model reduction for uncertainty quantification in computational structural dynamics. Comput. Mech. 59(2), 219\u2013246 (2017)","journal-title":"Comput. Mech."},{"key":"9749_CR9","doi-asserted-by":"crossref","unstructured":"Feng, L., Benner, P.: A robust algorithm for parametric model order reduction based on implicit moment matching. Reduced order methods for modeling and computational reduction. In: Quarteroni, A., Rozza, G. (eds.) MS&A Series. Springer, Cham (2014)","DOI":"10.1007\/978-3-319-02090-7_6"},{"issue":"2","key":"9749_CR10","doi-asserted-by":"publisher","first-page":"203","DOI":"10.1006\/jath.1997.3216","volume":"95","author":"P Guillaume","year":"1998","unstructured":"Guillaume, P., Huard, A., Robin, V.: Generalized multivariate Pad\u00e9 approximants. J. Approx. Theory 95(2), 203\u2013214 (1998)","journal-title":"J. Approx. Theory"},{"key":"9749_CR11","doi-asserted-by":"publisher","first-page":"2191","DOI":"10.1007\/s10444-019-09675-z","volume":"45","author":"S Hain","year":"2019","unstructured":"Hain, S., Ohlberger, M, Radic, M., Urban, K.: A hierarchical a-posteriori error estimatorfor the reduced basis method. Adv. Comput. Math. 45, 2191\u20132214 (2019)","journal-title":"Adv. Comput. Math."},{"issue":"13","key":"9749_CR12","doi-asserted-by":"publisher","first-page":"1636","DOI":"10.1002\/nme.4271","volume":"90","author":"U Hetmaniuk","year":"2012","unstructured":"Hetmaniuk, U., Tezaur, R., Farhat, C.: Review and assessment of interpolatory model order reduction methods for frequency response structural dynamics and acoustics problems. Int. J. Numer. Methods Eng. 90(13), 1636\u20131662 (2012)","journal-title":"Int. J. Numer. Methods Eng."},{"issue":"10","key":"9749_CR13","doi-asserted-by":"publisher","first-page":"1109","DOI":"10.1002\/nme.4436","volume":"93","author":"U Hetmaniuk","year":"2013","unstructured":"Hetmaniuk, U., Tezaur, R., Farhat, C.: An adaptive scheme for a class of interpolatory model reduction methods for frequency response problems. Int. J. Numer. Methods Eng. 93(10), 1109\u20131124 (2013)","journal-title":"Int. J. Numer. Methods Eng."},{"key":"9749_CR14","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1016\/j.apnum.2012.12.004","volume":"79","author":"R Hiptmair","year":"2014","unstructured":"Hiptmair, R., Moiola, A., Perugia, I.: Trefftz discontinuous Galerkin methods for acoustic scattering on locally refined meshes. Appl. Numer. Math. 79, 79\u201391 (2014)","journal-title":"Appl. Numer. Math."},{"key":"9749_CR15","doi-asserted-by":"publisher","first-page":"1475","DOI":"10.1007\/s10444-018-9594-8","volume":"44","author":"R Hiptmair","year":"2018","unstructured":"Hiptmair, R., Scarabosio, L., Schillings, C., Schwab, C.: Large deformation shape uncertainty quantification in acoustic scattering. Adv. Comput. Math. 44, 1475\u20131518 (2018)","journal-title":"Adv. Comput. Math."},{"issue":"1\u20132","key":"9749_CR16","doi-asserted-by":"publisher","first-page":"255","DOI":"10.1016\/S0377-0427(99)00177-6","volume":"115","author":"A Huard","year":"2000","unstructured":"Huard, A., Robin, V.: Continuity of approximation by least-squares multivariate Pad\u00e9 approximants. J. Comput. Appl. Math. 115(1\u20132), 255\u2013268 (2000)","journal-title":"J. Comput. Appl. Math."