{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,24]],"date-time":"2025-09-24T09:00:03Z","timestamp":1758704403641},"reference-count":25,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2020,3,23]],"date-time":"2020-03-23T00:00:00Z","timestamp":1584921600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,3,23]],"date-time":"2020-03-23T00:00:00Z","timestamp":1584921600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv Comput Math"],"published-print":{"date-parts":[[2020,4]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Quantifying the error that is induced by numerical approximation techniques is an important task in many fields of applied mathematics. Two characteristic properties of error bounds that are desirable are reliability and efficiency. In this article, we present an error estimation procedure for general nonlinear problems and, in particular, for parameter-dependent problems. With the presented auxiliary linear problem (ALP)-based error bounds and corresponding theoretical results, we can prove large improvements in the accuracy of the error predictions compared with existing error bounds. The application of the procedure in parametric model order reduction setting provides a particularly interesting setup, which is why we focus on the application in the reduced basis framework. Several numerical examples illustrate the performance and accuracy of the proposed method.<\/jats:p>","DOI":"10.1007\/s10444-020-09741-x","type":"journal-article","created":{"date-parts":[[2020,3,23]],"date-time":"2020-03-23T16:04:01Z","timestamp":1584979441000},"update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Rigorous and effective a-posteriori error bounds for nonlinear problems\u2014application to RB methods"],"prefix":"10.1007","volume":"46","author":[{"given":"Andreas","family":"Schmidt","sequence":"first","affiliation":[]},{"given":"Dominik","family":"Wittwar","sequence":"additional","affiliation":[]},{"given":"Bernard","family":"Haasdonk","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,3,23]]},"reference":[{"key":"9741_CR1","doi-asserted-by":"crossref","unstructured":"Ainsworth, M., Oden, J.T.: A Posteriori Error Estimation in Finite Element Analysis. Wiley (2000)","DOI":"10.1002\/9781118032824"},{"key":"9741_CR2","doi-asserted-by":"publisher","first-page":"430","DOI":"10.1016\/j.laa.2015.09.027","volume":"488","author":"P Benner","year":"2016","unstructured":"Benner, P., Bujanovi\u0107, Z.: On the solution of large-scale algebraic Riccati equations by using low-dimensional invariant subspaces. Linear Algebra Appl. 488, 430\u2013459 (2016). https:\/\/doi.org\/10.1016\/j.laa.2015.09.027","journal-title":"Linear Algebra Appl."},{"issue":"4","key":"9741_CR3","doi-asserted-by":"publisher","first-page":"483","DOI":"10.1137\/130932715","volume":"57","author":"P Benner","year":"2015","unstructured":"Benner, P., Gugercin, S., Willcox, K.: A survey of projection-based model reduction methods for parametric dynamical systems. SIAM Rev. 57(4), 483\u2013531 (2015). https:\/\/doi.org\/10.1137\/130932715","journal-title":"SIAM Rev."},{"key":"9741_CR4","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611974829","volume-title":"Model Reduction and Approximation","author":"P Benner","year":"2017","unstructured":"Benner, P., Ohlberger, M., Cohen, A., Willcox, K.: Model Reduction and Approximation. Society for Industrial and Applied Mathematics, Philadelphia (2017). http:\/\/epubs.siam.org\/doi\/abs\/10.1137\/1.9781611974829"},{"key":"9741_CR5","doi-asserted-by":"crossref","unstructured":"Benner, P., Saak, J.: A semi-discretized heat transfer model for optimal cooling of steel profiles. In: Dimension Reduction of Large-Scale Systems, pp 353\u2013356. Springer (2005)","DOI":"10.1007\/3-540-27909-1_19"},{"key":"9741_CR6","unstructured":"Braess, D.: