{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T14:07:31Z","timestamp":1776866851462,"version":"3.51.2"},"reference-count":46,"publisher":"American Mathematical Society (AMS)","issue":"267","license":[{"start":{"date-parts":[[2009,11,25]],"date-time":"2009-11-25T00:00:00Z","timestamp":1259107200000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>With respect to space-time tensor-product wavelet bases, parabolic initial boundary value problems are equivalently formulated as bi-infinite matrix problems. Adaptive wavelet methods are shown to yield sequences of approximate solutions which converge at the optimal rate. In case the spatial domain is of product type, the use of spatial tensor product wavelet bases is proved to overcome the so-called curse of dimensionality, i.e., the reduction of the convergence rate with increasing spatial dimension.<\/p>","DOI":"10.1090\/s0025-5718-08-02205-9","type":"journal-article","created":{"date-parts":[[2009,4,27]],"date-time":"2009-04-27T13:47:33Z","timestamp":1240840053000},"page":"1293-1318","source":"Crossref","is-referenced-by-count":146,"title":["Space-time adaptive wavelet methods for parabolic evolution problems"],"prefix":"10.1090","volume":"78","author":[{"given":"Christoph","family":"Schwab","sequence":"first","affiliation":[]},{"given":"Rob","family":"Stevenson","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2008,11,25]]},"reference":[{"issue":"2","key":"1","doi-asserted-by":"publisher","first-page":"829","DOI":"10.1016\/j.jcp.2005.10.009","article-title":"Simultaneous space-time adaptive wavelet solution of nonlinear parabolic differential equations","volume":"214","author":"Alam, Jahrul M.","year":"2006","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"key":"2","doi-asserted-by":"publisher","first-page":"322","DOI":"10.1007\/BF02165003","article-title":"Error-bounds for finite element method","volume":"16","author":"Babu\u0161ka, Ivo","year":"1970","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"7","key":"3","doi-asserted-by":"publisher","first-page":"793","DOI":"10.1051\/m2an\/1992260707931","article-title":"A wavelet based space-time adaptive numerical method for partial differential equations","volume":"26","author":"Bacry, E.","year":"1992","journal-title":"RAIRO Mod\\'{e}l. Math. Anal. Num\\'{e}r.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-583X","issn-type":"print"},{"key":"4","unstructured":"[Bar05] A. Barinka. Fast Evaluation Tools for Adaptive Wavelet Schemes. Ph.D. thesis, RTWH Aachen, March 2005."},{"key":"5","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1017\/S0962492904000182","article-title":"Sparse grids","volume":"13","author":"Bungartz, Hans-Joachim","year":"2004","journal-title":"Acta Numer.","ISSN":"https:\/\/id.crossref.org\/issn\/0962-4929","issn-type":"print"},{"key":"6","isbn-type":"print","volume-title":"Finite elements","author":"Braess, Dietrich","year":"2001","ISBN":"https:\/\/id.crossref.org\/isbn\/0521011957","edition":"2"},{"issue":"233","key":"7","doi-asserted-by":"publisher","first-page":"27","DOI":"10.1090\/S0025-5718-00-01252-7","article-title":"Adaptive wavelet methods for elliptic operator equations: convergence rates","volume":"70","author":"Cohen, Albert","year":"2001","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3","key":"8","doi-asserted-by":"publisher","first-page":"203","DOI":"10.1007\/s102080010027","article-title":"Adaptive wavelet methods. II. Beyond the elliptic case","volume":"2","author":"Cohen, A.","year":"2002","journal-title":"Found. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1615-3375","issn-type":"print"},{"issue":"247","key":"9","doi-asserted-by":"publisher","first-page":"1167","DOI":"10.1090\/S0025-5718-04-01634-5","article-title":"An adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems","volume":"73","author":"Chen, Zhiming","year":"2004","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"10","isbn-type":"print","first-page":"53","article-title":"Wavelets on a bounded interval","author":"Chui, Charles K.","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/3764327464"},{"issue":"4","key":"11","doi-asserted-by":"publisher","first-page":"717","DOI":"10.1093\/imanum\/drl035","article-title":"Adaptive frame methods for elliptic operator equations: the steepest descent approach","volume":"27","author":"Dahlke, Stephan","year":"2007","journal-title":"IMA J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"},{"issue":"6","key":"12","doi-asserted-by":"publisher","first-page":"1791","DOI":"10.1137\/S0036141094276160","article-title":"Intertwining multiresolution analyses and the construction of piecewise-polynomial wavelets","volume":"27","author":"Donovan, George C.","year":"1996","journal-title":"SIAM J. Math. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"issue":"5","key":"13","doi-asserted-by":"publisher","first-page":"1029","DOI":"10.1137\/S0036141096313112","article-title":"Orthogonal polynomials and the construction of piecewise polynomial smooth wavelets","volume":"30","author":"Donovan, G. C.","year":"1999","journal-title":"SIAM J. Math. