×

zbMATH — the first resource for mathematics

On the reduced basis method. (English) Zbl 0832.65047
The reduced basis method is being used successfully to approximate large systems of nonlinear equations arising in several types of applications by much smaller systems usually of the order of two to ten equations, (see e.g. the recent survey by A. K. Noor [Appl. Mech. Rev. 47, No. 5, 126-146 (1994)]). This paper analyzes the effect of various choices in the construction of the reduced basis on the constants appearing in the error bounds.
Error estimates, especially near turning points of parametrized problems, are obtained from a projection formulation of the form considered by T. A. Porsching [Math. Comput. 45, 487-496 (1985; Zbl 0586.65040)]. In particular, norm-bounds of an implicitly defined projection are discussed. Moreover, it is shown how Sloan corrections [I. Sloan, Math. Comput. 30, 758-764 (1976; Zbl 0343.45010)] can be used, in certain cases, to improve the approximations.

MSC:
65H10 Numerical computation of solutions to systems of equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] BREZZI, Finite dimensional approximation of non-linear problems. Part 11: Limit points, Numer. Math. 37 pp 1– (1981) · Zbl 0525.65036 · doi:10.1007/BF01396184
[2] FINK, On the error behavior of the reduced basis technique for nonlinear finite element approximations, ZAMM 63 pp 21– (1983) · Zbl 0533.73071 · doi:10.1002/zamm.19830630105
[3] GREWANK, Characterization and computation of generalized turning points, SIAM J. Numer. Anal. 21 pp 176– (1984) · Zbl 0536.65031 · doi:10.1137/0721012
[4] GRIEWANK, The approximation of generalized turning points by projection methods with superconvergence to the critical parameter, Numer. Math. 48 pp 591– (1986) · Zbl 0596.65040 · doi:10.1007/BF01389452
[5] GRIEWANK, The calculation of Hopf points by a direct method, IMA J. Numer. Anal. 3 pp 295– (1983) · Zbl 0521.65070 · doi:10.1093/imanum/3.3.295
[6] HEINEMANN, Multiplicity, stability and oscillatory dynamics of the tubular reactor, Chemical Eng. Sci. 36 pp 1411– (1981) · doi:10.1016/0009-2509(81)80175-3
[7] JARAIJSCH, Numerical methods for bifurcation problems. ISNM 70 pp 296– (1984) · doi:10.1007/978-3-0348-6256-1_20
[8] JARAUSCH, Solving large nonlinear systems by an adaptive condensation process, Numer. Math. 50 pp 633– (1987) · Zbl 0647.65036 · doi:10.1007/BF01398377
[9] KELLER, Numerical methods in bifurcation problems (1987)
[10] MACKENS , W. Condensation of large sets of nonlinear equations using the reduced basis method 1988 · Zbl 0702.65056
[11] MACKENS , W. Numerical differentiation of implicitly defined spaces curves · Zbl 0666.65016
[12] NOOR, Recent advances in reduction methods for nonlinearproblems, Computersand Structures 13 pp 31– (1981) · Zbl 0455.73080 · doi:10.1016/0045-7949(81)90106-1
[13] NOOR, Global-local methodologies and their application to nonlinear analysis, Finite Elem. in Anal. and Design 2 pp 333– (1986) · doi:10.1016/0168-874X(86)90020-X
[14] NOOR, On making large nonlinear problems small, Computat. Meth. in Appl. Mech. and Eng. 34 pp 955– (1982) · Zbl 0478.65031 · doi:10.1016/0045-7825(82)90096-2
[15] NOOR, Reduced basis technique for nonlinear analysis ofstructures, AIAA J. 18 pp 455– (1981) · doi:10.2514/3.50778
[16] NOOR, Recent advanced in reduction methods for instability analysis of structures, Computers and Structures 16 pp 67– (1983) · Zbl 0498.73094 · doi:10.1016/0045-7949(83)90148-7
[17] NOOR, Multiple-parameter reduced basis technique for bifurcation and post-buckling analysis of composite plates, Internat. J. Numer. Meth. in Eng. 19 pp 1783– (1983) · Zbl 0557.73070 · doi:10.1002/nme.1620191206
[18] NOOR, Reduction methods for nonlinear steady-state thermal analysis, Internat. J . Numer. Meth. in Eng. 20 pp 1323– (1984) · Zbl 0557.65076 · doi:10.1002/nme.1620200711
[19] NOOK, Mixed models and reduction techniques for large-rotation nonlinear problems, Computat. Meth. in Appl. Mech. and Eng. 44 pp 67– (1984) · Zbl 0517.73073 · doi:10.1016/0045-7825(84)90120-8
[20] NOOR, Exploiting symmetries in the modeling and analysis of tires, Computat. Meth in Appl. Mech. and Eng. 63 pp 37– (1987) · Zbl 0634.73077 · doi:10.1016/0045-7825(87)90123-X
[21] PORSCHING, Estimation of the error in the reduced basis method solution of nonlinear equations, Math. of Comput. 45 pp 487– (1985) · doi:10.1090/S0025-5718-1985-0804937-0
[22] RHEINBOLDT, Numerical analysis of parameterized nonlinear equations. Univ. of Arkansas Lecture Notes in the Math. Sciences 7 (1986)
[23] RHEINBOLDT , W. C. On the theory and error estimation ofthe reduced basis method for multi-parameter problems · Zbl 0802.65068
[24] SLOAN, Improvement by iteration for compact operation equations, Math. Comput. 30 pp 758– (1976) · Zbl 0343.45010 · doi:10.1090/S0025-5718-1976-0474802-4
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.