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Sensitivity Analysis and Optimization for LargeScale DifferentialAlgebraic Equation Systems

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Yang Cao and Linda Petzold. University of California Santa Barbara. Shengtai Li ... DASSL (Petzold (1982)), DASPK (Brown, Hindmarsh and Petzold (1994) ... – PowerPoint PPT presentation

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Title: Sensitivity Analysis and Optimization for LargeScale DifferentialAlgebraic Equation Systems


1
Sensitivity 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
2
Outline
  • 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

3
DAE 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

4
Sensitivity 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)
5
Generating the Sensitivity Residuals
6
Background DAE Software for Simulation
7
DAE 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)

8
Limitations of Forward Sensitivity Method
9
Forward vs. Adjoint Sensitivity Analysis
Forward Model
t0
local perturbation
Adjoint Model
area of possible origin
t0
10
Basic 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

11
DAE 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.
12
DAE 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
13
Properties 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
14
Properties 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)

15
Numerical 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)

16
DAE 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

17
Optimal Problem and Applications
Basic Problem
Applications Parameter estimation, design
optimization, and optimal control for nonlinear
DAEs and PDEs
18
Solution 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


19
Computation 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

20
Modified 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
21
Modified 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,
22
Modified 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

23
Chemical 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

24
Spacecraft 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

25
Tissue 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

26
Summary 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)
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