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"issue":"8","key":"14","doi-asserted-by":"publisher","first-page":"3758","DOI":"10.1016\/j.jcp.2007.11.046","article-title":"An adaptive multiresolution scheme with local time stepping for evolutionary PDEs","volume":"227","author":"Domingues, Margarete O.","year":"2008","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"2","key":"15","doi-asserted-by":"publisher","first-page":"132","DOI":"10.1006\/acha.1998.0247","article-title":"Biorthogonal spline wavelets on the interval\u2014stability and moment conditions","volume":"6","author":"Dahmen, Wolfgang","year":"1999","journal-title":"Appl. Comput. Harmon. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/1063-5203","issn-type":"print"},{"key":"16","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-58090-1","volume-title":"Mathematical analysis and numerical methods for science and technology. Vol. 5","author":"Dautray, Robert","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/354050205X"},{"issue":"3-4","key":"17","doi-asserted-by":"publisher","first-page":"255","DOI":"10.1007\/BF03322055","article-title":"Wavelets with complementary boundary conditions\u2014function spaces on the cube","volume":"34","author":"Dahmen, Wolfgang","year":"1998","journal-title":"Results Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0378-6218","issn-type":"print"},{"issue":"1","key":"18","doi-asserted-by":"publisher","first-page":"184","DOI":"10.1137\/S0036141098333451","article-title":"Wavelets on manifolds. I. Construction and domain decomposition","volume":"31","author":"Dahmen, Wolfgang","year":"1999","journal-title":"SIAM J. Math. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"key":"19","doi-asserted-by":"crossref","unstructured":"[DSS08] T J. Dijkema, Ch. Schwab, and R.P. Stevenson. An adaptive wavelet method for solving high-dimensional elliptic PDEs. Technical report, January 2008. To appear.","DOI":"10.1007\/s00365-009-9064-0"},{"issue":"1","key":"20","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1137\/0728003","article-title":"Adaptive finite element methods for parabolic problems. I. A linear model problem","volume":"28","author":"Eriksson, Kenneth","year":"1991","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"3","key":"21","doi-asserted-by":"publisher","first-page":"706","DOI":"10.1137\/0732033","article-title":"Adaptive finite element methods for parabolic problems. II. Optimal error estimates in \ud835\udc3f_{\u221e}\ud835\udc3f\u2082 and \ud835\udc3f_{\u221e}\ud835\udc3f_{\u221e}","volume":"32","author":"Eriksson, Kenneth","year":"1995","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"4","key":"22","doi-asserted-by":"publisher","first-page":"611","DOI":"10.1051\/m2an\/1985190406111","article-title":"Time discretization of parabolic problems by the discontinuous Galerkin method","volume":"19","author":"Eriksson, Kenneth","year":"1985","journal-title":"RAIRO Mod\\'{e}l. Math. Anal. Num\\'{e}r.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-583X","issn-type":"print"},{"issue":"1","key":"23","doi-asserted-by":"publisher","first-page":"90","DOI":"10.1016\/j.cam.2006.11.013","article-title":"An optimal adaptive wavelet method for nonsymmetric and indefinite elliptic problems","volume":"211","author":"Gantumur, Tsogtgerel","year":"2008","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"issue":"258","key":"24","doi-asserted-by":"publisher","first-page":"615","DOI":"10.1090\/S0025-5718-06-01917-X","article-title":"An optimal adaptive wavelet method without coarsening of the iterands","volume":"76","author":"Gantumur, Tsogtgerel","year":"2007","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"254","key":"25","doi-asserted-by":"publisher","first-page":"697","DOI":"10.1090\/S0025-5718-05-01807-7","article-title":"Computation of differential operators in wavelet coordinates","volume":"75","author":"Gantumur, Tsogtgerel","year":"2006","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1-2","key":"26","doi-asserted-by":"publisher","first-page":"77","DOI":"10.1007\/s00607-005-0135-1","article-title":"Computation of singular integral operators in wavelet coordinates","volume":"76","author":"Gantumur, T.","year":"2006","journal-title":"Computing","ISSN":"https:\/\/id.crossref.org\/issn\/0010-485X","issn-type":"print"},{"issue":"4","key":"27","doi-asserted-by":"publisher","first-page":"525","DOI":"10.1007\/s003650010010","article-title":"Optimized tensor-product approximation spaces","volume":"16","author":"Griebel, M.","year":"2000","journal-title":"Constr. Approx.","ISSN":"https:\/\/id.crossref.org\/issn\/0176-4276","issn-type":"print"},{"issue":"1-2","key":"28","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1007\/BF02123478","article-title":"Tensor product type subspace splittings and multilevel iterative methods for anisotropic problems","volume":"4","author":"Griebel, M.","year":"1995","journal-title":"Adv. