Derivative Based vs. Derivative Free Optimization Methods for Nonlinear Optimum Experimental Design

Transcription

1 Derivative Based vs. Derivative Free Optimization Methods for Nonlinear Optimum Experimental Design Stefan Körkel 1, Huiqin Qu 2, Gerd Rücker 3, and Sebastian Sager 1 1 Interdisciplinary Center for Scientific Computing, University of Heidelberg, Im Neuenheimer Feld 368, D Heidelberg, Germany 2 Intelligent Information Processing Laboratory, Fudan University, Shanghai 3 Deutsche Börse AG, Frankfurt am Main 1 Introduction An important task in the procedure of the validation of dynamic process models is nonlinear optimum experimental design. It aims at computing experimental layouts, setups and controls in order to optimize the statistical reliability of parameter estimates from the resulting experimental data. The models we consider usually arise from applications in chemistry or chemical engineering and consist of nonlinear systems of differential equations, e.g. ordinary differential equations (ODE) or differential algebraic equations (DAE). In this paper we sketch our numerical approach (implemented in our software package VPLAN) which is based on sequential quadratic programming with a tailored derivative computation and compare it to the easy to implement but much less powerful derivative free approach. 2 Problem Statement 2.1 Model Validation In many practical and industrial applications dynamic processes play an important role. To simulate, understand, control and optimize these processes, they are described by dynamic mathematical models, usually systems of ordinary or partial differential equations. In this paper we concentrate on differential algebraic equations (DAE): 0 = g(t, y, z, p, q)

2 2 Stefan Körkel, Huiqin Qu, Gerd Rücker, and Sebastian Sager where t [t 0 ; t end ] is the time and x = (y, z) : [t 0 ; t end ] nx are the system states. The values of the quantities p np, the parameters, are known only roughly. The controls q describe the layout, the setup and the processing of experiments, we distinguish between time independent control variables q 1 nq 1 and time dependent control functions q 2 : [t 0 ; t end ] n q2, q = (q 1, q 2 ). To estimate the parameters p from experimental data, we minimize the weighted sum of the squares of the residuals between measurement values η i and model responses h i (t i, x(t i ), p, q)), i = 1,..., M: min p,x M i=1 w i (η i h i (t i, x(t i ), p, q)) 2 σ 2 i 0 = g(t, y, z, p, q) 0 = d(x(t 0 ),..., x(t f ), p, q) (1) The quantities σ i are the variances of the measurement errors. The weight w i [0; 1] specifies if measurement i is actually carried out or not. In parameter estimation we can assume that all the weights are 1, later in experimental design we will use the w i as variables to choose the placement of the measurements. For the numerical solution of this kind of problems, we use the boundary value problem optimization approach suggested by Bock [Boc87]. 2.2 Optimum Experimental Design The parameter estimation problem (1) depends on the randomly distributed experimental data, hence the solution ˆp are also random variables. Their uncertainty can be described by the variance-covariance matrix [Boc87] C = ( I 0 ) ( J T 1 J 1 J T 2 J 2 0 ) 1 ( ) ( J T 1 J 1 0 J T 1 J 1 J2 T 0 0 J 2 0 ) T ( ) I, 0 where J 1 resp. J 2 are the derivatives w.r.t. the parameters of the least squares terms resp. the constraints of problem (1) evaluated in the solution point. Our aim is to compute an experimental design, i.e. controls q for layout, setup and processing and weights w for measurement selection, which yields by parameter estimation from the experimental data estimates with minimal statistical uncertainty. For this purpose we minimize functions on the variance-covariance matrix, e.g. φ(c) = trace(c) or det(c) or max{λ : λ eigenvalue of C} or max C ii Let ξ := (q, w) denote the experimental design variables. Then we can formulate the experimental design optimization problem

