{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T05:00:38Z","timestamp":1780376438419,"version":"3.54.1"},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2025,8,9]],"date-time":"2025-08-09T00:00:00Z","timestamp":1754697600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,8,9]],"date-time":"2025-08-09T00:00:00Z","timestamp":1754697600000},"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":["10.55776\/F65"],"award-info":[{"award-number":["10.55776\/F65"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["10.55776\/P33477"],"award-info":[{"award-number":["10.55776\/P33477"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100009244","name":"Stockholm University","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100009244","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2025,9]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>This paper describes novel algorithms for the identification of (almost-)resonant behavior in scattering problems. Our methods, relying on rational approximation, aim at building surrogate models of what we call \u201cfield amplification\u201d, defined as the norm of the solution operator of the scattering problem, which we express through boundary-integral equations. To provide our techniques with theoretical foundations, we first derive results linking the field amplification to the spectral properties of the operator that defines the scattering problem. Such results are then used to justify the use of rational approximation in the surrogate-modeling task. Some of our proposed methods apply rational approximation in a \u201cstandard\u201d way, building a rational approximant for either the solution operator directly or, in the interest of computational efficiency, for a randomly \u201csketched\u201d version of it. Our other \u201chybrid\u201d approaches are more innovative, combining rational-approximation-assisted root-finding with approximation using radial basis functions. Three key features of our methods are that (i) they are agnostic of the strategy used to discretize the scattering problem, (ii) they do not require any computations involving non-real wavenumbers, and (iii) they can adjust to different settings through the use of adaptive sampling strategies. We carry out some numerical experiments involving 2D scatterers to compare our approaches. In our tests, two of our approaches (one standard, one hybrid) emerge as the best performers, with one or the other being preferable, depending on whether emphasis is placed on accuracy or efficiency.<\/jats:p>","DOI":"10.1007\/s10915-025-03020-x","type":"journal-article","created":{"date-parts":[[2025,8,9]],"date-time":"2025-08-09T03:23:01Z","timestamp":1754709781000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Surrogate Modeling of Resonant Behavior in Scattering Problems Through Adaptive Rational Approximation and Sketching"],"prefix":"10.1007","volume":"104","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0398-1580","authenticated-orcid":false,"given":"Davide","family":"Pradovera","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1378-2668","authenticated-orcid":false,"given":"Ralf","family":"Hiptmair","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1368-2883","authenticated-orcid":false,"given":"Ilaria","family":"Perugia","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,8,9]]},"reference":[{"key":"3020_CR1","doi-asserted-by":"crossref","unstructured":"Baker, G.A., Graves-Morris, P.: Pad\u00e9 approximants, 2nd edn. Encyclopedia of Mathematics and its Applications, Cambridge University Press, (1996)","DOI":"10.1017\/CBO9780511530074"},{"issue":"2","key":"3020_CR2","doi-asserted-by":"publisher","first-page":"700","DOI":"10.1093\/imanum\/drt002","volume":"34","author":"T Betcke","year":"2014","unstructured":"Betcke, T., Phillips, J., Spence, E.A.: Spectral decompositions and nonnormality of boundary integral operators in acoustic scattering. IMA J. Numer. Anal. 34(2), 700\u2013731 (2014). https:\/\/doi.org\/10.1093\/imanum\/drt002","journal-title":"IMA J. Numer. Anal."