Finite Elements. Cambridge University Press. https:\/\/books.google.de\/books?id=PizECgOWoGgC (2007)"},{"key":"9741_CR7","doi-asserted-by":"crossref","unstructured":"Caloz, G., Rappaz, J.: Handbook of Numerical Analysis, vol. 5, chap Numerical Analysis for Nonlinear and Bifurcation Problems, pp. 487\u2013637 (1997)","DOI":"10.1016\/S1570-8659(97)80004-X"},{"issue":"5","key":"9741_CR8","doi-asserted-by":"publisher","first-page":"2737","DOI":"10.1137\/090766498","volume":"32","author":"S Chaturantabut","year":"2010","unstructured":"Chaturantabut, S., Sorensen, D.: Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32(5), 2737\u20132764 (2010). https:\/\/doi.org\/10.1137\/090766498","journal-title":"SIAM J. Sci. Comput."},{"key":"9741_CR9","doi-asserted-by":"publisher","unstructured":"Dolej\u0161\u00ed, V., Feistauer, M.: Discontinuous Galerkin Method Springer International Publishing. https:\/\/doi.org\/10.1007\/978-3-319-19267-3 (2015)","DOI":"10.1007\/978-3-319-19267-3"},{"key":"9741_CR10","doi-asserted-by":"crossref","unstructured":"Haasdonk, B.: Reduced basis methods for parametrized PDEs \u2013 a tutorial introduction for stationary and instationary problems. In: Benner, P., Cohen, A., Ohlberger, M., Willcox, K. (eds.) Model Reduction and Approximation: Theory and Algorithms. http:\/\/www.simtech.uni-stuttgart.de\/publikationen\/prints.php?ID=938, pp 65\u2013136. SIAM, Philadelphia (2017)","DOI":"10.1137\/1.9781611974829.ch2"},{"key":"9741_CR11","doi-asserted-by":"publisher","unstructured":"Hain, S., Ohlberger, M., Radic, M., Urban, K.: A hierarchical a posteriori error estimator for the reduced basis method. Advances in Computational Mathematics. https:\/\/doi.org\/10.1007\/s10444-019-09675-z (2019)","DOI":"10.1007\/s10444-019-09675-z"},{"key":"9741_CR12","doi-asserted-by":"publisher","first-page":"1963","DOI":"10.1016\/j.cma.2010.02.011","volume":"199","author":"D Huynh","year":"2010","unstructured":"Huynh, D., Knezevic, D., Chen, Y., Hesthaven, J., Patera, A.: A natural-norm successive constraint method for inf-sup lower bounds. Comput. Methods Appl. Mech. Eng. 199, 1963\u20131975 (2010). https:\/\/doi.org\/10.1016\/j.cma.2010.02.011. http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0045782510000691","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"1","key":"9741_CR13","doi-asserted-by":"publisher","first-page":"50","DOI":"10.1137\/0328003","volume":"28","author":"C Kenney","year":"1990","unstructured":"Kenney, C., Hewer, G.: The sensitivity of the algebraic and differential Riccati equations. SIAM J. Control. Optim. 28(1), 50\u201369 (1990). https:\/\/doi.org\/10.1137\/0328003","journal-title":"SIAM J. Control. Optim."},{"key":"9741_CR14","volume-title":"Linear Optimal Control Systems, vol. 1","author":"H Kwakernaak","year":"1972","unstructured":"Kwakernaak, H., Sivan, R.: Linear Optimal Control Systems, vol. 1. Wiley-interscience, New York (1972)"},{"key":"9741_CR15","doi-asserted-by":"crossref","unstructured":"Lancaster, P., Rodman, L.: Algebraic Riccati Equations. Oxford University Press (1995)","DOI":"10.1093\/oso\/9780198537953.001.0001"},{"issue":"1534-0392-2009-","key":"9741_CR16","doi-asserted-by":"publisher","first-page":"383","DOI":"10.3934\/cpaa.2009.8.383","volume":"8","author":"Y Maday","year":"2009","unstructured":"Maday, Y., Nguyen, N.C., Patera, A.T., Pau, S.H.: A general multipurpose interpolation procedure: The magic points. Commun. Pure Appl. Anal. 8(1534-0392-2009-1-383), 383 (2009). https:\/\/doi.org\/10.3934\/cpaa.2009.8.383. http:\/\/aimsciences.org\/\/article\/id\/30a3894d-b0c8-4e29-8b8d-2611be32876f","journal-title":"Commun. Pure Appl. Anal."