},{"issue":"8","key":"9749_CR17","doi-asserted-by":"publisher","first-page":"473","DOI":"10.1016\/j.crma.2007.09.019","volume":"345","author":"DBP Huynh","year":"2007","unstructured":"Huynh, D.B.P., Rozza, G., Sen, S., Patera, A.T.: A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants. Comptes Rendus Mathematique 345(8), 473\u2013478 (2007)","journal-title":"Comptes Rendus Mathematique"},{"issue":"03","key":"9749_CR18","doi-asserted-by":"publisher","first-page":"1343010","DOI":"10.1142\/S021987621343010X","volume":"11","author":"DBP Huynh","year":"2014","unstructured":"Huynh, D.B.P., Knezevic, D.J., Patera, A.T.: A static condensation reduced basis element approximation: Application to three-dimensional acoustic muffler analysis. Int. J. Comput. Methods 11(03), 1343010 (2014)","journal-title":"Int. J. Comput. Methods"},{"key":"9749_CR19","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-55682-1","volume-title":"Probability Essentials","author":"J Jacod","year":"2004","unstructured":"Jacod, J., Protter, P.: Probability Essentials. Springer, Berlin (2004)"},{"issue":"12","key":"9749_CR20","doi-asserted-by":"publisher","first-page":"1170","DOI":"10.1002\/nme.5497","volume":"111","author":"E Jacquelin","year":"2017","unstructured":"Jacquelin, E., Dessombz, O., Sinou, J.-J., Adhikari, S., Friswell, M.I.: Polynomial chaos-based extended Pad\u00e9 expansion in structural dynamics. Int. J. Numer. Methods Eng. 111(12), 1170\u20131191 (2017)","journal-title":"Int. J. Numer. Methods Eng."},{"key":"9749_CR21","doi-asserted-by":"publisher","first-page":"1104","DOI":"10.1016\/j.proeng.2017.09.212","volume":"199","author":"E Jacquelin","year":"2017","unstructured":"Jacquelin, E., Dessombz, O., Sinou, J.-J., Adhikari, S., Friswell, M.I.: Steady-state response of a random dynamical system described with Pad\u00e1pproximants and random eigenmodes. Procedia Eng. 199, 1104\u20131109 (2017). X International Conference on Structural Dynamics, EURODYN 2017","journal-title":"Procedia Eng."},{"issue":"3","key":"9749_CR22","doi-asserted-by":"publisher","first-page":"A1525","DOI":"10.1137\/140967696","volume":"37","author":"S Kapita","year":"2015","unstructured":"Kapita, S., Monk, P., Warburton, T.: Residual-based adaptivity and PWDG methods for the Helmholtz equation. SIAM J. Sci. Comput. 37(3), A1525\u2013A1553 (2015)","journal-title":"SIAM J. Sci. Comput."},{"issue":"6","key":"9749_CR23","doi-asserted-by":"publisher","first-page":"1555","DOI":"10.1051\/m2an\/2012016","volume":"46","author":"T Lassila","year":"2012","unstructured":"Lassila, T., Manzoni, A., Rozza, G.: On the approximation of stability factors for general parametrized partial differential equations with a two-level affine decomposition. ESAIM: Math. Modell. Numer. Anal. 46(6), 1555\u20131576 (2012)","journal-title":"ESAIM: Math. Modell. Numer. Anal."},{"issue":"3","key":"9749_CR24","doi-asserted-by":"publisher","first-page":"579","DOI":"10.1137\/0523030","volume":"23","author":"M Lenoir","year":"1992","unstructured":"Lenoir, M., Vullierme-Ledard, M., Hazard, C.: Variational formulations for the determination of resonant states in scattering problems. SIAM J. Math. Anal. 23(3), 579\u2013608 (1992)","journal-title":"SIAM J. Math. Anal."},{"key":"9749_CR25","doi-asserted-by":"crossref","unstructured":"Manetti, M.: Topology. Springer International Publishing (2015)","DOI":"10.1007\/978-3-319-16958-3"},{"key":"9749_CR26","unstructured":"McLean, W.C.H.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press (2000)"},{"key":"9749_CR27","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1016\/j.cma.2015.03.026","volume":"295","author":"D Modesto","year":"2015","unstructured":"Modesto, D., Zlotnik, S., Huerta, A.: Proper generalized decomposition for parameterized Helmholtz problems in heterogeneous and unbounded domains: Application to harbor agitation. Comput. Methods Appl. Mech. Eng. 295, 127\u2013149 (2015)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"9749_CR28","doi-asserted-by":"crossref","unstructured":"Ohayon, R., Soize, C.: Computational vibroacoustics in low- and medium- frequency bands: damping, ROM, and UQ modeling. Appl. Sci.