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Certified error bounds for uncertain elliptic equations. (English) Zbl 1145.65088

In many applications, partial differential equations depend on parameters which are only approximately known. Using tools from functional analysis and global optimization, methods are presented for obtaining certificates for rigorous and realistic error bounds on the solution of linear elliptic partial differential equations in arbitrary domains, either in the energy norm, or of key functionals of the solution. Uncertainty in the parameters can be taken into account, either in a worst case setting, or given limited probalistic information in terms of clouds.

MSC:

65N15 Error bounds for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations

Software:

INTOPT_90; PENNON; AMPL; NEOS
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ainsworth, M.; Oden, J. T., A Posteriori Error Estimation in Finite Element Analysis (2000), Wiley: Wiley New York · Zbl 1008.65076
[2] Babuška, I.; Nobile, F.; Tempone, R., Worst case scenario analysis for elliptic problems with uncertainty, Numer. Math., 101, 185-219 (2005) · Zbl 1082.65115
[3] I. Babuška, T. Strouboulis, The Finite Element Method and Its Reliability, Oxford, 2001.; I. Babuška, T. Strouboulis, The Finite Element Method and Its Reliability, Oxford, 2001. · Zbl 0995.65501
[4] Becker, R.; Rannacher, R., An optimal control approach to a posteriori error estimation in finite element methods, (Iserles, A., Acta Numerica 2001 (2001), Cambridge University Press: Cambridge University Press Cambridge), 1-102 · Zbl 1105.65349
[5] Behnke, H.; Goerisch, F., Inclusions for eigenvalues of selfadjoint problems, (Herzberger, J., Topics in Validated Computations (1994), North-Holland: North-Holland Amsterdam), 277-322 · Zbl 0838.65060
[6] Bertsimas, D.; Caramanis, C., Bounds on linear PDEs via semidefinite optimization, Math. Programming Ser. A, 108, 135-158 (2006) · Zbl 1099.90063
[7] COCONUT Environment, An open source solver platform for global optimization problems, Web site: \( \langle;\) http://www.mat.univie.ac.at/coconut-environment/\( \rangle;\).; COCONUT Environment, An open source solver platform for global optimization problems, Web site: \( \langle;\) http://www.mat.univie.ac.at/coconut-environment/\( \rangle;\).
[8] Eiermann, M., Automatic, guaranteed integration of analytic functions, BIT, 29, 270-282 (1989) · Zbl 0688.65017
[9] Fourer, R.; Gay, D. M.; Kernighan, B. W., AMPL: A Modeling Language for Mathematical Programming (1993), Duxbury Press, Brooks/Cole Publishing Company
[10] Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (2001), Springer: Springer Berlin · Zbl 0691.35001
[11] Griewank, A.; Corliss, G. F., Automatic Differentiation of Algorithms (1991), SIAM: SIAM Philadelphia, PA · Zbl 0747.00030
[12] A. Hannukainen, S. Korotov, Computational technologies for reliable control of global and local errors for linear elliptic type boundary value problems, Research Report A495, Institute of Mathematics, Helsinki University of Technology, 2006.; A. Hannukainen, S. Korotov, Computational technologies for reliable control of global and local errors for linear elliptic type boundary value problems, Research Report A495, Institute of Mathematics, Helsinki University of Technology, 2006. · Zbl 1161.65081
[13] Kearfott, R. B., Rigorous Global Search: Continuous Problems (1996), Kluwer: Kluwer Dordrecht · Zbl 0876.90082
[14] Knyazev, A. V.; Osborn, J., New a priori FEM error estimates for eigenvalues, SIAM J. Numer. Anal., 43, 2647-2667 (2006) · Zbl 1111.65100
[15] Kočvara, M.; Stingl, M., PENNON: a code for convex nonlinear and semidefinite programming, Optim. Meth. Software, 18, 317-333 (2003) · Zbl 1037.90003
[16] Lebbah, Y.; Michel, C.; Rueher, M., A rigorous global filtering algorithm for quadratic constraints, Constraints J., 10, 47-65 (2005) · Zbl 1066.90090
[17] Möller, B.; Beer, M., Fuzzy Randomness (2004), Springer: Springer Berlin · Zbl 1080.74003
[18] Moore, R. E., Methods and Applications of Interval Analysis (1979), SIAM: SIAM Philadelphia, PA · Zbl 0417.65022
[19] Muhanna, R. L.; Mullen, R. L.; Zhang, H., Interval finite element as a basis for generalized models of uncertainty in engineering mechanics, (Muhanna, R. L.; Mullen, R. L., Proceedings of the NSF Workshop Reliable Engineering Computing. Proceedings of the NSF Workshop Reliable Engineering Computing, Savannah, GA (September 15-17, 2004)), \(353-370, \langle\) http://www.gtsav.gatech.edu/rec/recworkshop/presentations/Hao.ppt \(\rangle \)
[20] Nakao, M. T., A numerical approach to the proof of existence of solutions for elliptic problems I, Japan J. Appl. Math., 5, 313-332 (1988) · Zbl 0694.35051
[21] Nakao, M. T., A numerical approach to the proof of existence of solutions for elliptic problems II, Japan J. Appl. Math., 7, 477-488 (1990) · Zbl 0717.35019
