Large Scale Scientific Computing pp 257-268 | Cite as

# A Two-Grid-Approach to Identification and Control Problems for Partial Differential Equations

## Abstract

Both parameter identification and control problems for partial differential equations (PDE) can be treated by minimization of functionals the values of which depend on the unknown parameters or control functions. Each evaluation of these functionals requires an approximate solution of a boundary value or an initial boundary value problem (IBVP) for the PDE. To get an acceptable trade-off between accuracy and the amount of computation, we preferably use a coarse-grid-approximation of the IBVP for minimizing these functionals. Only at a few points of the parameter or control function space a fine-grid-approximation of the IBVP has to be calculated in order to reduce the systematic error created by the coarse grid.

This approach will be illustrated by the hand of parabolic equations, but it seems to be quite general even for large scale identification and control problems.

## Keywords

Control Problem Coarse Grid Partial Differential Equation Initial Boundary Value Problem Radial Temperature Profile## Preview

Unable to display preview. Download preview PDF.

## References

- [1]G. Erikson and G. Dahlquist. On an inverse non-linear diffusion problem, in P. Deuflhard and E. Hairer (eds.) Numerical Treatment of Inverse Problems in Differential and Integral Equations, Birkhäuser, 1983, pp. 238–245.CrossRefGoogle Scholar
- [2]V. Friedrich and B. Hofmann. A predictor-corrector technique for constrained least-squares regularization, Wiss. Information 46, Techn. Hochschule Karl-Marx-Stadt, 1984.Google Scholar
- [3]G. H. Golub, M. Heath and G. Wahba. Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics 21 (1979). 215–223.MathSciNetMATHGoogle Scholar
- [4]B. Hofmann. Regularization for Applied Inverse and Ill-Posed Problems, Teubner, Leipzig, 1986.MATHGoogle Scholar
- [5]Phillips, D. L. A technique for the numerical solution of certain integral equations of the first kind, JACM 9(1). 84–87 (1962)MATHCrossRefGoogle Scholar
- [6]A. N. Tikhonov and V. Y. Arsenin. Methods for the Solution of Ill-Posed Problems (2nd Russian edition), Nauka, Moscow, 1979.MATHGoogle Scholar
- [7]A. N. Tikhonov and V. A. Morozov. Regularization methods for ill-posed problems (Russian), in V. A. Morozov and E. S. Nikolaev (eds.) Numerical Methods and Programming 35, Moscow UniversityGoogle Scholar
- [8]Wahba, G., Wold, S. A complete automatic French curve: fitting spline functions by cross validation. Communications in Statistics 4(1), 1–17 (1975).MathSciNetMATHCrossRefGoogle Scholar