{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:26:50Z","timestamp":1787333210074,"version":"build-2736575974"},"reference-count":20,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>We compute $u(t)=\\exp(-tA)\\varphi$ using rational Krylov subspace reduction for $0\\leq t&lt;\\infty$, where $u(t),\\varphi\\in\\mathbf{R}^N$ and $0&lt;A=A^*\\in\\mathbf{R}^{N\\times N}$. A priori optimization of the rational Krylov subspaces for this problem was considered in [V. Druskin, L. Knizhnerman, and M. Zaslavsky, SIAM J. Sci. Comput., 31 (2009), pp. 3760\u20133780]. There was suggested an algorithm generating sequences of equidistributed shifts, which are asymptotically optimal for the cases with uniform spectral distributions. Here we develop a recursive greedy algorithm for choice of shifts taking into account nonuniformity of the spectrum. The algorithm is based on an explicit formula for the residual in the frequency domain allowing adaptive shift optimization at negligible cost. The effectiveness of the developed approach is demonstrated in an example of the three-dimensional diffusion problem for Maxwell's equation arising in geophysical exploration. We compare our approach with the one using the above-mentioned equidistributed sequences of shifts. Numerical examples show that our algorithm is able to adapt to the spectral density of operator A. For examples with near-uniform spectral distributions, both algorithms show the same convergence rates, but the new algorithm produces superior convergence for cases with nonuniform spectra.<\/jats:p>","DOI":"10.1137\/090774082","type":"journal-article","created":{"date-parts":[[2010,8,20]],"date-time":"2010-08-20T18:48:15Z","timestamp":1282330095000},"page":"2485-2496","source":"Crossref","is-referenced-by-count":68,"title":["On Adaptive Choice of Shifts in Rational Krylov Subspace Reduction of Evolutionary Problems"],"prefix":"10.1137","volume":"32","author":[{"given":"Vladimir","family":"Druskin","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chad","family":"Lieberman","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mikhail","family":"Zaslavsky","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,8,19]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-69-03614-X"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142999360688"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1137\/080741744"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/070694855"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/080742403"},{"key":"R6","unstructured":"V. Druskin and M. Zaslavsky,\n                      On convergence of Krylov subspace approximations of time-invariant Hermitian dynamical systems\n                      , Linear Algebra Appl., to appear."},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1137\/040605461"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492902000120"},{"key":"R9","unstructured":"E. Grimme,\n                      Krylov Projection Methods for Model Reduction\n                      , Ph.D. thesis, University of Illinois at Urbana-Champaign, Urbana, IL, 1997."},{"key":"R10","unstructured":"S. Gugercin, A. Antoulas, and C. Beattie,\n                      A rational Krylov iteration for optimal $\\mathcal{H}_2$ model reduction\n                      , in Proceedings of the 17th International Symposium on Math, Theory of Networks and Systems, Kyoto, Japan, 2006, pp. 1665\u20131667."},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1137\/080715159"},{"key":"R12","unstructured":"V. Lebedev,\n                      On Zolotarev problems in the alternating direction method\n                      . II, in Trudy Semin. S. L. Sobolev, No. 1, Novosibirsk, Nauka, Moscow, 1976, pp. 51\u201359 (in Russian)."},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.1967.1098680"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1023\/A:1023659813269"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1070\/SM2007v198n03ABEH003842"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1137\/060672856"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1007\/BF01935024"},{"key":"R18","doi-asserted-by":"crossref","unstructured":"E. Saff and V. Totik,\n                      Logarithmic Potentials with External Fields\n                      , Springer-Verlag, Berlin, 1997.","DOI":"10.1007\/978-3-662-03329-6"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1007\/BF02575706"},{"key":"R20","doi-asserted-by":"crossref","unstructured":"M. Zaslavsky, S. Davydycheva, V. Druskin, A. Abubakar, T. Habashy, and L. Knizhnerman,\n                      Finite-difference solution of the 3D electromagnetic problem using divergence-free preconditioners\n                      , in Proceedings of the SEG Annual Meeting, New Orleans, LA, 2006, pp. 775\u2013778.","DOI":"10.1190\/1.2370372"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/090774082","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:41:23Z","timestamp":1787330483000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/090774082"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":20,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/090774082"],"URL":"https:\/\/doi.org\/10.1137\/090774082","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}