3 Optimization and Derivatives for Nonlinear Experimental Design 3 min φ(c) q,w,x C is the variance-covariance matrix in the solution point of the problem M min w i (η i h i (t i, x(t i ), p, q)) 2 p,x σ 2 i=1 i 0 = g(t, y, z, p, q) 0 = d(x(t 0 ),..., x(t f ), p, q) Constraints: 0 = g(t, y, z, p, q) lo ψ(t, x, p, q, w) up 0 = χ(t, x, p, q, w) w {0, 1} M. (2) Remark 1( The ) objective function of problem (2) is defined on the Jacobian J = of the parameter estimation problem (1) which depends J1 on J 2 derivatives of the solution of the dynamic system w.r.t. the parameters p, see [Kör02]. The experimental design problem (2) is a nonlinear inequality-constrained optimal control problem. The objective function is implicitly defined on derivatives of the solution of the dynamic model equations and not separable. For the numerical solution, we apply the direct approach of optimal control, which consists of parameterization of the control functions q 2, discretization of the state constraints, relaxation of the 0-1 variables, and a finite-dimensional parameterization of the solution of the dynamic system. For details we refer to [Kör02]. We obtain a finite dimensional nonlinear constrained optimization problem: min φ(ξ) s.t. 0 = χ(ξ), 0 ψ(ξ) (3) 3 Derivative Based Optimization To solve the experimental design optimization problem (3) we choose the Newton-type method of Sequential Quadratic Programming (SQP). For details on this method we want to refer e.g. to the textbook [NW99]. We employ the SQP implementation SNOPT [GMS02]. For the optimization, first derivatives of objective and constraints with respect to the optimization variables ξ are required. We consider directional derivatives for directions ξ and apply the chain rule

4 4 Stefan Körkel, Huiqin Qu, Gerd Rücker, and Sebastian Sager φ := dφ φ(ξ + h ξ) φ(ξ) ξ := lim dξ h 0 h = dφ dc C, σ i dc C := dj J, dj J := ξ. (4) dξ The steps in (4) require intricate derivative computations. C and dc dj J mean the differentiation of functions on matrices w.r.t matrices, see [Kör02]. dj dξ ξ is the derivative w.r.t. ξ of the derivative w.r.t. p of the parameter estimation problem, e.g. ( ) ( dj 1 wi hi 2 x i q = diag dq x p q q + 2 h i x i x i x x p q q + 2 h i x i x q p q + 2 h i x i p x q q + 2 h i p q q ) i=1,...,m where x i := x(t i, p, q). Note that for the computation of (5) we not only need the solution x of the DAE, but also first and mixed second derivatives: dφ dc (5) x p (t i, p, q), x q (t i, p, q), 2 x p q (t i, p, q). We apply the backward differentiation formulae (BDF), a multistep integration method implemented in the code DAESOL [BBKS99], to solve the DAE systems. The derivatives of x are solutions of variational DAEs (VDAE). BDF schemes for these VDAE can also be considered as derivatives of the BDF scheme for the DAE if the same stepsize and order control is used. Hence we can compute the exact derivatives of the numerical approximation of x. Moreover, all these BDF schemes have the same structure, thus the matrix decompositions for solution of the implicit problems can be applied simultaneously to all required first and second derivatives. For details on this approach of internal numerical differentiation see [BBKS99] or [Kör02]. The diverse VDAEs for the first and second derivatives contain first and second derivatives of the model functions f and g of the right hand side of the DAE. Further, first and second derivatives of the measurement model response functions h i and of the nonlinear constraints are required. Usually, these functions are given as user defined subroutines. In our software VPLAN, we apply techniques of automatic differentiation based on the package ADI- FOR [BCKM94] to compute the needed derivatives automatically. 4 Derivative Free Optimization We use the derivative free multidirectional search method developed by Torczon [Tor89], which is based on the iterative change of a simplex with k + 1