},{"issue":"10","key":"3020_CR3","doi-asserted-by":"publisher","first-page":"3839","DOI":"10.1016\/j.laa.2011.03.030","volume":"436","author":"WJ Beyn","year":"2012","unstructured":"Beyn, W.J.: An integral method for solving nonlinear eigenvalue problems. Linear Algebra Appl. 436(10), 3839\u20133863 (2012). https:\/\/doi.org\/10.1016\/j.laa.2011.03.030","journal-title":"Linear Algebra Appl."},{"issue":"323","key":"3020_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(323), 1229\u20131257 (2020). https:\/\/doi.org\/10.1090\/mcom\/3511","journal-title":"Math. Comput."},{"issue":"4","key":"3020_CR5","doi-asserted-by":"publisher","first-page":"1","DOI":"10.3934\/mine.2023074","volume":"5","author":"F Bonizzoni","year":"2023","unstructured":"Bonizzoni, F., Pradovera, D., Ruggeri, M.: Rational-approximation-based model order reduction of Helmholtz frequency response problems with adaptive finite element snapshots. Math. Eng. 5(4), 1\u201338 (2023). https:\/\/doi.org\/10.3934\/mine.2023074","journal-title":"Math. Eng."},{"key":"3020_CR6","doi-asserted-by":"publisher","unstructured":"Bruno, O.P., Santana, M.A., Trefethen, L.N.: Evaluation of resonances via AAA rational approximation of randomly scalarized boundary integral resolvents (2024). https:\/\/doi.org\/10.48550\/arXiv.2405.19582","DOI":"10.48550\/arXiv.2405.19582"},{"issue":"2","key":"3020_CR7","doi-asserted-by":"publisher","first-page":"970","DOI":"10.1137\/09075250X","volume":"32","author":"Y Chen","year":"2010","unstructured":"Chen, Y., Hesthaven, J., 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). https:\/\/doi.org\/10.1137\/09075250X","journal-title":"SIAM J. Sci. Comput."},{"key":"3020_CR8","doi-asserted-by":"publisher","unstructured":"Claeys, X., Hiptmair, R., Jerez-Hanckes, C.: Multitrace boundary integral equations. In: Graham, I., Langer, U., Melenk, J., Sini, M. (eds.) Direct and inverse problems in wave propagation and applications, pp. 51\u2013100. De Gruyter (2013). https:\/\/doi.org\/10.1515\/9783110282283.51","DOI":"10.1515\/9783110282283.51"},{"issue":"4","key":"3020_CR9","doi-asserted-by":"publisher","first-page":"812","DOI":"10.1137\/0720053","volume":"20","author":"JD Dixon","year":"1983","unstructured":"Dixon, J.D.: Estimating extremal eigenvalues and condition numbers of matrices. SIAM J. Numer. Anal. 20(4), 812\u2013814 (1983). https:\/\/doi.org\/10.1137\/0720053","journal-title":"SIAM J. Numer. Anal."},{"issue":"2","key":"3020_CR10","doi-asserted-by":"publisher","first-page":"A929","DOI":"10.1137\/23M1570508","volume":"46","author":"TA Driscoll","year":"2024","unstructured":"Driscoll, T.A., Nakatsukasa, Y., Trefethen, L.N.: AAA rational approximation on a continuum. SIAM J. Sci. Comput. 46(2), A929\u2013A952 (2024). https:\/\/doi.org\/10.1137\/23M1570508","journal-title":"SIAM J. Sci. Comput."},{"key":"3020_CR11","doi-asserted-by":"publisher","unstructured":"Gohberg, I., Kre\u012dn, M.: Introduction to the theory of linear nonselfadjoint operators in Hilbert space, Translations of Mathematical Monographs, vol.