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Computationally, we approximate the finite element solution of the optimality system and estimate its error through the discretizations with respect to both spatial and random parameter spaces.<\/jats:p>","DOI":"10.1137\/100801731","type":"journal-article","created":{"date-parts":[[2011,7,28]],"date-time":"2011-07-28T18:06:26Z","timestamp":1311876386000},"page":"1532-1552","source":"Crossref","is-referenced-by-count":46,"title":["Error Estimates of Stochastic Optimal Neumann Boundary Control Problems"],"prefix":"10.1137","volume":"49","author":[{"given":"Max D.","family":"Gunzburger","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hyung-Chun","family":"Lee","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jangwoon","family":"Lee","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2011,7,28]]},"reference":[{"key":"R1","unstructured":"R. Adams,\n                      Sobolev Spaces\n                      , Academic, New York, 1975."},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1016\/S0045-7825(02)00354-7"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1142\/S021820250300257X"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/050645142"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142902418680"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2004.02.026"},{"key":"R7","doi-asserted-by":"crossref","unstructured":"S. Bonaccorsi, F. Confortola, and E. Mastrogiacomo,\n                      Optimal control of stochastic differential equations with dynamical boundary conditions\n                      , J. Math. Anal. Appl., 344 (2008), pp 667\u2013681.","DOI":"10.1016\/j.jmaa.2008.03.013"},{"key":"R8","doi-asserted-by":"crossref","unstructured":"S. C. Brenner and L. R. Scott,\n                      The Mathematical Theory of Finite Element Methods\n                      , 2nd ed., Springer, New York, 2002.","DOI":"10.1007\/978-1-4757-3658-8"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1007\/BF01395985"},{"key":"R10","unstructured":"R. Courant and D. Hilbert,\n                      Methods of Mathematical Physics\n                      , Interscience, New York, 1953."},{"key":"R11","unstructured":"M. Crouzeix and J. Rappaz,\n                      On Numerical Approximation in Bifurcation Theory\n                      , Masson, Paris, 1990."},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1007\/s00245-006-0866-1"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1016\/S0045-7825(01)00237-7"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1051\/cocv:2007001"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012996311307"},{"key":"R16","unstructured":"L. C. Evans,\n                      Partial Differential Equations\n                      , American Mathematical Society, Providence, RI, 1998."},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2004.04.008"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1137\/080717924"},{"key":"R19","doi-asserted-by":"crossref","unstructured":"R. G. Ghanem and P. D. Spanos,\n                      Stochastic Finite Elements: A Spectral Approach\n                      , Springer-Verlag, Berlin, 1991.","DOI":"10.1007\/978-1-4612-3094-6"},{"key":"R20","doi-asserted-by":"crossref","unstructured":"V. Girault and P. Raviart,\n                      Finite Element Methods for Navier-Stokes Equations\n                      , Springer, Berlin, 1986.","DOI":"10.1007\/978-3-642-61623-5"},{"key":"R21","unstructured":"P. Grisvard,\n                      Elliptic Problems in Nonsmooth Domains\n                      , Longman Higher Education, 1986."},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1991-1079020-5"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/1991250607111"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012994262361"},{"key":"R25","doi-asserted-by":"crossref","unstructured":"H. Holden, B. \u00d8ksendal, J. Ub\u00f8e, and T. Zhang,\n                      Stochastic Partial Differential Equations; A Modeling, White Noise Functional Approach\n                      , Birkh\u00e4user, Boston, Cambridge, MA, 1996.","DOI":"10.1007\/978-1-4684-9215-6"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012996304870"},{"key":"R27","first-page":"443","volume":"8","author":"Hou L. S.","year":"2011","journal-title":"Int. J. Numer. Anal. Model."},{"key":"R28","unstructured":"L. S. Hou, J. Lee, and H. Manouzi,\n                      Finite element approximations of stochastic optimal control problems constrained by stochastic elliptic PDEs\n                      , J. Math. Anal. Appl., to appear."},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.1137\/0503013"},{"key":"R30","unstructured":"M. Loeve,\n                      Probability Theory Vols. $1 + 2$\n                      , Springer, New York, 1978."},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1080\/00207160701210133"},{"key":"R32","doi-asserted-by":"crossref","unstructured":"H. Manouzi and L. S. Hou,\n                      An optimal control problem for stochastic linear PDE's driven by a Gaussian white noise\n                      , Numerical Mathematics and Advanced Applications, Springer, Berlin, Heidelberg, 2008, pp. 629\u2013636.","DOI":"10.1007\/978-3-540-69777-0_75"},{"key":"R33","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-003-0455-z"},{"key":"R34","unstructured":"V. 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