-Basel, 7(6) (2017)","DOI":"10.3390\/app7060586"},{"key":"9749_CR29","doi-asserted-by":"crossref","unstructured":"Pradovera, D.: Interpolatory rational model order reduction of parametric problems lacking uniform inf-sup stability, ArXiv e-prints (2019)","DOI":"10.1137\/19M1269695"},{"key":"9749_CR30","doi-asserted-by":"publisher","first-page":"291","DOI":"10.1017\/S0962492911000055","volume":"20","author":"C Schwab","year":"2011","unstructured":"Schwab, C., Gittelson, C.J.: Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs. Acta Numerica 20, 291\u2013467 (2011)","journal-title":"Acta Numerica"},{"key":"9749_CR31","unstructured":"Sen, S.: Reduced basis approximation and a posteriori error estimation for non-coercive elliptic problems: Applications to acoustics. PhD thesis, Massachusetts Institute of Technology (2007)"},{"issue":"1","key":"9749_CR32","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1016\/j.jcp.2006.02.012","volume":"217","author":"S Sen","year":"2006","unstructured":"Sen, S., Veroy, K., Huynh, D.B.P., Deparis, S., Nguyen, N.C., Patera, A.T.: \u201cNatural norm\u201d a posteriori error estimators for reduced basis approximations. J. Comput. Phys. 217(1), 37\u201362 (2006)","journal-title":"J. Comput. Phys."},{"issue":"5","key":"9749_CR33","doi-asserted-by":"publisher","first-page":"372","DOI":"10.1007\/BF00251419","volume":"31","author":"S Steinberg","year":"1968","unstructured":"Steinberg, S.: Meromorphic families of compact operators. Arch. Ration. Mech. Anal. 31(5), 372\u2013379 (1968)","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"4","key":"9749_CR34","doi-asserted-by":"publisher","first-page":"355","DOI":"10.1080\/13873954.2011.547678","volume":"17","author":"T Tonn","year":"2011","unstructured":"Tonn, T., Urban, K., Volkwein, S.: Comparison of the reduced-basis and POD a posteriori error estimators for an elliptic linear-quadratic optimal control problem. Math. Comput. Model. Dyn. Syst. 17(4), 355\u2013369 (2011)","journal-title":"Math. Comput. Model. Dyn. Syst."},{"key":"9749_CR35","doi-asserted-by":"crossref","unstructured":"Veroy, K., Prud\u2019Homme, C, Rovas, D.V., Patera, A.T.: A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations (2003)","DOI":"10.2514\/6.2003-3847"}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-020-09749-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10444-020-09749-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-020-09749-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,5,8]],"date-time":"2021-05-08T02:00:42Z","timestamp":1620439242000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10444-020-09749-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,5,8]]},"references-count":35,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2020,6]]}},"alternative-id":["9749"],"URL":"https:\/\/doi.org\/10.1007\/s10444-020-09749-3","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"value":"1019-7168","type":"print"},{"value":"1572-9044","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,5,8]]},"assertion":[{"value":"20 July 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 November 2019","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 May 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"46"}}