{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,19]],"date-time":"2025-11-19T06:50:44Z","timestamp":1763535044936},"reference-count":31,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2010,4,1]],"date-time":"2010-04-01T00:00:00Z","timestamp":1270080000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Comput. Visual Sci."],"published-print":{"date-parts":[[2010,4]]},"DOI":"10.1007\/s00791-010-0134-4","type":"journal-article","created":{"date-parts":[[2010,4,22]],"date-time":"2010-04-22T12:01:53Z","timestamp":1271937713000},"page":"153-160","source":"Crossref","is-referenced-by-count":18,"title":["Multigrid and sparse-grid schemes for elliptic control problems with random coefficients"],"prefix":"10.1007","volume":"13","author":[{"given":"A.","family":"Borz\u00ec","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2010,4,23]]},"reference":[{"issue":"3","key":"134_CR1","doi-asserted-by":"crossref","first-page":"1005","DOI":"10.1137\/050645142","volume":"45","author":"I. Babu\u0161ka","year":"2007","unstructured":"Babu\u0161ka I., Nobile F., Tempone R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 45(3), 1005\u20131034 (2007)","journal-title":"SIAM J. Numer. Anal."},{"key":"134_CR2","volume-title":"Mathematical Problems from Combustion Theory","author":"J. Bebernes","year":"1988","unstructured":"Bebernes J., Eberly D.: Mathematical Problems from Combustion Theory. Springer, New York (1988)"},{"key":"134_CR3","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/j.cam.2005.12.023","volume":"200","author":"A. Borz\u00ec","year":"2007","unstructured":"Borz\u00ec A.: High-order discretization and multigrid solution of elliptic nonlinear constrained optimal control problems. J. Comput. Appl. Math. 200, 67\u201385 (2007)","journal-title":"J. Comput. Appl. Math."},{"key":"134_CR4","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1007\/s00791-006-0057-2","volume":"11","author":"A. Borz\u00ec","year":"2008","unstructured":"Borz\u00ec A.: Smoothers for control- and state-constrained optimal control problems. Comput. Vis. Sci. 11, 59\u201366 (2008)","journal-title":"Comput. Vis. Sci."},{"issue":"3\/4","key":"134_CR5","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1007\/s00791-004-0148-x","volume":"7","author":"A. Borz\u00ec","year":"2004","unstructured":"Borz\u00ec A., Borz\u00ec G.: An efficient algebraic multigrid method for solving optimality systems. Comput. Vis. Sci. 7(3\/4), 183\u2013188 (2004)","journal-title":"Comput. Vis. Sci."},{"issue":"1","key":"134_CR6","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1137\/S1064827599360194","volume":"22","author":"A. Borz\u00ec","year":"2000","unstructured":"Borz\u00ec A., Kunisch K.: The numerical solution of the steady state solid fuel ignition model and its optimal control. SIAM J. Sci. Comput. 22(1), 263\u2013284 (2000)","journal-title":"SIAM J. Sci. Comput."},{"key":"134_CR7","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1137\/060671590","volume":"51","author":"A. Borz\u00ec","year":"2009","unstructured":"Borz\u00ec A., Schulz V.: Multigrid methods for PDE optimization. SIAM Rev. 51, 361\u2013395 (2009)","journal-title":"SIAM Rev."},{"key":"134_CR8","doi-asserted-by":"crossref","first-page":"2172","DOI":"10.1137\/070711311","volume":"31","author":"A. Borz\u00ec","year":"2009","unstructured":"Borz\u00ec A., von Winckel G.: Multigrid methods and sparse-grid collocation techniques for parabolic optimal control problems with random coefficients. SIAM J. Sci. Comput. 31, 2172\u20132192 (2009)","journal-title":"SIAM J. Sci. Comput."},{"key":"134_CR9","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1090\/S0025-5718-1977-0431719-X","volume":"31","author":"A. Brandt","year":"1977","unstructured":"Brandt A.: Multi-level adaptive solutions to boundary-value problems. Math. Comput. 31, 333\u2013390 (1977)","journal-title":"Math. Comput."