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A nonlinear preconditioner for optimum experimental design problems. (English) Zbl 1317.90281

Summary: We show how to efficiently compute A-optimal experimental designs, which are formulated in terms of the minimization of the trace of the covariance matrix of the underlying regression process, using quasi-Newton sequential quadratic programming methods. In particular, we introduce a modification of the problem that leads to significantly faster convergence. To derive this modification, we model each iteration in terms of an initial experimental design that is to be improved, and show that the absolute condition number of the model problem grows without bounds as the quality of the initial design improves. As a remedy, we devise a preconditioner that ensures that the absolute condition number will instead stay uniformly bounded. Using numerical experiments, we study the effect of this reformulation on relevant cases of the general problem class, and find that it leads to significant improvements in both stability and convergence behavior.

MSC:

90C30 Nonlinear programming
90C55 Methods of successive quadratic programming type
62K99 Design of statistical experiments

Software:

LBNL; VPLAN; RADAU; QPOPT
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References:

[1] Bard Y (1974) Nonlinear parameter estimation. Academic Press, New York · Zbl 0345.62045
[2] Bauer I, Bock H, Körkel S, Schlöder J (2000) Numerical methods for optimum experimental design in DAE systems. J Comput Appl Math 120(1—-2):1-15 · Zbl 0998.65083
[3] Birken P, Jameson A (2010) On nonlinear preconditioners in Newton-Krylov methods for unsteady flows. Int J Numer Meth Fl 62(5):565-573 · Zbl 1423.76330
[4] Bock, H.; Ebert, K. (ed.); Deuflhard, P. (ed.); Jäger, W. (ed.), Numerical treatment of inverse problems in chemical reaction kinetics, No. 18, 102-125 (1981), Heidelberg
[5] Cai XC, Keyes DE (2002) Nonlinearly preconditioned inexact Newton algorithms. SIAM J Sci Comp 24(1):183-200 · Zbl 1015.65058
[6] Demmel JW (1987) On condition numbers and the distance to the nearest ill-posed problem. Numer Math 51:251-289 · Zbl 0597.65036
[7] Fateman R (2006) Building algebra systems by overloading lisp: automatic differentiation. University of California, Berkeley (submitted)
[8] Fedorov V (1972) Theory of optimal experiments. Academic Press, New York
[9] FitzHugh R (1961) Impulses and physiological states in theoretical models of nerve membrane. Biophys J 1(6):445-466
[10] Franceschini G, Macchietto S (2008) Model-based design of experiments for parameter precision: state of the art. Chem Eng Sci 63(19):4846-4872 · Zbl 1381.93095
[11] Gill P, Murray W, Saunders M (1995) User’s guide for QPOPT 1.0: a fortran package for quadratic programming. http://www.sbsi-sol-optimize.com/manuals/QPOPT
[12] Griewank A (1989) On automatic differentiation. In: Proceedings of mathematical programming: recent developments and applications. Kluwer Academic Publishers, Dordrecht · Zbl 0696.65015
[13] Griewank A, Walther A (2008) Evaluating derivatives: principles and techniques of algorithmic differentiation, 2nd edn. SIAM, Philadelphia · Zbl 1159.65026
[14] Hairer E, Wanner G (1999) Stiff differential equations solved by Radau methods. J Comput Appl Math 111:93-111 · Zbl 0945.65080
[15] Han S (1977) A globally convergent method for nonlinear programming. JOTA 22:297-310 · Zbl 0336.90046
[16] Hida Y, Li XS, Bailey DH (2001) Algorithms for quad-double precision floating point arithmetic. In: Proceedings of the 15th symposium on computer arithmetic, IEEE Computer Society Press, New York, pp 155-162 · Zbl 0945.65080
[17] Körkel S (2002) Numerische Methoden für optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen. PhD thesis, Universität Heidelberg, Heidelberg · Zbl 1011.62076
[18] Lohmann T, Bock H, Schlöder J (1992) Numerical methods for parameter estimation and optimal experimental design in chemical reaction systems. Ind Eng Chem 31:54-57
[19] Mommer MS, Sommer A, Schlöder JP, Bock HG (2011) Differentiable evaluation of objective functions in sampling design with variance-covariance matrices. PAMM 11(1):727-728
[20] Nagumo J, Arimoto S, Yoshizawa S (1962) An active pulse transmission line simulating nerve axon. Proc IRE 50(10):2061-2070
[21] Nocedal J, Wright SJ (2006) Numerical optimization, 2nd edn. Springer, New York · Zbl 1104.65059
[22] Powell, M.; Watson, G. (ed.), A fast algorithm for nonlinearly constrained optimization calculations, No. 630 (1978), Berlin · Zbl 0374.65032
[23] Pronzato L (2008) Survey paper: optimal experimental design and some related control problems. Automatica 44(2):303-325 · Zbl 1283.93154
[24] Pukelsheim F (2006) Optimal design of experiments. In: Proceedings of classics in applied mathematics, vol 50. SIAM, Philadelphia (2006) · Zbl 1101.62063
[25] Pukelsheim F, Rieder S (1992) Efficient rounding of approximate designs. Biometrika 79(4):763-770
[26] Schittkowski K (1982) The nonlinear programming method of Wilson, Han, and Powell with an augmented lagrangian type line search function. Numer Math 38:83-114. doi:10.1007/BF01395810 · Zbl 0534.65030
[27] Schöneberger J, Arellano-Garcia H, Thielert H, Körkel S, Wozny G (2008) Optimal experimental design of a catalytic fixed bed reactor. In: Braunschweig B, Joulia X (eds) Proceedings of 18th European symposium on computer aided process engineering, ESCAPE 18. Elsevier. ISBN: 978-0-444-53227-5 · Zbl 1283.93154
[28] Zolezzi T (2002) On the distance theorem in quadratic optimization. J Convex Anal 9(2):693-700 · Zbl 1034.49030
[29] Zolezzi T (2003) Condition number theorems in optimization. SIAM J Optim 14(2):507-516 · Zbl 1041.49026
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