Digital Filter Stepsize Control
in DASPK
and its Effect on
Control Optimization Performance
Kirsten Meeker
University 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
F (t , y, y ' )  0
y (t0 )  y0
 Backward differentiation formula
 Approximates y' using past y values
k
h 0 y ' n    i yn i
i 0
 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
Original Stepsize Control
 
hn   
 rˆn 
1
k 1
hn 1
error tole rance  , local error rˆn , order k
New Digital Filter Stepsize Control
 
hn 1   
ˆ
r
 n
1
  


ˆ
r
 n 1 
2
k1  k 2   2  1 4
 hn 


 hn 1 
 2
hn
Controller Frequency Response

h
Controller
Process
r̂n
 Simple controller - emphasizes high frequencies
 stepsize and local error rougher than disturbance
 Digital filter - uniform frequency response
 smoother stepsize and local error
Optimizer - KNITRO
 Given DAE system F (t , y, y ' , p, u (t ))  0
y (t0 , p)  y0
 Minimize objective function
 (t , y (t ), p, u (t )) dt

 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
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 unfolded proteins (scaled by 1010)
J ( ) 
t1
t1
t0
t0
2
[chaperones]
dt  

2
[P
]
 un dt
100
Pareto Optimal Curve
80
Various nonoptimal values
of parameters
60
40
20
0
Wild type heat shock
10
11
12
Cost of chaperones (scaled by 1010)
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
1
F (t , y,
h 0
k
 y
i 0
i
n i
)  0
g ( yn  1)  g ( yn)  hg ' ( yn)  0
1
 1
yn
 g 



 yn    g ( yn )   0,1,
 y 
g ' ( yn)h   g ( yn)

Jy   g ( yn)