Title: Sensitivity Analysis and Optimization for LargeScale DifferentialAlgebraic Equation Systems
1Sensitivity Analysis and Optimization for
Large-Scale Differential-Algebraic Equation
Systems
Yang Cao and Linda Petzold University of
California Santa Barbara Shengtai Li Los Alamos
National Laboratory Radu Serban Lawrence
Livermore National Laboratory
http//www.engineering.ucsb.edu/cse
2Outline
- Sensitivity analysis for DAEs
- Forward method and software
- Properties of the DAE adjoint system and
numerical solution - Adjoint software
- Design optimization of DAE systems
- Method
- Applications
- Summary, conclusions and future plans
3DAE Sensitivity Analysis
- Given the DAE depending on parameter p,
- sensitivity analysis finds the change in the
solution with respect to perturbations - in the parameters, dx/dpi
- Uses of sensitivity analysis
- Gain physical insight into governing processes
- Parameter estimation
- Design optimization
- Optimal control
- Determine nonlinear reduced order models
- Assess uncertainty and range of validity of
reduced order models
4Sensitivity Analysis (Forward Mode)
General DAE problem with parameters
Differentiate with respect to each parameter to
obtain sensitivity system
where
Sometimes called Tangent Linear Model (TLM)
5Generating the Sensitivity Residuals
6Background DAE Software for Simulation
7DAE Sensitivity Software Forward Method
- DASPK3.0
- Solution and forward sensitivity analysis using
methods of DASPK (Petzold and Li, 2000) - Applicable for DAE index up to two (Hessenberg)
- Consistent initialization for solutions and
sensitivities - General formulation leaves some variables fixed
and varies others for consistent initialization - Requires very little additional input from user
- Exploit structure
- Evaluation of sensitivity residuals
- Automatic differentiation (ADIFOR)
- Adaptive-increment directional-derivative finite
difference approximation - Naturally parallel (MPI)
8Limitations of Forward Sensitivity Method
9Forward vs. Adjoint Sensitivity Analysis
Forward Model
t0
local perturbation
Adjoint Model
area of possible origin
t0
10Basic Idea and Derivation of the Adjoint Method
- Given the nonlinear system
- with derived function
- We wish to compute
- We have
- Linearizing the original nonlinear system,
- The forward sensitivity method computes
for each p. But this is too - costly if p is large.
- To derive the adjoint method, first multiply by
to obtain - Now let solve
- Then
11DAE Sensitivity Analysis (Adjoint Method)
Given the DAE depending on parameters p, and
a function or a function at the end point
(tT) g(x,p,T) Sensitivity analysis finds the
change dG/dp or dg/dp of these functions with
respect to perturbations in the parameters p.
The function we choose depends on the application
problem. Usually the dimension of G or g is much
smaller than that of x or p.
12DAE Adjoint Equations
For G, we solve The corresponding
sensitivities are For g, we solve Here
we need to get the boundary condition from the
end point of but we need not solve for
. The corresponding sensitivities are
13Properties of the DAE Adjoint System - Stability
If the original system is stable, will the
adjoint system also be stable? Consider
This system is equivalent to the stable system
The adjoint system is
Which is equivalent to the unstable (backwards)
system
14Properties of the DAE Adjoint System - Stability
Original DAE system
Augmented adjoint system
- Results
- If the original DAE system is stable then
- The adjoint DAE system is stable (ODE,
semi-explicit index-1 DAE, index-2 Hessenberg DAE
and combinations) - The adjoint DAE system may not be stable, however
the augmented adjoint system is stable
(fully-implicit index 0 and index 1 DAE)
15Numerical Stability
If a numerical method with a given stepsize is
stable for the original DAE system, will it also
be stable for the adjoint system?
