Sequential quadratic programming for certain parameter identification problems. (English) Zbl 0743.65070

The authors show how the method of sequential quadratic programming can be applied to elliptic parameter identification problems (and their large scale finite dimensional discretizations) by treating in detail the following problem: To estimate the coefficient function \(q(x)\) in the one-dimensional elliptic equation \(-(qu')'=f\) in \(I\), \(I\) denoting the unit interval in \(R\), with \(u=0\) on the boundary, provided the function \(f(x)\) and an observation \(z(x)\) of \(u\) in \(I\) are known.
The paper also contains a discussion of implementation aspects of the proposed method as well as a presentation of some numerical results.


65L10 Numerical solution of boundary value problems involving ordinary differential equations
90C20 Quadratic programming
34A55 Inverse problems involving ordinary differential equations
65K10 Numerical optimization and variational techniques
34B05 Linear boundary value problems for ordinary differential equations
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[1] E.L. Allgower, K. Böhmer, F.A. Potra and W.C. Rheinboldt, ”A mesh-independence principle for operator equations and their discretizations,”SIAM Journal on Numerical Analysis 23 (1986) 160–169. · Zbl 0591.65043
[2] P.T. Boggs and J.W. Tolle, ”A family of descent functions for constrained optimization,”SIAM Journal on Numerical Analysis 21 (1984) 1146–1161. · Zbl 0575.65066
[3] F. Colonius and K. Kunisch, ”Output least squares stability for parameter estimation in two point boundary value problems,”Journal für die Reine und Angewandte Mathematik 270 (1986) 1–29. · Zbl 0584.34009
[4] R. Fletcher,Practical Methods of Optimization (Wiley, New York, 1987). · Zbl 0905.65002
[5] S.-P. Han, ”A globally convergent method for nonlinear programming,”Journal of Optimization Theory and Applications 22 (1977) 297–309. · Zbl 0336.90046
[6] K. Ito and K. Kunisch, ”The augmented Lagrangian method for equality and inequality constrained problems in Hilbert spaces, to appear in:SIAM Journal on Control and Optimization. · Zbl 0706.90096
[7] A.D. Ioffe, ”Necessary and sufficient conditions for a local minimum. 1: A reduction theorem and first order conditions,”SIAM Journal on Control and Optimization 17 (1979) 245–250. · Zbl 0417.49027
[8] A.D. Ioffe, ”Necessary and sufficient conditions for a local minimum. 2: Conditions of Levitin–Miljutin–Osmolovskii type,”SIAM Journal on Control and Optimization 17 (1979) 251–265. · Zbl 0417.49028
[9] A.D. Ioffe, ”Necessary and sufficient conditions for a local minimum. 3: Second order conditions and augmented duality,”SIAM Journal on Control and Optimization 17 (1979) 266–288. · Zbl 0417.49029
[10] K. Ito and K. Kunisch, ”The augmented Lagrangian method for parameter estimation in elliptic systems,” Technical Report #87-37, Lefschetz Center for Dynamical Systems and Center for Control Sciences (Lefschetz, 1987). · Zbl 0709.93021
[11] L.V. Kantorovich and G.P. Akilov,Functional Analysis (Pergamon, New York, 1982), 2nd ed.). · Zbl 0484.46003
[12] C. Kravaris and J. Seinfeld, ”Identification of parameters in distributed parameter systems by regularization,”SIAM Journal on Control and Optimization 23 (1985) 217–241. · Zbl 0563.93018
[13] C. Kravaris and J. Sinfeld, ”Identification of spatially varying parameters in distributed parameter systems by discrete regularization,”Journal of Mathematical Analysis and Applications 119 (1986) 128–152. · Zbl 0614.35095
[14] M. Kroller and K. Kunisch, ”A numerical study of an augmented Lagrangian method for the estimation of parameters in a two point boundary value problem,” Technische Universität Graz Technical Report #87, Institute für Mathematik, (Graz, 1987).
[15] K. Kunisch and L. White, ”Regularity properties in parameter estimation of diffusion coefficients in one dimensional elliptic boundary value problems,”Applicable Analysis 21 (1986) 71–87. · Zbl 0585.49008
[16] M.J.D. Powell and Y. Yuan, ”A recursive quadratic programming algorithm that uses differentiable penalty functions,”Mathematical Programming 35 (1986) 265–278. · Zbl 0598.90079
[17] R.T. Rockafellar, ”Augmented Lagrane multiplier functions and duality in nonlinear programming,”SIAM Journal on Control 12 (1974) 268–285. · Zbl 0285.90063
[18] R. Temam,Navier–Stokes Equations, Theory and Numerical Analysis (North-Holland, New York, 1985).
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