\u00a018. American Mathematical Society, Providence, Rhode Island (1969). https:\/\/doi.org\/10.1090\/mmono\/018","DOI":"10.1090\/mmono\/018"},{"issue":"3","key":"3020_CR12","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1007\/s10915-023-02284-5","volume":"96","author":"L Grubi\u0161i\u0107","year":"2023","unstructured":"Grubi\u0161i\u0107, L., Hiptmair, R., Renner, D.: Detecting near resonances in acoustic scattering. J. Sci. Comput. 96(3), 81 (2023). https:\/\/doi.org\/10.1007\/s10915-023-02284-5","journal-title":"J. Sci. Comput."},{"issue":"3","key":"3020_CR13","doi-asserted-by":"publisher","first-page":"1052","DOI":"10.1109\/61.772353","volume":"14","author":"B Gustavsen","year":"1999","unstructured":"Gustavsen, B., Semlyen, A.: Rational approximation of frequency domain responses by vector fitting. IEEE Trans. Power Delivery 14(3), 1052\u20131061 (1999). https:\/\/doi.org\/10.1109\/61.772353","journal-title":"IEEE Trans. Power Delivery"},{"key":"3020_CR14","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1017\/S0962492917000034","volume":"26","author":"S G\u00fcttel","year":"2017","unstructured":"G\u00fcttel, S., Tisseur, F.: The nonlinear eigenvalue problem. Acta Numer 26, 1\u201394 (2017). https:\/\/doi.org\/10.1017\/S0962492917000034. (Publisher: Cambridge University Press)","journal-title":"Acta Numer"},{"issue":"2","key":"3020_CR15","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1137\/090771806","volume":"53","author":"N Halko","year":"2011","unstructured":"Halko, N., Martinsson, P.G., Tropp, J.A.: Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions. SIAM Rev. 53(2), 217\u2013288 (2011). https:\/\/doi.org\/10.1137\/090771806","journal-title":"SIAM Rev."},{"issue":"1","key":"3020_CR16","doi-asserted-by":"publisher","first-page":"465","DOI":"10.1137\/110823985","volume":"5","author":"M Hanke","year":"2012","unstructured":"Hanke, M.: One shot inverse scattering via rational approximation. SIAM J. Imag. Sci. 5(1), 465\u2013482 (2012). https:\/\/doi.org\/10.1137\/110823985","journal-title":"SIAM J. Imag. Sci."},{"issue":"4","key":"3020_CR17","doi-asserted-by":"publisher","first-page":"1446","DOI":"10.1137\/21M1447052","volume":"82","author":"R Hiptmair","year":"2022","unstructured":"Hiptmair, R., Moiola, A., Spence, E.A.: Spurious quasi-resonances in boundary integral equations for the Helmholtz transmission problem. SIAM J. Appl. Math. 82(4), 1446\u20131469 (2022). https:\/\/doi.org\/10.1137\/21M1447052","journal-title":"SIAM J. Appl. Math."},{"issue":"2","key":"3020_CR18","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1007\/s10543-023-00959-x","volume":"63","author":"D Huybrechs","year":"2023","unstructured":"Huybrechs, D., Trefethen, L.N.: AAA interpolation of equispaced data. BIT Numer. Math. 63(2), 21 (2023). https:\/\/doi.org\/10.1007\/s10543-023-00959-x","journal-title":"BIT Numer. Math."},{"key":"3020_CR19","doi-asserted-by":"publisher","unstructured":"Johnson, W.B., Lindenstrauss, J.: Extensions of Lipschitz mappings into a Hilbert space. In: Beals, R., Beck, A., Bellow, A., Hajian, A. (eds.) Contemporary Mathematics, vol.\u00a026, pp. 189\u2013206. American Mathematical Society (1984). https:\/\/doi.org\/10.1090\/conm\/026\/737400","DOI":"10.1090\/conm\/026\/737400"},{"issue":"2","key":"3020_CR20","doi-asserted-by":"publisher","first-page":"52","DOI":"10.1007\/s10915-021-01467-2","volume":"87","author":"K Li","year":"2021","unstructured":"Li, K., Huang, T.Z., Li, L., Lanteri, S.: Non-intrusive reduced-order modeling of parameterized electromagnetic scattering problems using cubic spline interpolation. J. Sci. Comput. 87(2), 52 (2021). https:\/\/doi.org\/10.1007\/s10915-021-01467-2","journal-title":"J. Sci. Comput."