},{"key":"134_CR10","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1017\/S0962492904000182","volume":"13","author":"H.-J. Bungartz","year":"2004","unstructured":"Bungartz H.-J., Griebel M.: Sparse grids. Acta Numerica 13, 147\u2013269 (2004)","journal-title":"Acta Numerica"},{"key":"134_CR11","doi-asserted-by":"crossref","first-page":"675","DOI":"10.1093\/imanum\/drm006","volume":"27","author":"H. Elman","year":"2007","unstructured":"Elman H., Furnival D.: Solving the stochastic steady-state diffusion problem using multigrid. IMA J. Numer. Anal. 27, 675\u2013688 (2007)","journal-title":"IMA J. Numer. Anal."},{"key":"134_CR12","doi-asserted-by":"crossref","first-page":"652","DOI":"10.1016\/j.jcp.2006.12.014","volume":"225","author":"B. Ganapathysubramanian","year":"2007","unstructured":"Ganapathysubramanian B., Zabaras N.: Sparse grid collocation schemes for stochastic natural convection problems. J. Comput. Phys. 225, 652\u2013685 (2007)","journal-title":"J. Comput. Phys."},{"key":"134_CR13","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1023\/A:1018981505752","volume":"12","author":"M. Gasca","year":"2000","unstructured":"Gasca M., Sauer T.: Polynomial interpolation in several variables. Adv. Comput. Math. 12, 377\u2013410 (2000)","journal-title":"Adv. Comput. Math."},{"key":"134_CR14","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-3094-6","volume-title":"Stochastic Finite Elements: A Spectral Approach","author":"R.G. Ghanem","year":"1991","unstructured":"Ghanem R.G., Spanos P.D.: Stochastic Finite Elements: A Spectral Approach. Springer, New York (1991)"},{"key":"134_CR15","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-02427-0","volume-title":"Multi-grid Methods and Applications","author":"W. Hackbusch","year":"1985","unstructured":"Hackbusch W.: Multi-grid Methods and Applications. Springer, New York (1985)"},{"key":"134_CR16","doi-asserted-by":"crossref","first-page":"194","DOI":"10.1016\/j.probengmech.2006.11.004","volume":"22","author":"S.P. Huang","year":"2007","unstructured":"Huang S.P., Mahadevan S., Rebba R.: Collocation-based stochastic finite element analysis for random field problems. Probabilistic Eng. Mech. 22, 194\u2013205 (2007)","journal-title":"Probabilistic Eng. Mech."},{"key":"134_CR17","doi-asserted-by":"crossref","DOI":"10.1137\/1.9780898718614","volume-title":"Lagrange Multiplier Approach to Variational Problems and Applications","author":"K. Ito","year":"2008","unstructured":"Ito K., Kunisch K.: Lagrange Multiplier Approach to Variational Problems and Applications. SIAM, Philadelphia (2008)"},{"key":"134_CR18","doi-asserted-by":"crossref","first-page":"561","DOI":"10.1145\/1114268.1114275","volume":"31","author":"A. Klimke","year":"2005","unstructured":"Klimke A., Wohlmuth B.: Algorithm 847: Spinterp: piecewise multilinear hierarchical sparse grid interpolation in MATLAB. ACM Trans. Math. Softw. 31, 561\u2013579 (2005)","journal-title":"ACM Trans. Math. Softw."},{"key":"134_CR19","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-65024-6","volume-title":"Optimal Control of Systems Governed by Partial Differential Equations","author":"J.L. Lions","year":"1971","unstructured":"Lions J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)"},{"key":"134_CR20","volume-title":"Control of Distributed Singular Systems","author":"J.-L. Lions","year":"1985","unstructured":"Lions J.-L.: Control of Distributed Singular Systems. Gauthier-Villars, Paris (1985)"},{"key":"134_CR21","volume-title":"Probability Theory. Vols. I & II, IV edn","author":"M. Loeve","year":"1978","unstructured":"Loeve M.: Probability Theory. Vols. I & II, IV edn. Springer, New York (1978)"},{"key":"134_CR22","doi-asserted-by":"crossref","first-page":"560","DOI":"10.1016\/j.jcp.2006.10.010","volume":"224","author":"Y.M. Marzouk","year":"2007","unstructured":"Marzouk Y.M., Najm H.N., Rahn L.A.: Stochastic spectral methods for efficient Bayesian solution of inverse problems. J. Comput. Phys. 224, 560\u2013586 (2007)","journal-title":"J. Comput. Phys."