| Title:
|
Uncertain input data problems and the worst scenario method (English) |
| Author:
|
Hlaváček, Ivan |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
52 |
| Issue:
|
3 |
| Year:
|
2007 |
| Pages:
|
187-196 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
An introduction to the worst scenario method is given. We start with an example and a general abstract scheme. An analysis of the method both on the continuous and approximate levels is discussed. We show a possible incorporation of the method into the fuzzy set theory. Finally, we present a survey of applications published during the last decade. (English) |
| Keyword:
|
uncertain input data |
| Keyword:
|
the worst-case approach |
| Keyword:
|
fuzzy sets |
| MSC:
|
49J20 |
| MSC:
|
74C10 |
| MSC:
|
74K20 |
| MSC:
|
93C20 |
| MSC:
|
93C25 |
| MSC:
|
93C41 |
| idZBL:
|
Zbl 1164.93354 |
| idMR:
|
MR2316152 |
| DOI:
|
10.1007/s10492-007-0010-9 |
| . |
| Date available:
|
2009-09-22T18:29:19Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134671 |
| . |
| Reference:
|
[1] I. Babuška, F. Nobile, R. Tempone: Worst case scenario analysis for elliptic problems with uncertainty.Numer. Math. 101 (2005), 185–219. MR 2195342, 10.1007/s00211-005-0601-x |
| Reference:
|
[2] Y. Ben-Haim: Information-Gap Decision Theory: Decisions Under Severe Uncertainty.Academic Press, San Diego, 2001. Zbl 0985.91013, MR 1856675 |
| Reference:
|
[3] Y. Ben-Haim, I. Elishakoff: Convex Models of Uncertainties in Applied Mechanics.Elsevier, Amsterdam, 1990. |
| Reference:
|
[4] A. Bernardini: What are the random and fuzzy sets and how to use them for uncertainty modeling in engineering systems?.In: CISM Courses and Lectures No. 388, I. Elishakoff (ed.), Springer-Verlag, Wien, 1999. |
| Reference:
|
[5] B. V. Bulgakov: Fehleranhäufung bei Kreiselapparaten.Ingenieur-Archiv 11 (1940), 461–469. 10.1007/BF02088988 |
| Reference:
|
[6] J. Chleboun: Reliable solution for a 1D quasilinear elliptic equation with uncertain coefficients.J. Math. Anal. Appl. 234 (1999), 514–528. Zbl 0944.35027, MR 1689404, 10.1006/jmaa.1998.6364 |
| Reference:
|
[7] J. Chleboun: On a reliable solution of a quasilinear elliptic equation with uncertain coefficients: Sensitivity analysis and numerical examples.Nonlinear Anal., Theory Methods Appl. 44 (2001), 375–388. Zbl 1002.35041, MR 1817101, 10.1016/S0362-546X(99)00274-6 |
| Reference:
|
[8] J. Chleboun: On fuzzy input data and the worst scenario method.Appl. Math. 48 (2003), 487–496. Zbl 1099.90081, MR 2025958, 10.1023/B:APOM.0000024488.86492.fb |
| Reference:
|
[9] J. Chleboun, P. Kocna: Isotope selective nondispersive infrared spectrometry can compete with isotope ratio mass spectrometry in cumulative ${}^{13} \text{CO}_2$ breath tests: Assessment of accuracy.Klin. Biochem. Metab. 13 (2005), 92–97. |
| Reference:
|
[10] E. J. Haug, K. K. Choi, V. Komkov: Design Sensitivity Analysis of Structural Systems.Academic Press, Orlando, 1986. MR 0860040 |
| Reference:
|
[11] I. Hlaváček: Reliable solution of problems in the deformation theory of plasticity with respect to uncertain material function.Appl. Math. 41 (1996), 447–466. MR 1415251 |
| Reference:
|
[12] I. Hlaváček: Reliable solutions of elliptic boundary value problems with respect to uncertain data. Proceedings of the WCNA-96.Nonlin. Anal., Theory Methods Appl. 30 (1997), 3879–3890. MR 1602891, 10.1016/S0362-546X(96)00236-2 |
| Reference:
|
[13] I. Hlaváček: Reliable solution of a quasilinear nonponential elliptic problem of a nonmonotone type with respect to the uncertainty in coefficients.J. Math. Anal. Appl. 212 (1997), 452–466. MR 1464890, 10.1006/jmaa.1997.5518 |
| Reference:
|
[14] I. Hlaváček: Reliable solution of linear parabolic problems with respect to uncertain coefficients.ZAMM, Z. Angew. Math. Mech. 79 (1999), 291–301. MR 1695286, 10.1002/(SICI)1521-4001(199905)79:5<291::AID-ZAMM291>3.0.CO;2-N |
| Reference:
|
[15] I. Hlaváček: Reliable solution of an elasto-plastic Reissner-Mindlin beam for the Hencky’s model with uncertain yield function.Appl. Math. 43 (1998), 223–237. MR 1620616, 10.1023/A:1023228608356 |
| Reference:
|
[16] I. Hlaváček: Reliable solution of a Signorini contact problem with friction, considering uncertain data.Numer. Linear Algebra Appl. 6 (1999), 411–434. MR 1731015, 10.1002/(SICI)1099-1506(199909)6:6<411::AID-NLA178>3.0.CO;2-W |
| Reference:
|
