{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T01:36:17Z","timestamp":1780450577981,"version":"3.54.1"},"reference-count":41,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2016,9,19]],"date-time":"2016-09-19T00:00:00Z","timestamp":1474243200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2016,9,19]],"date-time":"2016-09-19T00:00:00Z","timestamp":1474243200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DE-EE0002750"],"award-info":[{"award-number":["DE-EE0002750"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1413454"],"award-info":[{"award-number":["DMS-1413454"]}],"id":[{"id":"10.13039\/100000001","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":[[2017,2]]},"DOI":"10.1007\/s10444-016-9482-z","type":"journal-article","created":{"date-parts":[[2016,9,19]],"date-time":"2016-09-19T09:42:24Z","timestamp":1474278144000},"page":"171-193","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Null space correction and adaptive model order reduction in multi-frequency Maxwell\u2019s problem"],"prefix":"10.1007","volume":"43","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6474-9828","authenticated-orcid":false,"given":"Michal","family":"Kordy","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Elena","family":"Cherkaev","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Philip","family":"Wannamaker","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2016,9,19]]},"reference":[{"key":"9482_CR1","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1090\/conm\/280\/04630","volume":"280","author":"AC Antoulas","year":"2001","unstructured":"Antoulas, A.C., Sorensen, D.C., Gugercin, S.: A survey of model reduction methods for large-scale systems. Contemp. Math. 280, 193\u2013220 (2001)","journal-title":"Contemp. Math."},{"key":"9482_CR2","doi-asserted-by":"publisher","first-page":"9","DOI":"10.1016\/S0168-9274(02)00116-2","volume":"43","author":"Z Bai","year":"2002","unstructured":"Bai, Z.: Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems. Appl. Numer. Math. 43, 9\u201344 (2002)","journal-title":"Appl. Numer. Math."},{"key":"9482_CR3","doi-asserted-by":"crossref","unstructured":"Benner, P., Mehrmann, V., Sorensen, D.C.: Dimension Reduction of Large-Scale Systems. Springer (2005)","DOI":"10.1007\/3-540-27909-1"},{"key":"9482_CR4","doi-asserted-by":"crossref","unstructured":"Bodendiek, A., Bollh\u00f6fer, M.: Adaptive-order rational Arnoldi-type methods in computational electromagnetism. BIT Numer. Math. 1\u201324 (2013)","DOI":"10.1007\/s10543-013-0458-9"},{"key":"9482_CR5","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1007\/s10712-009-9087-x","volume":"31","author":"RU Boerner","year":"2010","unstructured":"Boerner, R.U.: Numerical modelling in geo-electromagnetics: advances and challenges. Surv. Geophys. 31, 225\u2013245 (2010)","journal-title":"Surv. Geophys."},{"issue":"5","key":"9482_CR6","doi-asserted-by":"publisher","first-page":"2485","DOI":"10.1137\/090774082","volume":"32","author":"V Druskin","year":"2010","unstructured":"Druskin, V., Lieberman, C., Zaslavsky, M.: On adaptive choice of shifts in rational Krylov subspace reduction of evolutionary problems. SIAM J. Sci. Comput. 32(5), 2485\u20132496 (2010)","journal-title":"SIAM J. Sci. Comput."},{"issue":"8","key":"9482_CR7","doi-asserted-by":"publisher","first-page":"546","DOI":"10.1016\/j.sysconle.2011.04.013","volume":"60","author":"V Druskin","year":"2011","unstructured":"Druskin, V., Simoncini, V.: Adaptive rational Krylov subspaces for large-scale dynamical systems. Syst. Control Lett. 60(8), 546\u2013560 (2011)","journal-title":"Syst. Control Lett."},{"issue":"3","key":"9482_CR8","doi-asserted-by":"publisher","first-page":"A1621","DOI":"10.1137\/110852607","volume":"35","author":"V Druskin","year":"2013","unstructured":"Druskin, V., Simoncini, V., Zaslavsky, M.: Solution of the time-domain inverse resistivity prolem in the model reduction framework part I: one-dimensional problem with SISO data. SIAM J. Sci. Comput. 35(3), A1621\u2013A1640 (2013)","journal-title":"SIAM J. Sci. Comput."