inter. conf. on numer. anal. & optim. theory and appl. king fahd university of petroleum and...

Inter. Conf. On Numer. Anal. & Optim. Theory and Appl.King Fahd University of Petroleum and Minerals

Dhahran, Saudi Arabia, Dec. 17 – 21, 2011

A Glimpse on

Optimal Control of Partial Differential Equations: Theory, Numerics, and Applications

Hans Josef Pesch

Chair of Mathematics in Engineering SciencesUniversity of Bayreuth, Bayreuth, Germany

[email protected]



multi-beam welding

weld seam

hot crack

main laser beam

mushy zone

weld pool

compression

additional beamssolidification

Motivation: Optimal placement of laser beams to avoid hot cracking

Semi-infinite optimization problemfor an elliptic PDE with state constraints

[Karkin, Ploshikin]




op

enin

g d

isp

lace

men

t

weld pool region

hot crack criterium

limit

zoom

isotherms surroundingthe mushy zone

[Petzet]



Motivation: Optimal load changes for fuel cell systems

Molten Carbonate Fuel Cell

cellstack




German Federal Pollution Control Act: Air

CO32-

O2

N2

½O2 + CO2 + 2e- CO32-

CH4 + H2O CO + 3H2

CO + H2O CO2 + H2H2 + CO3

2- H2O + CO2 + 2e-

CO + CO32- 2CO2 + 2e-

U

e-

Recirculation

Exhaust

CH4

H2O

Cathode

Anode

Elektrolyte

Mixer

Catalyticburner

Anode gas channel

Cathode gas channel

1D counter-flow design

Air inlet

only ions can movethrough electrolyte



catalyticburner

mixeranode

anode inlet

cathode inlet

anode exhaust

cathodeexhaust

exhaust air inletrecirculation

cathode

2D cross-flow design

CO32-

solid


28 semi-linear partial integro-differential equations with non-standard non-linear boundary conditions

[Sundmacher][Heidebrecht]




0 0.1 1.1 11.1 111.1 1111.1

using control

optimal controlsimulation

0.8 sec

0 0.1 1.1 11.1 111.1 1111.1scaled time

using controls

optimal controlsimulation

0.4 sec

scaled time

cell voltage 0.7 0.6for a load change

[Sternberg]



Motivation: Minimum fuel transcontinental flights at hypersonic speeds

Europe - USA in 2 hrs / Europe - Australia in 4.5 hrs

ODE

PDE

2 box constraints1 control-state constraint1 state constraint

quasilinear heat equationnon-linear boundary conditionscoupled with ODE



Motivation: Minimum fuel transcontinental flights at hypersonic speeds

velocity [m/s] altitude [10,000 m] flight path angle [deg]

temperature [K] temperature [K] temperature [K]

1st layer 2nd layer 3rd layer

limittemperature

1000 Kon a boundary arc

[s]

[s]

[Wächter, Chudej, LeBras]



Outline

A glimpse on the theory

A glimpse on the numerics

An application

Conclusions



Elliptic optimal control problemwith distributed control

An example: optimal stationary temperature distribution

subject to

tracking functional Tikhonov regularizationset of admissible controls

A simple elliptic optimal control problems

Lions (since 1970s), Casas (1987-), Tröltzsch (1980-)



An example: optimal stationary temperature distribution

subject to

A simple elliptic optimal control problems

Elliptic optimal control problemwith boundary control

tracking functional Tikhonov regularizationset of admissible controls



Necessary condition: variational inequality

with linear and continuous solution operatorsubject to

An example: Optimal stationary temperature distribution

Elliptic optimal control problemwith distributed control

Optimization problem in Hilbert space

Necessary conditions



Necessary condition: variational inequalityDescripton with the adjoint solution operator

Optimization problem in Hilbert space




Description with the adjoint solution operator

Description with the adjoint state




Optimality system: semi-linear elliptic, distributed + boundary control



Optimal Controlof PDE

FunctionalAnalysis

Partial DifferentialEquations

Optimization

in Banach spaces

Numerics ofPDE

Numerical Methods

of Optimization

High PerformanceScientific Computing

Numerical Methods of

Linear Algebra

ParallelNumerical Methods

Challenges in PDE constrained optimization



Outline



An application

Conclusions



Methods for PDE constrained optimization

The general problem

The aims

concepts for real-life application

small constanteffort of simulation

effort of optimization



capture as much structure

of ( P )as possibleon discrete

level( Ph )

First Discretize then Optimize vs. First Optimize then Discretize

First discretize then optimze (fDtO)

First optimze then discretize (fOtD)

Questions

appropriate choice of and ansatz for ?

appropriate choice of and ansatz for ?

appropriate ansatz for adjoint variables and multipliers?