[17] I. Hlaváček: Reliable solution of a torsion problem in Hencky plasticity with uncertain yield function.Math. Models Methods Appl. Sci. 11 (2001), 855–865. Zbl 1037.74028, MR 1842230, 10.1142/S0218202501001148 |
| Reference:
|
[18] I. Hlaváček: Reliable solution of a perfect plastic problem with uncertain stress-strain law and yield function.SIAM J. Numer. Anal. 39 (2001), 1539–1555. Zbl 1014.74015, MR 1885706 |
| Reference:
|
[19] I. Hlaváček: Worst scenario approach for elastoplasticity with hardening and uncertain input data.ZAMM, Z. Angew. Math. Mech. 82 (2002), 671–684. Zbl 1099.74505, MR 1903014, 10.1002/1521-4001(200210)82:10<671::AID-ZAMM671>3.0.CO;2-2 |
| Reference:
|
[20] I. Hlaváček: Reliable solution in strain space of elastoplastic problems with isotropic hardening and uncertain input data.Math. Models Methods Appl. Sci. 12 (2002), 1337–1357. MR 1927028, 10.1142/S0218202502002082 |
| Reference:
|
[21] I. Hlaváček: Post-buckling range of plates in axial compression with uncertain initial imperfections.Appl. Math. 47 (2002), 25–44. MR 1876490, 10.1023/A:1021702816894 |
| Reference:
|
[22] I. Hlaváček: Buckling of a Timoshenko beam on elastic foundation with uncertain input data.IMA J. Appl. Math. 68 (2003), 185–204. Zbl 1037.74018, MR 1968311, 10.1093/imamat/68.2.185 |
| Reference:
|
[23] I. Hlaváček: Plate bending problems with uncertain input data. I. Classical problems., Submitted. |
| Reference:
|
[24] I. Hlaváček, J. Chleboun: Reliable analysis of transverse vibrations of Timoshenko-Mindlin beams with respect to uncertain shear correction factor.Comput. Methods Appl. Mech. Eng. 190 (2000), 903–918. MR 1797723, 10.1016/S0045-7825(99)00452-1 |
| Reference:
|
[25] I. Hlaváček, J. Chleboun, and I. Babuška: Uncertain Input Data Problems and the Worst Scenario Method.Elsevier, Amsterdam, 2004. MR 2285091 |
| Reference:
|
[26] I. Hlaváček, J. Lovíšek: Control in obstacle-pseudoplate problems with friction on the boundary. Optimal design and problems with uncertain data.Appl. Math. (Warsaw) 28 (2001), 407–426. MR 1873903, 10.4064/am28-4-3 |
| Reference:
|
[27] I. Hlaváček, M. Křížek, and J. Malý: On Galerkin approximations of a quasilinear nonpotential elliptic problem of a nonmonotone type.J. Math. Anal. Appl. 184 (1994), 168–189. MR 1275952, 10.1006/jmaa.1994.1192 |
| Reference:
|
[28] I. Hlaváček, J. Lovíšek: Semi-coercive variational inequalities with uncertain input data.Applications to shallow shells. Math. Models Methods Appl. Sci. 15 (2005), 273–299. MR 2120000, 10.1142/S0218202505000364 |
| Reference:
|
[29] I. Hlaváček, J. Nedoma: Reliable solution of a unilateral contact problem with friction and uncertain input data in thermoelasticity.Math. Comput. Simul. 67 (2005), 559–580. MR 2111780, 10.1016/j.matcom.2004.08.001 |
| Reference:
|
[30] I. Hlaváček: Unilateral contact with Coulomb friction and uncertain input data.Numer. Funct. Anal. Optimization 24 (2003), 509–530. Zbl 1049.49007, MR 1995999, 10.1081/NFA-120023866 |
| Reference:
|
[31] I. Hlaváček, J. Plešek, and D. Gabriel: Validation and sensitivity study of an elastoplastic problem using the worst scenario method.Comput. Methods Appl. Mech. Eng. 195 (2006), 763–774. MR 2183622, 10.1016/j.cma.2005.02.010 |
| Reference:
|
[32] V. Krištof: An elliptic problem with a nonlinear Newton boundary condition and a double uncertainty.PhD. Thesis, Palacký University, Olomouc, 2004. (Czech) |
| Reference:
|
[33] V. G. Litvinov: Optimization in Elliptic Problems with Applications to Mechanics of Deformable Bodies and Fluid Mechanics.Birkhäuser-Verlag, Berlin, 2000. Zbl 0947.49001, MR 1774123 |
| Reference:
|
[34] J. Lovíšek: Reliable solution of parabolic obstacle problems with respect to uncertain data.Appl. Math. 48 (2003), 321–351. Zbl 1099.35054, MR 2008888, 10.1023/B:APOM.0000024480.06960.ea |
| Reference:
|
[35] L. Nechvátal: Worst scenario method in homogenization. Linear case.Appl. Math. 51 (2006), 263–294. Zbl 1164.35317, MR 2228666, 10.1007/s10492-006-0015-9 |
| Reference:
|
[36] T. Roubíček: Relaxation in Optimization Theory and Variational Calculus.Walter de Gruyter, Berlin, 1997. MR 1458067 |
| Reference:
|
[37] M. Tužilová: Reliable solutions of a thermoelastic beam model with uncertain coefficients.PhD. Thesis, Palacký University, Olomouc, 2003. (Czech) |
| Reference:
|
[38] H.-J. Zimmermann: Fuzzy Set Theory—and Its Applications.Kluwer Academic Publishers, Boston, 2001. MR 1882395 |
| . |
