Estimation of the error in the reduced basis method solution of nonlinear equations

Author:
T. A. Porsching

Journal:
Math. Comp. **45** (1985), 487-496

MSC:
Primary 65H10; Secondary 65G99

MathSciNet review:
804937

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Abstract | References | Similar Articles | Additional Information

Abstract: The reduced basis method is a projection technique for approximating the solution curve of a finite system of nonlinear algebraic equations by the solution curve of a related system that is typically of much lower dimension. In this paper, the reduced basis error is shown to be dominated by an approximation error. This, in turn, leads to error estimates for projection onto specific subspaces; for example, subspaces related to Taylor, Lagrange and discrete least-squares approximation.

**[1]**B. O. Almroth, P. Stern & F. A. Brogan, "Automatic choice of global shape functions in structural analysis,"*AIAA J.*, v. 16, 1978, p. 525.**[2]**B. O. Almroth, P. Stehlin & F. A. Brogan,*Use of Global Functions for Improvement in Efficiency of Nonlinear Analysis*, AIAA National Conf. 1981, Paper No. 81-0575, p. 286.**[3]**James P. Fink and Werner C. Rheinboldt,*On the discretization error of parametrized nonlinear equations*, SIAM J. Numer. Anal.**20**(1983), no. 4, 732–746. MR**708454**, 10.1137/0720049**[4]**J. P. Fink and W. C. Rheinboldt,*On the error behavior of the reduced basis technique for nonlinear finite element approximations*, Z. Angew. Math. Mech.**63**(1983), no. 1, 21–28 (English, with German and Russian summaries). MR**701832**, 10.1002/zamm.19830630105**[5]**Peter Lancaster,*Theory of matrices*, Academic Press, New York-London, 1969. MR**0245579****[6]**A. K. Noor, C. M. Anderson & J. M. Peters,*Global-Local Approach for Non-Linear Shell Analysis*, Proc. Seventh ASCE Conf. Electronic Comp., St. Louis, MO, 1979, p. 634.**[7]**Ahmed K. Noor,*Recent advances in reduction methods for nonlinear problems*, Comput. & Structures**13**(1981), no. 1-3, 31–44. MR**616719**, 10.1016/0045-7949(81)90106-1**[8]**A. K. Noor & J. M. Peters, "Tracing post-limit-point path with reduced basis technique,"*Comput. Methods Appl. Mech. Engrg.*, v. 28, 1981, p. 217.**[9]**A. K. Noor & J. M. Peters, "Bifurcation and post-buckling analysis of laminated composite plates via reduced basis technique,"*Comput. Methods Appl. Mech. Engrg.*, v. 29, 1981, p. 271.**[10]**J. S. Peterson,*High Reynolds Number Solutions for Incompressible Viscous Flow Using the Reduced Basis Technique*, Technical Report ICMA-83-49, University of Pittsburgh, Pittsburgh, PA, 1983.**[11]**Werner C. Rheinboldt,*Solution fields of nonlinear equations and continuation methods*, SIAM J. Numer. Anal.**17**(1980), no. 2, 221–237. MR**567270**, 10.1137/0717020**[12]**Theodore J. Rivlin,*An introduction to the approximation of functions*, Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1969. MR**0249885**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1985-0804937-0

Keywords:
Nonlinear equations,
continuation methods,
projection methods

Article copyright:
© Copyright 1985
American Mathematical Society