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1019-7168","issn-type":"print"},{"issue":"1","key":"29","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00607-007-0241-3","article-title":"A sparse grid space-time discretization scheme for parabolic problems","volume":"81","author":"Griebel, M.","year":"2007","journal-title":"Computing","ISSN":"https:\/\/id.crossref.org\/issn\/0010-485X","issn-type":"print"},{"key":"30","unstructured":"[Goo00] T.N.T. Goodman. Biorthogonal refinable spline functions. In A. Cohen, C. Rabut, and L.L. Schumaker, editors, Curve and Surface Fitting: Saint-Malo 1999, pages 1\u20138, Nashville, TN, 2000. Vanderbilt University Press."},{"issue":"256","key":"31","doi-asserted-by":"publisher","first-page":"1871","DOI":"10.1090\/S0025-5718-06-01867-9","article-title":"Wavelets with patchwise cancellation properties","volume":"75","author":"Harbrecht, Helmut","year":"2006","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"255","key":"32","doi-asserted-by":"publisher","first-page":"1319","DOI":"10.1090\/S0025-5718-06-01828-X","article-title":"Wavelets on manifolds: an optimized construction","volume":"75","author":"Kunoth, Angela","year":"2006","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"33","series-title":"Lecture Notes in Computational Science and Engineering","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-04484-1","volume-title":"Adaptive multilevel solution of nonlinear parabolic PDE systems","volume":"16","author":"Lang, Jens","year":"2001","ISBN":"https:\/\/id.crossref.org\/isbn\/3540679006"},{"key":"34","unstructured":"[Met02] A. Metselaar. Handling Wavelet Expansions in Numerical Methods. Ph.D. thesis, University of Twente, 2002."},{"issue":"3","key":"35","doi-asserted-by":"publisher","first-page":"493","DOI":"10.1007\/s10915-006-9102-z","article-title":"Fully adaptive multiscale schemes for conservation laws employing locally varying time stepping","volume":"30","author":"M\u00fcller, Siegfried","year":"2007","journal-title":"J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0885-7474","issn-type":"print"},{"issue":"164","key":"36","doi-asserted-by":"publisher","first-page":"321","DOI":"10.2307\/2007679","article-title":"Numerical approximations to nonlinear conservation laws with locally varying time and space grids","volume":"41","author":"Osher, Stanley","year":"1983","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3-4","key":"37","doi-asserted-by":"publisher","first-page":"223","DOI":"10.1016\/S0045-7825(98)00121-2","article-title":"Adaptive finite elements for a linear parabolic problem","volume":"167","author":"Picasso, Marco","year":"1998","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"key":"38","unstructured":"[Pri06] M. Primbs. Stabile biorthogonale Spline-Waveletbasen auf dem Intervall. Ph.D. thesis, Universit\u00e4t Duisburg, 2006."},{"key":"39","unstructured":"[Raa07] T. Raasch. Adaptive Wavelet and Frame Schemes for Elliptic and Parabolic Equations. Ph.D. thesis, Philipps-Universit\u00e4t Marburg, 2007."},{"key":"40","unstructured":"[Rei08] N. Reich. Wavelet Compression of Anisotropic Integrodifferential Operators on Sparse Grids, Ph.D. Dissertation, ETH Z\u00fcrich, 2008."},{"issue":"261","key":"41","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1090\/S0025-5718-07-02019-4","article-title":"Adaptive wavelet algorithms for elliptic PDE\u2019s on product domains","volume":"77","author":"Schwab, Christoph","year":"2008","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3","key":"42","doi-asserted-by":"publisher","first-page":"1074","DOI":"10.1137\/S0036142902407988","article-title":"Adaptive solution of operator equations using wavelet frames","volume":"41","author":"Stevenson, Rob","year":"2003","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"43","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","volume-title":"Galerkin finite element methods for parabolic problems","volume":"25","author":"Thom\u00e9e, Vidar","year":"2006","ISBN":"https:\/\/id.crossref.org\/isbn\/9783540331216","edition":"2"},{"key":"44","unstructured":"[Ver96] R. Verf\u00fcrth. A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, Chichester, 1996."},{"key":"45","doi-asserted-by":"crossref","unstructured":"[vPS04] T. von Petersdorff and Ch. Schwab. Numerical solution of parabolic equations in high dimensions. M2AN Math. Model. Numer. Anal., 38(1):93\u2013127, 2004.","DOI":"10.1051\/m2an:2004005"},{"key":"46","doi-asserted-by":"crossref","unstructured":"[Wlo82] J. Wloka. Partielle Differentialgleichungen. B.G. Teubner, Stuttgart, 1982. Sobolevr\u00e4ume und Randwertaufgaben.","DOI":"10.1007\/978-3-322-96662-9"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2009-78-267\/S0025-5718-08-02205-9\/S0025-5718-08-02205-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2009-78-267\/S0025-5718-08-02205-9\/S0025-5718-08-02205-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:07:34Z","timestamp":1776787654000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2009-78-267\/S0025-5718-08-02205-9\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,11,25]]},"references-count":46,"journal-issue":{"issue":"267","published-print":{"date-parts":[[2009,7]]}},"alternative-id":["S0025-5718-08-02205-9"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-08-02205-9","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2008,11,25]]}}}