Results If the original DAE system is
numerically stable then
- The adjoint DAE system is numerically stable
(ODE, semi-explicit index-1 DAE, index-2
Hessenberg DAE and combinations) - The adjoint DAE system may not be numerically
stable, however the augmented adjoint DAE system
is numerically stable (fully-implicit ODE and DAE)
16DAE Adjoint Sensitivity Software
- DASPKADJOINT (Li and Petzold, 2001)
- Generation of adjoint sensitivity residuals
- Currently TAMC, soon ADIFOR 3.0
-
- Consistent initialization of adjoint DAE system
- Predictable and compact storage of solution
values needed for adjoint - Research Issue
- Matrix-free preconditioners for the adjoint
system
17Optimal Problem and Applications
Basic Problem
Applications Parameter estimation, design
optimization, and optimal control for nonlinear
DAEs and PDEs
18Solution Strategy and Software
- COOPT (Serban and Petzold, 2001)
- Basic Approach
- Original time interval divided into subintervals
(multiple shooting) - DAEs solved numerically on subintervals at each
optimization iteration - Continuity constraints across subintervals
- Optimization problem solved by sparse SQP method
- Partial derivative matrices for optimization
generated by DAE sensitivity software
19Computation of Fundamental Matrix
Problem Optimization requires sensitivities with
respect to control parameters and initial
conditions. For a large-scale PDE problem solved
via standard multiple shooting method, this is
O(nx) sensitivity solves, where nx is the
dimension of the discretized PDE
Solution Use structure of continuity constraints
to reduce computational complexity
- Considerations
- Inequality constraints
- Global convergence
- Software complexity and maintenance
20Modified SQP Method
QPk subproblem
where
Most of the constraints arise from the continuity
conditions in the multiple shooting method
where fi is the solution to the DAE at ti1,
given initial conditions xi at ti, and controls
ui over ti, ti1
21Modified SQP Method (cont.)
The first (N x nx) constraints for QPk arise from
linearizing the continuity constraints
Fi is the fundamental matrix of the DAE over
interval i,
Hi is the partial derivative matrix of the
continuity constraint with respect to the control
variables,
22Modified SQP Method (cont.)
Using its special structure, we can invert the
first Jacobian explicitly, to obtain
- There is no need to form Fi. Instead, compute
the products Fi H0, via nu sensitivity solves - Very efficient if nu ltlt nx
- We replace the original QPk constraints arising
from the continuity conditions by the above. The
continuity constraints are multiplied by (J11)-1
outside of the optimization code, thus avoiding
major changes to the optimizer - The original optimality conditions no longer
yield a descent direction. However, we can show
that a modified l1-merit function yields descent - Treatment of inequality constraints is
unaffected - Efficiency comes at a price of stability for
general ODE/DAE systems. However, for dissipative
PDE systems this is not an issue - Modified method is more robust and efficient
than single shooting for nonlinear problems
23Chemical Vapor Deposition Processes
- Dynamic model
- Compressible, chemically-reacting stagnation
flow - Partial differential-algebraic equations (PDAE)
- Wafer temperature trajectory is specified
- Promote nucleation and film initiation
- Different conditions for mature growth
- Controls
- Inlet velocity
- Inlet precursor mole fractions
- Objectives
- Deposit films of desired thickness
- Minimize deposition time to increase throughput
- Minimize precursor loss due to bypass
24Spacecraft Trajectory Design
- Dynamic model
- Equations of motion of CR3BP
- Ordinary differential equations (ODE)
- Controls
- Maneuver times and magnitudes
- Impulsive optimal control problem
- Objectives
- Minimize fuel consumption
- Insert on the halo orbit around the libration
point L1 - Investigate influence of delay in first
maneuver and of perturbations in launching
velocity
25Tissue Engineering Bioartificial Artery
- Dynamic model
- Partial differential-algebraic equations (PDAE)
- Anisotropic biphasic theory
- Objectives
- Determine cell traction forces from experimental
data - Determine optimal growth conditions
26Summary and Conclusions
- DASPK3.0 solution and sensitivity analysis
(forward method) for large-scale DAE systems - DASPKADJOINT sensitivity analysis by adjoint
method more efficient than forward method when
there are more than a few parameters - COOPT design optimization and optimal control
for large-scale DAE systems - Additional capabilities and future plans
- COOPTAM COOPT with adaptive mesh refinement for
PDE - Adjoint sensitivity analysis for PDE with
adaptive mesh refinement (ADDA method) - COOPT using adjoint sensitivities (soon)
aggregates the constraints - Adjoint method in combination with small sample
statistical estimation for condition estimation
of linear systems and linear matrix equations,
global error estimation for ODEs and DAEs (soon),
error estimation for reduced/simplified models
(soon) -