[22] Nakao, M. T.; Yamamoto, N., A guaranteed bound of the optimal constant in the error estimates for linear triangular element, Comput. Suppl., 15, 165-173 (2001) · Zbl 1013.65119
[23] Nakao, M. T.; Yamamoto, N.; Kimura, S., On the best constant in the error bound for the \(H_0^1\)-projection into piecewise polynomial spaces, J. Approx. Theory, 93, 491-500 (1998) · Zbl 0907.65100
[24] NEOS server for optimization. Web site: \( \langle;\) http://www-neos.mcs.anl.gov/\( \rangle;\).; NEOS server for optimization. Web site: \( \langle;\) http://www-neos.mcs.anl.gov/\( \rangle;\).
[25] Neumaier, A., Interval Methods for Systems of Equations (1990), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0706.15009
[26] Neumaier, A., Global, rigorous and realistic bounds for the solution of dissipative differential equations. Part I: theory, Computing, 52, 315-336 (1994) · Zbl 0809.65077
[27] Neumaier, A., Generalized Lyapunov-Schmidt reduction for parametrized equations at near singular points, Linear Algebra Appl., 324, 119-131 (2001) · Zbl 0974.65058
[28] Neumaier, A., Introduction to Numerical Analysis (2001), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0980.65001
[29] Neumaier, A., Clouds, fuzzy sets and probability intervals, Reliable Comput., 10, 249-272 (2004) · Zbl 1055.65062
[30] Neumaier, A., Complete search in continuous global optimization and constraint satisfaction, (Iserles, A., Acta Numerica 2004 (2004), Cambridge University Press: Cambridge University Press Cambridge), 271-369 · Zbl 1113.90124
[31] Neumaier, A.; Pownuk, A., Linear systems with large uncertainties, with applications to truss structures, Reliable Comput., 13, 149-172 (2007) · Zbl 1117.65063
[32] Pares, N.; Bonet, J.; Huerta, A.; Peraire, J., The computation of bounds for linear-functional outputs of weak solutions to the two-dimensional elasticity equations, Comput. Methods Appl. Mech. Eng., 195, 406-429 (2006) · Zbl 1193.74041
[33] Petras, K., Gaussian versus optimal integration of analytic functions, Constr. Approx., 14, 231-245 (1998) · Zbl 0893.41019
[34] Petras, K., Self-validating integration and approximation of piecewise analytic functions, J. Comput. Appl. Math., 145, 345-359 (2002) · Zbl 1002.65030
[35] Plum, M., Explicit \(H_2\)-estimates and pointwise bounds for solutions of second-order elliptic boundary value problems, J. Math. Anal. Appl., 165, 36-61 (1992) · Zbl 0780.35028
[36] Plum, M., Existence and enclosure results for continua of solutions of parameter-dependent nonlinear boundary value problems, J. Comput. Appl. Math., 60, 187-200 (1995) · Zbl 0834.65119
[37] Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P., Numerical Recipes (1986), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0587.65005
[38] Rackwitz, R., Reliability analysis—a review and some perspectives, Structural Safety, 23, 365-395 (2001)
[39] Repin, S., A posteriori error estimation for nonlinear variational problems by duality theory, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 243, 201-214 (1997)
[40] Repin, S.; Sauter, S.; Smolianski, A., A posteriori error estimation for the Dirichlet problem with account of the error in the approximation of boundary conditions, Computing, 70, 205-233 (2003) · Zbl 1128.35319
[41] Sauer-Budge, A. M.; Bonet, J.; Huerta, A.; Peraire, J., Computing bounds for linear functionals of exact weak solutions to Poisson’s equation, SIAM J. Numer. Anal., 42, 1610-1630 (2004) · Zbl 1084.65107
[42] Sauer-Budge, A. M.; Peraire, J., Computing bounds for linear functionals of exact weak solutions to the advection-diffusion-reaction equation, SIAM J. Sci. Comput., 26, 636-652 (2004) · Zbl 1082.65113
[43] H. Schichl, Mathematical Modeling and Global Optimization, Cambridge University Press, Cambridge, to appear, \( \langle;\) http://www.mat.univie.ac.at/\( \sim;\) herman/papers/habil.pdf \(\rangle;\).; H. Schichl, Mathematical Modeling and Global Optimization, Cambridge University Press, Cambridge, to appear, \( \langle;\) http://www.mat.univie.ac.at/\( \sim;\) herman/papers/habil.pdf \(\rangle;\).
[44] Schueller, G. I., Computational stochastic mechanics—recent advances, Comput. & Structures, 79, 2225-2234 (2001)
[45] Schueller, G. I., Developments in stochastic structural mechanics, Arch. Appl. Mech. (Ingenieur Archiv), 75, 755-773 (2006) · Zbl 1168.74398
[46] Yamamoto, N.; Nakao, M. T., Numerical verifications of solutions for elliptic equations in nonconvex polygonal domains, Numer. Math., 65, 503-521 (1993) · Zbl 0797.65081
[47] Yamamoto, N.; Nakao, M. T.; Watanabe, Y., Validated computation for a linear elliptic problem with a parameter, (Kawarada; etal., Advances in Numerical Mathematics, GAKUTO International Series, Mathematical Sciences and Applications, vol. 12 (1999)), 155-162 · Zbl 0953.65080
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