1. Digital Filter Stepsize Control in DASPK and its Effect on Control Optimization Performance Kirsten Meeker University of California, Santa Barbara

2. Introduction  Solutions vs. perturbed initial conditions not smooth for adaptive ODE/DAE solvers  In optimal control or parameter estimation of ODE/DAE systems, optimization performance depends on smoothness of solution vs. small perturbations in control parameters  Digital filter stepsize control  Smoother solution dependence  More efficient optimization search  Söderlind and Wang, Adaptive time-stepping and computational stability, ACM T Comp Logic, 2002

3. Outline  DAE solver - DASPK  Stepsize controllers  Optimizer - KNITRO  Test Results  Simulation  Sensitivity analysis  Optimization

4. DAE solver - DASPK  Backward differentiation formula  Approximates y' using past y values  Newton’s method  Find y n at each time step  Linear systems solved by direct method or preconditioned Krylov iteration Li and Petzold, Software and Algorithms for Sensitivity Analysis of Large-Scale Differential Algebraic Systems, UCSB, 2000

5. Original Stepsize Control

6. New Digital Filter Stepsize Control

7. Controller Frequency Response  Simple controller - emphasizes high frequencies  stepsize and local error rougher than disturbance  Digital filter - uniform frequency response  smoother stepsize and local error Controller Process

8.  Given DAE system  Minimize objective function  Sequential quadratic programming  Sensitivity derivatives from DASPK  Trust regions to solve non-convex problems  R. A. Waltz and J. Nocedal, KNITRO User's Manual Technical Report OTC 2003/05, Optimization Technology Center, Northwestern University, Evanston, IL Optimizer - KNITRO

9. Test Results  Simulation  Sensitivity analysis  Optimization

10. Simulation Test Results  36 - 54% fewer time steps  22 - 50% faster CPU time  Smoother stepsize changes  Larger stepsizes when solution near constant

11. Sensitivity Test Results  15 - 16% fewer time steps  34 - 65% more Newton iterations  0 - 40% slower CPU time

12. E. Coli Heat Shock  Heat causes unfolding, misfolding, or aggregation of cell proteins  Stress response is to produce heat-shock proteins to refold denatured proteins  Model first order kinetics (law of mass-action)  Stiff system of 31 equations  11 differential  20 algebraic constraints  H. El Samad and C. Homescu and M. Khammash and L.R.Petzold, The heat shock response: Optimization solved by evolution ?, ICSB 2004

13. Optimality of Heat Shock Response  For a given α, minimize J α with respect to θ Cost of chaperones (scaled by 10 10 ) Cost of unfolded proteins (scaled by 10 10 ) Wild type heat shock Various nonoptimal values of parameters Pareto Optimal Curve 100 80 60 40 20 0 101112

14. Heat Shock Performance Stage 1

15. Heat Shock Performance Stage 2

16. Summary of Optimization Test Results  E. Coli heat shock  95% fewer time steps  97% faster CPU time  2D heat, halo orbit insertion - no change

17. Summary and Conclusions  Implemented a Digital Filter Stepsize Controller into DASPK3.1  Tested on several problems involving simulation and sensitivity analysis, and found that:  Overall efficiency was roughly comparable to that of DASPK  Stepsize sequences used were smoother with the new digital filter stepsize controller

18. Summary and Conclusions  Tested on several problems involving optimization of DAE systems, and found that:  For two problems that are not very challenging, the performance was comparable to that using original DASPK  For a highly nonlinear heat shock problem involving a wide range of scales, the optimizer required dramatically fewer iterations when using DASPK3.1mod to solve the DAEs. We conjecture that this is due to the smoother dependence of the numerical solution on the parameters.

19. Thanks!  Linda Petzold, Thesis Advisor  John Gilbert, Committee  Mustafa Khammash, Committee  Söderlind and Wang, Digital filter stepsize controller  Chris Homescu, Hana El-Samad, Mustafa Khammash, E. Coli Heat Shock

20. Newton’s Method