Abstract
An important task in groundwater flow computations is to find distributed flow parameters from point measurements. The focus of this paper is on the development of new multigrid optimization methods for the numerical treatment of this practical problem class in the context of stationary flow problems. The new methods combine known multigrid approaches for regularized linear-quadratic control problems with reduced SQP methods for large-scale optimization problems. The efficiency of the resulting multigrid methods is demonstrated in comparison with conventional techniques applied to the above problem class.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bank, R.E., Welfert, B.D., Yserentant, H.: A class of iterative methods for solving saddle point problems. Numerische Mathematik 56 (1990) 645–666
Bastian, P., Birken, K., Johannsen, K., Lang, S., Neuß, N., Rentz-Reichert, H., Wieners, C: UG — a flexible Software toolbox for solving partial differential equations. Computing and Visualization in Science 1 (1997, to appear)
Bock, H.G.: Randwertproblemmethoden zur Parameteridentifizierung in Systemen nichtlinearer Differentialgleichungen. Bonner Mathematische Schriften 183, Bonn, 1987.
Bock, H.G., Schiöder, J., Schulz, V.: Numerik großer Differentiell-Algebraischer Gleichungen — Simulation und Optimierung. In Prozeßsimulation, VCH Verlagsgesellschaft mbH, Weinheim, S. 35–80, 1994.
Bornemann, F., Deuflhard, P.: The cascadic multigrid method for elliptic problems. Numerische Mathematik 75 (1996) 135–152
Bramble, J., Pasciak, J.: A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems. Math. Comp. 50 (1988) 1–17
Carrera, J., Neumann, S.P.: Estimation of aquifer parameters under transient and steady-state conditions: 1. Maximum Likelihood Method incorporating prior information. Water Resources Research 22 (1986) 199–210
Carrera, J., Neumann, S.P.: Estimation of aquifer parameters under transient and steady-state conditions: 2. Uniqueness, stability and solution algorithms. Water Resources Research 22 (1986) 211–227
Carrera, J., Neumann, S.P.: Estimation of aquifer parameters under transient and steady-state conditions: 3. Application to synthetic and field data. Water Resources Research 22 (1986) 228–242
Dennis, Jr., J.E., Schnabel, R.B.: Numerical Methods for Nonlinear Equations and Unconstrained Optimization (Prentice-Hall, Eaglewood Cliffs, N.J., 1983)
Dyn, N., Ferguson, W.: The numerical solution of equality-constrained quadratic programming problems. Math. Comp. 41 (1983) 165–170
Finsterle, S.: Inverse Modellierung zur Bestimmung hydrogeologischer Parameter eines Zweiphasensystems, Dissertation Nt. 9985, ETH Zürich
Fletcher, R.: Practical methods of optimization. John Wiley and Sons, New York, 1987.
Gill, O., Murray, W., Saunders, W., Wright, M.: User’s guide to NPSOL. Technical report SOL 83-9, Department of Operations Research, Stanford Univerity, Stanford, CA, 1983.
Gill, P., Murray, W., Wright, M.: Practical Optimization, Academic Press, New York, 1981
Hackbusch, W.: Fast solution of elliptic control problems. JOTA 31 (1980) 565–581
Heinkenschloss, M.: On the solution of a two ball trust region subproblem. Mathematical Programming 64 (1994) 249–276
Heinkenschloss, M.: The numerical solution of a control problem governed by a phase field model. ICAM report 95-12-01, Virginia Polytechnic Institute and State University, 1995.
Moré, J.J.: Recent developments in algorithms and software for trust region methods. In: A. Bachern, M. Grötschel and B. Korte (eds.): Mathematical Programming, The State of The Art (Springer, Berlin, 1983) 258–287
Moré, J.J., Sorensen, D.: Computing a trust region step. SIAM Journal on Scientific and Statistical Computations 4 (1983) 553–572
Schulz, V.: Reduced SQP methods for large-scale optimal control problems in DAE with application to path planning problems for satellite mounted robots. Dissertation, University of Heidelberg, 1996.
Schulz, V., Dreyer, Th., Speer, Th., Bock, H.G.: Optimum shape design of turbine blades. In P. Kleinschmidt, A. Bachern, U. Derigs, D. Fischer, U. Leopold-Wildburger, R. Möhring (eds.): Operations Research Proceedings 1995, p. 190–195, Springer, Heidelberg, 1996.
Ta’asan, S.: “One Shot” methods for optimal control of distributed parameter systems I: finite dimensional control. ICASE report No. 91-2, 1991.
Wittum, G.: Multi-grid methods for Stokes and Navier-Stokes equations. Transforming smoothers — algorithms and numerical results. Numerische Mathematik 54 (1989) 543–563
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Schulz, V., Wittum, G. (1998). Multigrid Optimization Methods for Stationary Parameter Identification Problems in Groundwater Flow. In: Hackbusch, W., Wittum, G. (eds) Multigrid Methods V. Lecture Notes in Computational Science and Engineering, vol 3. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-58734-4_16
Download citation
DOI: https://doi.org/10.1007/978-3-642-58734-4_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-63133-0
Online ISBN: 978-3-642-58734-4
eBook Packages: Springer Book Archive