},{"issue":"3","key":"9482_CR9","doi-asserted-by":"publisher","first-page":"A1584","DOI":"10.1137\/110827600","volume":"34","author":"HC Elman","year":"2012","unstructured":"Elman, H.C., Meerbergen, K., Spence, A., Wu, M.: Lyapunov inverse iteration for identifying Hopf bifurcations in models of incompressible flow. SIAM J. Sci. Comput. 34(3), A1584\u2013A1606 (2012)","journal-title":"SIAM J. Sci. Comput."},{"key":"9482_CR10","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1007\/s10712-011-9138-y","volume":"33","author":"ME Everett","year":"2012","unstructured":"Everett, M.E.: Theoretical developments in electromagnetic induction geophysics with selected applications in the near-surface. Surv. Geophys. 33, 29\u201363 (2012)","journal-title":"Surv. Geophys."},{"key":"9482_CR11","doi-asserted-by":"publisher","first-page":"699","DOI":"10.1016\/j.jappgeo.2011.09.025","volume":"75","author":"CG Farquharson","year":"2011","unstructured":"Farquharson, C.G., Meinsopust, M.: Three-dimensional finite-element modeling of magnetotelluric data with a divergence correction. J. Appl. Geophys. 75, 699\u2013710 (2011)","journal-title":"J. Appl. Geophys."},{"key":"9482_CR12","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1017\/S0962492902000120","volume":"12","author":"RW Freund","year":"2003","unstructured":"Freund, R.W.: Model reduction methods based on Krylov subspaces. Acta Numer. 12, 267\u2013319 (2003)","journal-title":"Acta Numer."},{"issue":"1","key":"9482_CR13","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1007\/BF02141740","volume":"12","author":"K Gallivan","year":"1996","unstructured":"Gallivan, K., Grimme, G., Van Dooren, P.: A rational Lanczos algorithm for model reduction. Numer. Algoritm. 12(1), 33\u201363 (1996)","journal-title":"Numer. Algoritm."},{"key":"9482_CR14","unstructured":"Grimme, E.J.: Krylov Projection Methods for Model Reduction. Ph.D. thesis, University of Illinois, Urbana-Champaigne (1997)"},{"issue":"2","key":"9482_CR15","doi-asserted-by":"publisher","first-page":"609","DOI":"10.1137\/060666123","volume":"30","author":"S Gugercin","year":"2008","unstructured":"Gugercin, S., Antoulas, A.C., Beattie, C.: H_2 model reduction for large-scale linear dynamical systems. SIAM J. Matrix Anal. Appl. 30(2), 609\u2013638 (2008)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"9482_CR16","doi-asserted-by":"publisher","DOI":"10.1007\/b13382","volume-title":"Least-Squares Finite Element Methods","author":"MD Gunzburger","year":"2009","unstructured":"Gunzburger, M.D., Bochev, P.B.: Least-Squares Finite Element Methods. Springer, New York (2009)"},{"issue":"1","key":"9482_CR17","doi-asserted-by":"publisher","first-page":"8","DOI":"10.1002\/gamm.201310002","volume":"36","author":"S G\u00fcttel","year":"2013","unstructured":"G\u00fcttel, S.: Rational Krylov approximation of matrix functions: Numerical methods and optimal pole selection. GAMM-Mitteilungen 36(1), 8\u201331 (2013)","journal-title":"GAMM-Mitteilungen"},{"issue":"3","key":"9482_CR18","doi-asserted-by":"publisher","first-page":"595","DOI":"10.1007\/s10543-013-0420-x","volume":"53","author":"S G\u00fcttel","year":"2013","unstructured":"G\u00fcttel, S., Knizhnerman, L.: A black-box rational Arnoldi variant for Cauchy\u2013Stieltjes matrix functions. BIT Numer. Math. 53(3), 595\u2013616 (2013)","journal-title":"BIT Numer. Math."},{"issue":"1","key":"9482_CR19","doi-asserted-by":"publisher","first-page":"204","DOI":"10.1137\/S0036142997326203","volume":"36","author":"R Hiptmair","year":"1998","unstructured":"Hiptmair, R.: Multigrid method for Maxwell\u2019s equations. SIAM J. Numer. Anal. 36(1), 204\u2013225 (1998)","journal-title":"SIAM J. Numer. Anal."},{"issue":"2","key":"9482_CR20","doi-asserted-by":"publisher","first-page":"953","DOI":"10.1137\/080715159","volume":"47","author":"L Knizhnerman","year":"2009","unstructured":"Knizhnerman, L., Druskin, V., Zaslavsky, M.: On optimal convergence rate of the rational Krylov subspace reduction for electromagnetic problems in unbounded domains. SIAM J. Numer. Anal. 47(2), 953\u2013971 (2009)","journal-title":"SIAM J. Numer. Anal."