},{"key":"134_CR23","doi-asserted-by":"crossref","first-page":"2309","DOI":"10.1137\/060663660","volume":"46","author":"F. Nobile","year":"2008","unstructured":"Nobile F., Tempone R., Webster C.: A sparse grid stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 46, 2309\u20132345 (2008)","journal-title":"SIAM J. Numer. Anal."},{"key":"134_CR24","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1115\/1.1446068","volume":"124","author":"M.M. Putko","year":"2002","unstructured":"Putko M.M., Newman P.A., Taylor A.C. III, Green L.L.: Approach for uncertainty propagation and robust design in CFD using sensitivity derivatives. J. Fluids Eng. 124, 60\u201369 (2002)","journal-title":"J. Fluids Eng."},{"key":"134_CR25","doi-asserted-by":"crossref","first-page":"646","DOI":"10.2514\/1.37955","volume":"47","author":"V. Schulz","year":"2009","unstructured":"Schulz V., Schillings C.: On the nature and treatment of uncertainties in aerodynamic design. AIAA J. 47, 646\u2013654 (2009)","journal-title":"AIAA J."},{"key":"134_CR26","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1007\/s00607-003-0024-4","volume":"71","author":"Ch Schwab","year":"2003","unstructured":"Schwab Ch, Todor R.A.: Sparse finite elements for stochastic elliptic problems higher order moments. Computing 71, 43\u201363 (2003)","journal-title":"Computing"},{"key":"134_CR27","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1016\/j.jcp.2006.12.011","volume":"224","author":"B. Seynaeve","year":"2007","unstructured":"Seynaeve B., Rosseel E., Nicola\u00ef B., Vandewalle S.: Fourier mode analysis of multigrid methods for partial differential equations with random coefficients. J. Comput. Phys. 224, 132\u2013149 (2007)","journal-title":"J. Comput. Phys."},{"key":"134_CR28","volume-title":"Multigrid","author":"U. Trottenberg","year":"2001","unstructured":"Trottenberg U., Oosterlee C., Sch\u00fcller A.: Multigrid. Academic Press, London (2001)"},{"issue":"3","key":"134_CR29","doi-asserted-by":"crossref","first-page":"1118","DOI":"10.1137\/040615201","volume":"27","author":"D. Xiu","year":"2005","unstructured":"Xiu D., Hesthaven J.S.: High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27(3), 1118\u20131139 (2005)","journal-title":"SIAM J. Sci. Comput."},{"key":"134_CR30","doi-asserted-by":"crossref","first-page":"619","DOI":"10.1137\/S1064827501387826","volume":"24","author":"D. Xiu","year":"2002","unstructured":"Xiu D., Karniadakis G.E.: The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24, 619\u2013644 (2002)","journal-title":"SIAM J. Sci. Comput."},{"key":"134_CR31","doi-asserted-by":"crossref","first-page":"4697","DOI":"10.1016\/j.jcp.2008.01.019","volume":"227","author":"N. Zabaras","year":"2008","unstructured":"Zabaras N., Ganapathysubramanian B.: A scalable framework for the solution of stochastic inverse problems using a sparse grid collocation approach. J. Comput. Phys. 227, 4697\u20134735 (2008)","journal-title":"J. Comput. Phys."}],"container-title":["Computing and Visualization in Science"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00791-010-0134-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00791-010-0134-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00791-010-0134-4","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T06:46:09Z","timestamp":1559112369000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00791-010-0134-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,4]]},"references-count":31,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2010,4]]}},"alternative-id":["134"],"URL":"https:\/\/doi.org\/10.1007\/s00791-010-0134-4","relation":{},"ISSN":["1432-9360","1433-0369"],"issn-type":[{"value":"1432-9360","type":"print"},{"value":"1433-0369","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,4]]}}}