Reduced-Order Modeling (ROM) for Simulation and Optimization

184 Pages·2018·6.299 MB·English

by  Winfried Keiper, Anja Milde, Stefan Volkwein

Winfried Keiper · Anja Milde 
Editors
Stefan Volkwein
Reduced-Order
Modeling (ROM)
for Simulation and
Optimization
Powerful Algorithms as Key Enablers for
Scientific Computing
Reduced-Order Modeling (ROM) for Simulation
and Optimization
Winfried Keiper Anja Milde
(cid:129)
Stefan Volkwein
Editors
Reduced-Order Modeling
(ROM) for Simulation
and Optimization
Powerful Algorithms as Key Enablers
fi
for Scienti c Computing
123
Editors
Winfried Keiper StefanVolkwein
Department ofCorporate Research Fachbereich Mathematik
RobertBosch GmbH UniversitätKonstanz
Renningen Konstanz
Germany Germany
AnjaMilde
Interdisciplinary Center
forScientificComputing
Heidelberg University
Heidelberg
Germany
ISBN978-3-319-75318-8 ISBN978-3-319-75319-5 (eBook)
https://doi.org/10.1007/978-3-319-75319-5
LibraryofCongressControlNumber:2018932542
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Preface
KoMSO Workshop
The growing demand for numerical solutions in scientific computing (modeling,
simulation, data analysis, and optimization problems in many application fields)
requires ever-higher algorithmic and computational performance.
Why is this so? To name just a few reasons: Models become larger and require
higher geometrical resolution and full 3D topology. Larger, more comprehensive
systemswithchallengingboundaryconditionsarebeingmodeledandaskforrobust
processcontrol.Inverseproblemsofmuchlargersizeneedtobesolved.Connected
components with more relevant physical effects are simulated simultaneously.
Optimizationwithparametricvariantsisperformed.Problemsrequiremulti-domain
and multi-scale modeling. An analysis of very large “big data” sets is required.
Highlycomplexandmuchfineroptimizationcriteriaareapplied…Allthesetrends
will persist for the foreseeable future.
There is a solution to the growing needs for scientific computing power on the
one hand: the progress in High-Performance Computing (HPC) hardware. We do
not know how much longer Moore’s Law will be valid, before microelectronics
reaches a physical limit. The other lever for improved computational performance,
with immense opportunities, is the algorithmic side of scientific computing.
It has been shown in many cases that the advancement of mathematical algo-
rithms has increased the algorithmic performance clearly more than the improve-
ment of computer hardware alone. In the past, approved examples of massive
reductions in computational efforts by fast algorithms include the fast Fourier
transform,orthesolutionofPoisson’sequation,ormorerecentlytheapplicationof
adjoint variable methods. In many cases, “Math can do more than Moore.”
As a side effect, new, faster simulation and optimization tools that can run on
existingcomputerhardwarewillenablemanyadditionalapplications,especiallyfor
problemsthatcannotbesolvednumericallytodayataffordablecostandeffort.This
will attract many more future users: a contribution to the digital transformation of
R&D.
v
vi Preface
Withthismotivationinmind,thisworkshopwasorganized.Itgatheredcreators
anddevelopersofnew,optimal,andfastmathematicalalgorithmsandtheindustrial
users of these tools.
The workshop was supported by the Forschungsverbund Wissenschaftliches
Rechnen, Baden-Württemberg (WiR-BaWü), a network of various departments in
the state of Baden-Württemberg ranging from mathematics, to engineering to
computer science. The state of Baden-Württemberg played a pioneering role in
advancing scientific computing. It started with the Interdisciplinary Center for
Scientific Computing (IWR) pursuing its vision to develop mathematical and
computational methods for applications in engineering, the sciences, and
humanities.