},{"key":"9482_CR21","doi-asserted-by":"crossref","unstructured":"Kolev, T.V., Vassilevski, P.S.: Some experience with a H1-based auxiliary space AMG for H(curl) problems. Lawrence Livermore Nat. Lab., Livermore, CA, Rep. UCRL-TR-221841 (2006)","DOI":"10.2172\/900464"},{"issue":"4","key":"9482_CR22","first-page":"731","volume":"12","author":"M Kordy","year":"2015","unstructured":"Kordy, M., Cherkaev, E., Wannamaker, P.: Variational formulation for Maxwell\u2019s equations with Lorenz gauge: Existence and uniqueness of solution. Int. J. Numer. Anal. Model. 12(4), 731\u2013749 (2015)","journal-title":"Int. J. Numer. Anal. Model."},{"key":"9482_CR23","doi-asserted-by":"crossref","unstructured":"Kordy, M., Cherkaev, E., Wannamaker, P.: Adaptive model order reduction for the Jacobian calculation in inverse multi-frequency problem for Maxwell\u2019s equations. Applied Numerical Mathematics 109, 1\u201318 (2016)","DOI":"10.1016\/j.apnum.2016.02.010"},{"issue":"1","key":"9482_CR24","doi-asserted-by":"publisher","first-page":"74","DOI":"10.1093\/gji\/ggv410","volume":"204","author":"M Kordy","year":"2016","unstructured":"Kordy, M., Wannamaker, P., Maris, V., Cherkaev, E., Hill, G.: Three-dimensional magnetotelluric inversion including topography using deformed hexahedral edge finite elements and direct solvers parallelized on SMP computers, Part I: Forward problem and parameter Jacobians. Geophys. J. Int. 204(1), 74\u201393 (2016)","journal-title":"Geophys. J. Int."},{"issue":"1","key":"9482_CR25","doi-asserted-by":"publisher","first-page":"94","DOI":"10.1093\/gji\/ggv411","volume":"204","author":"M Kordy","year":"2016","unstructured":"Kordy, M., Wannamaker, P., Maris, V., Cherkaev, E., Hill, G.: Three-dimensional magnetotelluric inversion including topography using deformed hexahedral edge finite elements and direct solvers parallelized on SMP computers, Part II: Direct data-space inverse solution. Geophys. J. Int. 204(1), 94\u2013110 (2016)","journal-title":"Geophys. J. Int."},{"key":"9482_CR26","doi-asserted-by":"crossref","unstructured":"Levin, E., Saff, E.B.: Potential theoretic tools in polynomial and rational approximation. In: Harmonic Analysis and Rational Approximation, pp 71\u201394. Springer (2006)","DOI":"10.1007\/11601609_5"},{"issue":"6","key":"9482_CR27","doi-asserted-by":"publisher","first-page":"431","DOI":"10.1002\/nla.625","volume":"16","author":"I Moret","year":"2009","unstructured":"Moret, I.: Rational Lanczos approximations to the matrix square root and related functions. Numer. Linear Algebra Appl. 16(6), 431\u2013445 (2009)","journal-title":"Numer. Linear Algebra Appl."},{"issue":"5","key":"9482_CR28","doi-asserted-by":"publisher","first-page":"633","DOI":"10.1111\/j.1365-2478.2006.00558.x","volume":"54","author":"WA Mulder","year":"2006","unstructured":"Mulder, W.A.: A multigrid solver for 3D electromagnetic diffusion. Geophys. Prospect. 54(5), 633\u2013649 (2006)","journal-title":"Geophys. Prospect."},{"issue":"1","key":"9482_CR29","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1007\/BF01389668","volume":"50","author":"JC N\u00e9d\u00e9lec","year":"1986","unstructured":"N\u00e9d\u00e9lec, J.C.: A new family of mixed finite elements in R\n                           3. Numer. Math. 50(1), 57\u201381 (1986)","journal-title":"Numer. Math."},{"key":"9482_CR30","doi-asserted-by":"publisher","first-page":"410","DOI":"10.1046\/j.1365-246x.2000.00007.x","volume":"140","author":"GA Newman","year":"2000","unstructured":"Newman, G.A., Alumbaugh, D.L.: Three-dimensional magnetotelluric inversion using non-linear conjugate gradients. Geophys. J. Int. 140, 410\u2013424 (2000)","journal-title":"Geophys. J. Int."},{"issue":"2","key":"9482_CR31","doi-asserted-by":"publisher","first-page":"657","DOI":"10.1137\/060672856","volume":"30","author":"M Popolizio","year":"2008","unstructured":"Popolizio, M., Simoncini, V.: Acceleration techniques for approximating the matrix exponential operator. SIAM J. Matrix Anal. Appl. 30(2), 657\u2013683 (2008)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"9482_CR32","doi-asserted-by":"publisher","first-page":"494","DOI":"10.1002\/nla.1894","volume":"21","author":"S Ragni","year":"2014","unstructured":"Ragni, S.: Rational Krylov methods in exponential integrators for European option pricing. Numer. Linear Algebra Appl. 21, 494\u2013512 (2014)","journal-title":"Numer. Linear Algebra Appl."