On the theory and error estimation of the reduced basis method for multi- parameter problems.

*(English)*Zbl 0802.65068The computational analysis of many problems in science and engineering involves the solutions of very large systems of nonlinear equations. Thus, there is considerable interest in reducing the size of these problems. One of these reduction techniques has become generally known as the reduced basis method. The aim of the reduced basis method is to construct special approximations that reduce the number or degrees of freedom without compromising error.

The idea of this method considered by the author may be reviewed as the design of approximations that do account of information about the manifold \(M\), which is the solution set of a nonlinear equation with multi-parameter, and its local coordinate systems. Therefore, the author thinks that a few basis vectors may already suffice to represent the manifold locally to a relatively high accuracy.

Moreover, the author also proves that with a special projection and a special linear map the approximated problem has a unique solution and the error estimate satisfies an inequality. Finally, the author gives a numerical example of the reduced basis method.

The idea of this method considered by the author may be reviewed as the design of approximations that do account of information about the manifold \(M\), which is the solution set of a nonlinear equation with multi-parameter, and its local coordinate systems. Therefore, the author thinks that a few basis vectors may already suffice to represent the manifold locally to a relatively high accuracy.

Moreover, the author also proves that with a special projection and a special linear map the approximated problem has a unique solution and the error estimate satisfies an inequality. Finally, the author gives a numerical example of the reduced basis method.

Reviewer: Yu Wenhuan (Tianjin)

##### MSC:

65H17 | Numerical solution of nonlinear eigenvalue and eigenvector problems |

##### Keywords:

error estimation; multi-parameter problems; large systems of nonlinear equations; reduction techniques; reduced basis method##### Software:

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\textit{W. C. Rheinboldt}, Nonlinear Anal., Theory Methods Appl. 21, No. 11, 849--858 (1993; Zbl 0802.65068)

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##### References:

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