},{"key":"9741_CR17","volume-title":"A Posteriori Error Estimates and Adaptive Methods for Convection Dominated Transport Processes","author":"M Ohlberger","year":"2001","unstructured":"Ohlberger, M.: A Posteriori Error Estimates and Adaptive Methods for Convection Dominated Transport Processes. Albert-Ludwigs-Universit\u00e4t Freiburg, Ph.D. thesis (2001)"},{"key":"9741_CR18","unstructured":"Patera, A., Rozza, G.: Reduced Basis Approximation and a Posteriori Error Estimation for Parametrized Partial Differential Equations. To appear in (tentative) MIT Pappalardo Graduate Monographs in Mechanical Engineering. MIT (2007) http:\/\/augustine.mit.edu\/methodology\/methodology_book.htm"},{"key":"9741_CR19","unstructured":"Schmidt, A.: Feedback Control for Parametric PDEs Using Reduced Basis Surrogate Models. Ph.D. thesis University of Stuttgart (2018)"},{"issue":"1","key":"9741_CR20","doi-asserted-by":"publisher","first-page":"129","DOI":"10.1051\/cocv\/2017011","volume":"24","author":"A Schmidt","year":"2018","unstructured":"Schmidt, A., Haasdonk, B.: Reduced basis approximation of large scale parametric algebraic Riccati equations. ESAIM: Control Optim. Calculus Variations 24(1), 129\u2013151 (2018). https:\/\/doi.org\/10.1051\/cocv\/2017011","journal-title":"ESAIM: Control Optim. Calculus Variations"},{"key":"9741_CR21","unstructured":"Steck, S., Urban, K.: A reduced basis method for the Hamilton-Jacobi-Bellmann equation with application to the european union emission trading scheme. Preprint, University of Ulm (2015)"},{"issue":"1-3","key":"9741_CR22","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1016\/S0024-3795(02)00255-0","volume":"349","author":"T Stykel","year":"2002","unstructured":"Stykel, T.: Numerical solution and perturbation theory for generalized Lyapunov equations. Linear Algebra Appl. 349(1-3), 155\u2013185 (2002). https:\/\/doi.org\/10.1016\/S0024-3795(02)00255-0","journal-title":"Linear Algebra Appl."},{"key":"9741_CR23","doi-asserted-by":"publisher","first-page":"773","DOI":"10.1002\/fld.867","volume":"47","author":"K Veroy","year":"2005","unstructured":"Veroy, K., Patera, A.: Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations: Rigorous reduced-basis a posteriori error bounds. Int. J. Numer. Methods Fluids 47, 773\u2013788 (2005)","journal-title":"Int. J. Numer. Methods Fluids"},{"key":"9741_CR24","doi-asserted-by":"publisher","unstructured":"Veroy, K., Prud\u2019homme, C., Rovas, D., Patera, A.: A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations. In: 16th AIAA Computational Fluid Dynamics Conference. Paper 2003-3847. American Institute of Aeronautics and Astronautics (2003), https:\/\/doi.org\/10.2514\/6.2003-3847","DOI":"10.2514\/6.2003-3847"},{"key":"9741_CR25","unstructured":"Volkwein, S.: Proper Orthogonal Decomposition: Theory and Reduced-Order Modelling. Lecture notes, Universit\u00e4t Konstanz (2013) http:\/\/www.math.uni-konstanz.de\/numerik\/personen\/volkwein\/teaching\/POD-Book.pdf"}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-020-09741-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10444-020-09741-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-020-09741-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,2]],"date-time":"2024-08-02T19:28:05Z","timestamp":1722626885000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10444-020-09741-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3,23]]},"references-count":25,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2020,4]]}},"alternative-id":["9741"],"URL":"https:\/\/doi.org\/10.1007\/s10444-020-09741-x","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"type":"print","value":"1019-7168"},{"type":"electronic","value":"1572-9044"}],"subject":[],"published":{"date-parts":[[2020,3,23]]},"assertion":[{"value":"23 March 2020","order":1,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"32"}}