},{"issue":"3","key":"9482_CR33","doi-asserted-by":"publisher","first-page":"766","DOI":"10.1111\/j.1365-246X.2008.03750.x","volume":"173","author":"B Ralph-Uwe","year":"2008","unstructured":"Ralph-Uwe, B., Ernst, O.G., Spitzer, K.: Fast 3-D simulation of transient electromagnetic fields by model reduction in the frequency domain using Krylov subspace projection. Geophys. J. Int. 173(3), 766\u2013780 (2008)","journal-title":"Geophys. J. Int."},{"key":"9482_CR34","doi-asserted-by":"crossref","unstructured":"Ransford, T.: Potential Theory in the Complex Plane. Cambridge University Press (1995)","DOI":"10.1017\/CBO9780511623776"},{"key":"9482_CR35","doi-asserted-by":"crossref","unstructured":"Saff, E.B., Totik, V.: Logarithmic Potentials with External Fields. Springer (1997)","DOI":"10.1007\/978-3-662-03329-6"},{"key":"9482_CR36","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1016\/S0926-9851(00)00038-0","volume":"46","author":"Y Sasaki","year":"2001","unstructured":"Sasaki, Y.: Full 3-D inversion of electromagnetic data on PC. J. Appl. Geophys. 46, 45\u201354 (2001)","journal-title":"J. Appl. Geophys."},{"key":"9482_CR37","doi-asserted-by":"publisher","first-page":"721","DOI":"10.1186\/BF03351724","volume":"54","author":"W Siripunvaraporn","year":"2002","unstructured":"Siripunvaraporn, W., Egbert, G., Lenbury, Y.: Numerical accuracy of magnetotelluric modeling: a comparison of finite difference approximations. Earth Planets Space 54, 721\u2013725 (2002)","journal-title":"Earth Planets Space"},{"key":"9482_CR38","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1016\/j.pepi.2004.08.023","volume":"150","author":"W Siripunvaraporn","year":"2005","unstructured":"Siripunvaraporn, W., Egbert, G., Lenbury, Y., Uyeshima, M.: Three-dimensional magnetotelluric inversion: data-space method. Phys. Earth Planet Inter. 150, 3\u201314 (2005)","journal-title":"Phys. Earth Planet Inter."},{"issue":"5","key":"9482_CR39","doi-asserted-by":"publisher","first-page":"1319","DOI":"10.1190\/1.1444055","volume":"61","author":"J Smith","year":"1996","unstructured":"Smith, J.: Conservative modeling of 3-D electromagnetic fields, Part II: Biconjugate gradient solution and an accelerator. Geophysics 61(5), 1319\u20131324 (1996)","journal-title":"Geophysics"},{"key":"9482_CR40","doi-asserted-by":"crossref","unstructured":"Villena, J.F., Silveira, L.M.: ARMS - automatic residue-minimization based sampling for multi-point modeling techniques. In: DAC\u201909 Proceedings of the 46th Annual Design Automation Conference, pp 951\u2013956 (2009)","DOI":"10.1145\/1629911.1630156"},{"issue":"4","key":"9482_CR41","doi-asserted-by":"publisher","first-page":"E161","DOI":"10.1190\/geo2012-0257.1","volume":"78","author":"M Zaslavsky","year":"2013","unstructured":"Zaslavsky, M., Druskin, V., Abubakar, A., Habashy, T., Simoncini, V.: Large-scale Gauss-Newton inversion of transient CSEM data using the model order reduction framework. Geophysics 78(4), E161\u2013E171 (2013)","journal-title":"Geophysics"}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-016-9482-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10444-016-9482-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-016-9482-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,5,17]],"date-time":"2020-05-17T15:36:58Z","timestamp":1589729818000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10444-016-9482-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,9,19]]},"references-count":41,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2017,2]]}},"alternative-id":["9482"],"URL":"https:\/\/doi.org\/10.1007\/s10444-016-9482-z","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"value":"1019-7168","type":"print"},{"value":"1572-9044","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,9,19]]},"assertion":[{"value":"7 August 2015","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 September 2016","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 September 2016","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}