5 Optimization and Derivatives for Nonlinear Experimental Design 5 points (where k is the number of the arguments in the function) so that the procedure converges to a minimal value point. Compared with the widely used simplex method [NM65, PTVF92], this method searches k distinct directions in parallel, i.e. only the best point is kept in each iteration, which makes it converge faster and more reliable than the simplex method in case the function has many arguments. The multidirectional search method uses only function values of the objective function and is only able to treat simple-bounds-constraints. 5 Numerical Results 5.1 The Test Problem We compare the two optimization approaches for an experimental design optimization problem for the Diels-Alder reaction [MB83]. It is a chemical reaction with a catalytic and non-catalytic reaction channel. Aim is to determine the reaction velocities of both reaction channels. The model of this process can be formulated as an ordinary differential equation system. where the state variables model the molar numbers of the species. The model contains 5 parameters, the steric factors and activation energies of the reaction velocities and catalyst deactivation rate. Experimental design variables are the initial molar numbers, the concentration of the catalyst and the temperature and the weights for the placement of 10 measurements. We want to plan two experiments for the most significant estimation of the model parameters. This leads to an optimization problem with simplebounds-constraints on the experimental design variables as only constraints. Thus it is also possible to treat it with the derivative free optimization method. For each experiment we have 17 degrees of freedom, thus altogether 34 optimization variables for two experiments. We start the optimization procedures with an experimental layout with objective value for the A criterion (trace(c)). 5.2 Results of the Derivative Based Optimization The computation with VPLAN using the SNOPT SQP optimizer needs 167 SQP iterations which require 168 function calls and 384 derivative evaluations of the objective function to achieve convergence to the optimal solution with A criterion = This computation runs 8.1 seconds user cpu time on a Pentium4 2.5 GHz under Linux. 5.3 Comparison to Derivative Free Optimization The multidirectional search method terminates with an objective value of for the A criterion. To achieve this result, function evaluations are necessary in a user cpu time of 5 minutes on the same computer as above.

6 6 Stefan Körkel, Huiqin Qu, Gerd Rücker, and Sebastian Sager The objective value of the derivative based result is slightly better than the objective value of the derivative free result. The computational time of the derivative based method is 8 seconds compared to 5 minutes for the derivative free method. 6 Conclusion The task of experimental design for dynamic processes yields complicated nonlinear optimization problems. Newton-type optimization methods such as the method of sequential quadratic programming require in particular derivatives of the objective function which is especially complicated for experimental design. Intricate computations are needed to provide the derivatives efficiently. Using this derivative based approach we have developed the software package VPLAN which can solve generally formulated problems of this class. It is inveigling to use easy to implement derivative free optimization methods instead. In this paper we have shown that, besides the drawback that such methods are restricted to problems with only simple-bounds-constraints, they require tremendously more computational time to achieve comparable results for nonlinear experimental design problems. References [BBKS99] I. Bauer, H. G. Bock, S. Körkel, and J. P. Schlöder. Numerical methods for initial value problems and derivative generation for DAE models with application to optimum experimental design of chemical processes. In F. Keil, W. Mackens, H. Voss, and J. Werther, editors, Scientific Computing in Chemical Engineering II, volume 2, pages , Berlin, Heidelberg, Springer-Verlag. [BCKM94] C. Bischof, A. Carle, P. Khademi, and A. Mauer. The ADIFOR 2.0 system for the automatic differentiation of fortran 77 programs. Technical Report CRPC-TR94491, Center for Research on Parallel Computation, Rice University, Houston, TX, [Boc87] H. G. Bock. Randwertproblemmethoden zur Parameteridentifizierung in Systemen nichtlinearer Differentialgleichungen. Bonner Mathematische [GMS02] Schriften 183, P. E. Gill, W. Murray, and M. A. Saunders. SNOPT: An SQP algorithm for large-scale constrained optimization. SIAM J. Opt., 12: , [Kör02] S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen. PhD thesis, Universität Heidelberg, available at [MB83] R. T. Morrison and R. N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983.

7 Optimization and Derivatives for Nonlinear Experimental Design 7 [NM65] J. A. Nelder and R. Mead. A simplex method for function minimization. Comput. J., 8: , [NW99] J. Nocedal and S. J. Wright. Numerical Optimization. Springer Series in Operations Research. Springer-Verlag, New York, [PTVF92] William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery. Numerical Recipes in C: The Art of Scientific Computing. Cambridge University Press, [Tor89] Virginia J. Torczon. Multi-Directional Search: A Direct Search Algorithm for Parallel Machines. PhD thesis, Houston, TX, USA, 1989.