This proceeding issue summarizes the successful meeting, which took place in
November2016,atthepremisesofRobertBoschCorporateResearchinRenningen
near Stuttgart. These are the major groups of contributions to the workshop:
Recent advances in Model Order Reduction (MOR) simulations: In many
applications(e.g.,incomputationalfluiddynamics),thespatialdiscretizationyields
quadratic-bilinear parameterized descriptor systems. Benner and Goyal extend
recent advances in interpolation-based optimal control MOR, ensuring bounded
errorsfortheirapproximation.Vectorialkernelapproximationsareveryimportant,
because they arise in MOR approximations for systems with vectorial inputs and
outputs. Haasdonk and Santin prove quasi-optimal rates of convergence for vec-
torial kernel greedy algorithms.
NewMORbasedoptimizationapproaches:Inmanypracticalapplications,many
objectives are equally important and also contradictory, such that one is forced to
find an optimal compromise between them. This results in multi-objective optimal
control problems where multiple objectives have to be minimized simultaneously.
This again requires many numerically expensive target evaluations. To overcome
the computational burden, Beermann et al. develop a MOR strategy, which is
utilized in a derivative-free set-oriented optimization solver. A-posteriori error
boundsensurethedesiredaccuracyoftheROM.HeinkenschlossandJandopresent
amethodtosolvelinearquadraticoptimalcontrolproblemsbyadjustingasuitable
ROM for the Hessian in each level of the iterative method. The computed optimal
controls have the same accuracy as the ones obtained by the high-fidelity
approximation while being computationally much faster.
Successfulbridgestotheindustrialworld:Inthecontextofenergyoptimization,
the simulation of gas networks has become more and more important recently.
However, many nonlinear Euler equations have to be solved on the edges of the
network. Due to the high complexity of the network, MOR offers the chance to
decrease the degrees of freedom dramatically. Benner et al. first approximate the
Euler equations by a differential–algebraic system of index 1, which can then be
successfullyreducedbyMOR.Thesimulationofelectricrotatingmachinesisboth
computationally expensive and memory-intensive. For that reason Bontinck et al.
develop a MOR strategy that handles thechallenging non-symmetric setting inthe
problem. The fast numerical simulation of elastic multi-body systems is of great
interest for industrial applications. Fehr et al. introduce the program package
Preface vii
Morembs, available in C++ and in MATLAB. The software is already used in
variousapplications.Hartmannetal.discussMORfordigitaltwins.Theypointout
thatMORisakeytechnologytotransfersophisticatedsimulationmodelsintoother
domains and life cycle phases.
The workshop was jointly organized by KoMSO, Konstanz University and the
Robert Bosch GmbH and co-sponsored by the Federal Ministry of Education and
Research (BMBF).
Renningen, Germany Winfried Keiper
Konstanz, Germany Stefan Volkwein
Contents
An Iterative Model Reduction Scheme for Quadratic-Bilinear
Descriptor Systems with an Application to Navier–Stokes
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Peter Benner and Pawan Goyal
Greedy Kernel Approximation for Sparse
Surrogate Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Bernard Haasdonk and Gabriele Santin
Set-Oriented Multiobjective Optimal Control of PDEs
Using Proper Orthogonal Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 47
Dennis Beermann, Michael Dellnitz, Sebastian Peitz
and Stefan Volkwein
Sequential Reduced-Order Modeling for Time-Dependent