},{"issue":"2","key":"3020_CR21","doi-asserted-by":"publisher","first-page":"1087","DOI":"10.1093\/imanum\/draa098","volume":"42","author":"P Lietaert","year":"2022","unstructured":"Lietaert, P., Meerbergen, K., P\u00e9rez, J., Vandereycken, B.: Automatic rational approximation and linearization of nonlinear eigenvalue problems. IMA J. Numer. Anal. 42(2), 1087\u20131115 (2022). https:\/\/doi.org\/10.1093\/imanum\/draa098","journal-title":"IMA J. Numer. Anal."},{"key":"3020_CR22","doi-asserted-by":"publisher","first-page":"403","DOI":"10.1017\/S0962492920000021","volume":"29","author":"PG Martinsson","year":"2020","unstructured":"Martinsson, P.G., Tropp, J.A.: Randomized numerical linear algebra: foundations and algorithms. Acta Numer 29, 403\u2013572 (2020). https:\/\/doi.org\/10.1017\/S0962492920000021","journal-title":"Acta Numer"},{"issue":"02","key":"3020_CR23","doi-asserted-by":"publisher","first-page":"317","DOI":"10.1142\/S0218202519500106","volume":"29","author":"A Moiola","year":"2019","unstructured":"Moiola, A., Spence, E.A.: Acoustic transmission problems: wavenumber-explicit bounds and resonance-free regions. Math. Models Methods Appl. Sci. 29(02), 317\u2013354 (2019). https:\/\/doi.org\/10.1142\/S0218202519500106","journal-title":"Math. Models Methods Appl. Sci."},{"issue":"3","key":"3020_CR24","doi-asserted-by":"publisher","first-page":"A1494","DOI":"10.1137\/16M1106122","volume":"40","author":"Y Nakatsukasa","year":"2018","unstructured":"Nakatsukasa, Y., S\u00e8te, O., Trefethen, L.N.: The AAA algorithm for rational approximation. SIAM J. Sci. Comput. 40(3), A1494\u2013A1522 (2018). https:\/\/doi.org\/10.1137\/16M1106122","journal-title":"SIAM J. Sci. Comput."},{"issue":"1","key":"3020_CR25","doi-asserted-by":"publisher","DOI":"10.1002\/pamm.202200032","volume":"22","author":"D Pradovera","year":"2023","unstructured":"Pradovera, D.: Adaptive approximation of nonlinear eigenproblems by minimal rational interpolation. PAMM 22(1), e202200032 (2023). https:\/\/doi.org\/10.1002\/pamm.202200032","journal-title":"PAMM"},{"issue":"6","key":"3020_CR26","doi-asserted-by":"publisher","first-page":"92","DOI":"10.1007\/s10444-023-10091-7","volume":"49","author":"D Pradovera","year":"2023","unstructured":"Pradovera, D.: Toward a certified greedy Loewner framework with minimal sampling. Adv. Comput. Math. 49(6), 92 (2023). https:\/\/doi.org\/10.1007\/s10444-023-10091-7","journal-title":"Adv. Comput. Math."},{"key":"3020_CR27","doi-asserted-by":"publisher","first-page":"2","DOI":"10.1186\/s13362-021-00117-4","volume":"12","author":"D Pradovera","year":"2022","unstructured":"Pradovera, D., Nobile, F.: A technique for non-intrusive greedy piecewise-rational model reduction of frequency response problems over wide frequency bands. J. Math. Ind. 12, 2 (2022). https:\/\/doi.org\/10.1186\/s13362-021-00117-4","journal-title":"J. Math. Ind."},{"key":"3020_CR28","doi-asserted-by":"publisher","unstructured":"Sauter, S.A., Schwab, C.: Boundary element methods, Springer series in Computational Mathematics, vol.