Solvelarge scale

NLP

Solvecoupled PDE

system



First Discretize then Optimize vs. First Optimize then Discretize

First discretize then optimze (fDtO):replace all quantities of the infinite dimensional optimization problemby finite dimensional substitutes and solve an NLP

First optimze then discretize (fOtD):Derive optimality conditions of the infinite dimensional system,discretize the optimality system and find solution of the discretizedoptimality system

In general

Ideal: discrete concept for which both approaches commuteDiscontinuous Galerkin methods



Mathematical Toolbox (incomplete list)

• Structure of optimality system allows one-shot-iterations Griewank, Schulz• Constraints require non-smooth solution techniques Ito, Hintermüller, Kunisch, M. Ulbrich• Structure of optimality system allows multigrid methods Borzi, Schulz• Structure of optimality system allows taylored discrete concepts Hinze, Meyer, Rösch• Relaxation of constraints by penalty or barrier methods Hintermüller, Kunisch, Schiela• State constraints: set optimal control problem with shape calculus Frey, Bechmann, Pesch, Rund• Adaptive algorithms Becker, Rannacher; Vexler; Hintermüller, Hoppe; Hinze, Günther; et.al.• Surrogate models for the PDE system in the optimality system Hinze et.al., Sachs et.al., Kunisch, Tröltzsch, S. Ulbrich, Volkwein• Shape calculus for shape optimization Sokolowski, Zolesio; Gauger, Schulz; Hintermüller, Ring; M. Ulbrich, S. Ulbrich• Automatic differentiation provides adjoints Griewank, Walther



Outline



An application

optimal control of a molten carbonate fuel cell

process control via model reduction techniques

Conclusions



catalyticburner

mixeranode

solid

anode inlet

cathode inlet

anode exhaust

cathodeexhaust

exhaust air inletrecirculation

cathode

Configuration and function of MCFC

2D cross-flow design

controllable

controllable

controllable

load changesinput

boundary conditionsby ODAE

slow

statevariable

fastvery fastalgebraic

[Heidebrecht] [Sundmacher]



anode gas temperature cathode gas temperature

[2.8 ≈ 560 °C]

[3.2 ≈ 680 °C]

Numerical results: simulation of load change

reforming reactions are endothermicoxidation reaction is exothermic

reduction reaction is exothermic

flow directions

[Chudej, Sternberg]



[2.8 ≈ 560 °C]

[3.2 ≈ 680 °C]

solid temperature


flow directionsin anode

and cathode

[Chudej, Sternberg]



[2.8 ≈ 560 °C]

[3.2 ≈ 680 °C]

solid temperature


state constraint would be desirable



with

Pareto performance index:

Numerical results: optimal control of fast load changewhile temperature gradients stay small

fast

slow

on

on

0.7 0.6

instead of state constraint



Aim for process control

How to apply optimal solutions in practise?



• measurable: cell voltage, gas temperatures and concentrations at anode and cathode outlet

• diserable for process control: information on spatial temperatur and concentration profiles

• solution ansatz: observer / state estimator

• Problem: complexity of model

Aim for process control

?

??Remedy: model reduction technique



Model reduction by POD (proper orthogonal decomposition)or Karhunen-Loève decomposition (K.: 1946, L: 1955, Lumley: 1967,…)

• good accuracy for a wide range of operation conditions

• suitable for describing the nonlinear behavior of the cell

• ability for extrapolation in case of varying parameter

Demands on model reduction techniques

German Industrial Partners:CFC Solutions GmbH,

offspring of MTU, Munich;IPF Berndt KG, Reilingen,

constructor and operator of power plants

2002-2005



Model reduction by POD: idea

Complete model:

Ansatz (separation of variables):

Reduced model:

orthogonalsnapshots

low order model: ODAE of index 1

Method of weighted residuals:



test signal

1. temperature basis function

Model reduction by POD: computation of snapshotsby the complete model

2. temperature basis function

orthogonalizationby singular value

decomposition



Model reduction by POD: comparison of reduced vs. complete model

random variation of cell current

perfect coincidencewith reference model

appropriate forprocess control

[Mangold, Sheng]

#eqs. 4759 vs. 1313200 sec vs. 82 sec

2 < N < 10



Model reduction by POD: comparison of reduced vs. complete model

example: response to changes

of the steam-to-carbon ratio in the feed

steam-to-carbon ratio temperature

complete

reduced

voltage



Scheme for state estimator for discrete measurements

process

input

MCFCsensors

y

Simulator

state

sensor models

y

measurement

?

observer

observer correction+

-

MCFC model



Temperature control at Hotmodule:nonlinear feed forward controller + PID controller

PID controller 1

PID controller 2

MCFCsystem

-

-

feed forward controller

[Sheng et al]

state estimator state estimator



Temperature control at Hotmodule:nonlinear feed forward controller + PID controller

feed forward controller only feed forward controller +

PID controller

significantlybetter process

behaviour



Focus on Theory:

Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods, and ApplicationsAMS, Graduate Studies in Mathematics, Vol. 112, 2010.

Focus on Methods:

Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE ConstraintsMathematical Modelling: Theorie and Applications, Vol. 23, 2008.

Focus on Applications:

See my homepage: google: Hans Josef Pesch

References



Conclusions

Concerning theory: already well developed

Concerning numerics: still improving

Concerning applications: has to be intensified

one always abuts against limits



Thank you for your attention

inter. conf. on numer. anal. & optim. theory and appl. king fahd university of petroleum and...

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