Digital Filter Stepsize Control

in DASPK and its Effect on

Control Optimization Performance

Kirsten MeekerUniversity of California, Santa Barbara

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

Outline

DAE solver - DASPK Stepsize controllers Optimizer - KNITRO Test Results

Simulation Sensitivity analysis Optimization

DAE solver - DASPK

Backward differentiation formula Approximates y' using past y values

Newton’s method Find yn 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

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Original Stepsize Control

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New Digital Filter Stepsize Control

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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 Processnr̂h

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

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Test Results

Simulation Sensitivity analysis Optimization

Simulation Test Results

36 - 54% fewer time steps 22 - 50% faster CPU time Smoother stepsize changes Larger stepsizes when solution

near constant

Sensitivity Test Results

15 - 16% fewer time steps 34 - 65% more Newton iterations 0 - 40% slower CPU time

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

Optimality of Heat Shock Response

For a given α, minimize Jα with respect to θ

Cost of chaperones (scaled by 1010)

Cost

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Wild type heat shock

Various nonoptimal valuesof parameters

Pareto Optimal Curve

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80

60

40

20

0

10 11 12

J () [chaperones]2dt t0

t1

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t0

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Heat Shock Performance Stage 1

Heat Shock Performance Stage 2

Summary of Optimization Test

Results E. Coli heat shock

95% fewer time steps 97% faster CPU time

2D heat, halo orbit insertion - no change

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

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.

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

Newton’s Method

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