\u00a039. Springer Berlin Heidelberg (2011). https:\/\/doi.org\/10.1007\/978-3-540-68093-2","DOI":"10.1007\/978-3-540-68093-2"},{"key":"3020_CR29","doi-asserted-by":"publisher","unstructured":"Simchowitz, M., Alaoui, A.E., Recht, B.: On the gap between strict-saddles and true convexity: An Omega(log d) lower bound for eigenvector approximation (2017). https:\/\/doi.org\/10.48550\/arXiv.1704.04548","DOI":"10.48550\/arXiv.1704.04548"},{"key":"3020_CR30","doi-asserted-by":"publisher","first-page":"287","DOI":"10.1017\/S0962492900002294","volume":"1","author":"IH Sloan","year":"1992","unstructured":"Sloan, I.H.: Error analysis of boundary integral methods. Acta Numer 1, 287\u2013339 (1992). https:\/\/doi.org\/10.1017\/S0962492900002294","journal-title":"Acta Numer"},{"issue":"3","key":"3020_CR31","doi-asserted-by":"publisher","first-page":"1587","DOI":"10.1007\/s13160-023-00599-2","volume":"40","author":"LN Trefethen","year":"2023","unstructured":"Trefethen, L.N.: Numerical analytic continuation. Jpn. J. Ind. Appl. Math. 40(3), 1587\u20131636 (2023). https:\/\/doi.org\/10.1007\/s13160-023-00599-2","journal-title":"Jpn. J. Ind. Appl. Math."},{"issue":"5","key":"3020_CR32","doi-asserted-by":"publisher","first-page":"1470","DOI":"10.1109\/TMAG.2010.2078800","volume":"47","author":"A Vion","year":"2011","unstructured":"Vion, A., Sabariego, R.V., Geuzaine, C.: A model reduction algorithm for solving multiple scattering problems using iterative methods. IEEE Trans. Magn. 47(5), 1470\u20131473 (2011). https:\/\/doi.org\/10.1109\/TMAG.2010.2078800","journal-title":"IEEE Trans. Magn."},{"key":"3020_CR33","unstructured":"Walsh, J.L.: Interpolation and approximation by rational functions in the complex domain. American Mathematical Society (1960)"},{"key":"3020_CR34","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511617539","volume-title":"Scattered data approximation","author":"H Wendland","year":"2004","unstructured":"Wendland, H.: Scattered data approximation. Cambridge University Press, Cambridge (2004). https:\/\/doi.org\/10.1017\/CBO9780511617539"},{"issue":"3","key":"3020_CR35","doi-asserted-by":"publisher","first-page":"335","DOI":"10.1016\/j.acha.2007.12.002","volume":"25","author":"F Woolfe","year":"2008","unstructured":"Woolfe, F., Liberty, E., Rokhlin, V., Tygert, M.: A fast randomized algorithm for the approximation of matrices. Appl. Comput. Harmon. Anal. 25(3), 335\u2013366 (2008). https:\/\/doi.org\/10.1016\/j.acha.2007.12.002","journal-title":"Appl. Comput. Harmon. Anal."},{"key":"3020_CR36","doi-asserted-by":"publisher","unstructured":"Xiong, H.: Generic simplicity of resonances in obstacle scattering (2022). https:\/\/doi.org\/10.48550\/arXiv.2105.07494","DOI":"10.48550\/arXiv.2105.07494"},{"issue":"2","key":"3020_CR37","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6420\/ad149d","volume":"40","author":"J Zimmerling","year":"2023","unstructured":"Zimmerling, J., Druskin, V., Guddati, M., Cherkaev, E., Remis, R.: Solving inverse scattering problems via reduced-order model embedding procedures. Inverse Prob. 40(2), 025002 (2023). https:\/\/doi.org\/10.1088\/1361-6420\/ad149d","journal-title":"Inverse Prob."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-025-03020-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-025-03020-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-025-03020-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,4]],"date-time":"2025-09-04T11:53:45Z","timestamp":1756986825000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-025-03020-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,9]]},"references-count":37,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,9]]}},"alternative-id":["3020"],"URL":"https:\/\/doi.org\/10.1007\/s10915-025-03020-x","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,8,9]]},"assertion":[{"value":"13 March 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 June 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 July 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 August 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"104"}}