Optimization Problems with Initial Value Controls . . . . . . . . . . . . . . . . 73
Matthias Heinkenschloss and Dörte Jando
A Direct Index 1 DAE Model of Gas Networks . . . . . . . . . . . . . . . . . . . 99
Peter Benner, Maike Braukmüller and Sara Grundel
Model Order Reduction for Rotating Electrical Machines . . . . . . . . . . . 121
Zeger Bontinck, Oliver Lass, Oliver Rain and Sebastian Schöps
Morembs—A Model Order Reduction Package for Elastic
Multibody Systems and Beyond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Jörg Fehr, Dennis Grunert, Philip Holzwarth, Benjamin Fröhlich,
Nadine Walker and Peter Eberhard
Model Order Reduction a Key Technology for Digital Twins . . . . . . . . 167
Dirk Hartmann, Matthias Herz and Utz Wever
ix
An Iterative Model Reduction Scheme
for Quadratic-Bilinear Descriptor
Systems with an Application to
Navier–Stokes Equations
PeterBennerandPawanGoyal
Abstract We discuss an interpolatory model reduction framework for quadratic-
bilinear(QB)descriptorsystems,arisingespeciallyfromthesemi-discretizationof
the Navier–Stokes equations. Several recent results indicate that directly applying
interpolatory model reduction frameworks, developed for systems of ordinary dif-
ferentialequations,todescriptorsystems,mayleadtoanunboundederrorbetween
theoriginalandreduced-ordersystems,e.g.,intheH -norm,duetoaninappropri-
2
atetreatmentofthepolynomialpartoftheoriginalsystem.Thus,themaingoalof
thisarticleistoextendtherecentlystudiedinterpolation-basedoptimalmodelreduc-
tionframeworkforQBordinarydifferentialequations(QBODEs)toaforementioned
descriptorsystemswhileensuringboundederror.Forthis,wefirstaimattransform-
ingthedescriptorsystemintoanequivalentODEsystembymeansofprojectorsfor
which standard model reduction techniques can be applied. Subsequently, we dis-
cusshowtoconstructoptimalreducedsystemscorrespondingtoanequivalentODE,
withoutrequiringexplicitcomputationoftheexpensiveprojectionusedintheanal-
ysis.Theefficiencyoftheproposedalgorithmisillustratedbymeansofanumerical
example,obtainedviasemi-discretizationoftheNavier–Stokesequations.
1 Introduction
High-fidelitymodelingofdynamicalsystemsisoftenrequiredtohaveabetterunder-
standing of the underlying dynamical behaviors of a system. However, numerical
simulationsofsuchhigh-fidelitysystemsareexpensiveandofteninefficient.Thus,
itisnotastraightforwardtask,orsometimesnotevenpossible,toperformengineer-
ing and control design studies using these high-fidelity systems. One approach to
circumvent this problem is model order reduction (MOR), aiming at constructing
B
P.Benner·P.Goyal ( )
MaxPlanckInstituteforDynamicsofComplexTechnicalSystems,
Sandtorstraße1,39106Magdeburg,Germany
e-mail:[e-mail]
P.Benner
e-mail:[e-mail]
©SpringerInternationalPublishingAG,partofSpringerNature2018 1
W.Keiperetal.(eds.),Reduced-OrderModeling(ROM)forSimulationandOptimization,
https://doi.org/10.1007/978-3-319-75319-5_1
2 P.BennerandP.Goyal
surrogatemodels(reduced-ordermodels)whicharelesscomplexandreplicatethe
importantdynamicsofthehigh-fidelitysystem.
Inthispaper,weinvestigateMORforquadratic-bilinear(QB)descriptorsystems
oftheform
(cid:2)m
E11v˙(t)= A11v(t)+A12p(t)+Hv(t)⊗v(t)+ Nkv(t)uk(t)+B1u(t), (1a)
k=1
0= A21v(t)+B2u(t), v(0)=v0, (1b)
y(t)=C1v(t)+C2p(t), (1c)
whereE11,A11,Nk ∈Rnv×nv,k ∈{1,...,m},H ∈Rnv×n2v,A12,A2T1 ∈Rnv×np,B1 ∈
Rnv×m, B2 ∈Rnp×m, C1 ∈Rq×nv, C2 ∈Rq×np; v(t)∈Rnv and p(t)∈Rnp are the
state vectors; u(t)∈Rm and y(t)∈Rq are the control input and measured output
vectors of the system, respectively, and u (t) is the k-th component of the input
k
vector; v ∈Rnv is an initial value for v(t). Furthermore, we assume that E and
0 11
A E−1A areinvertible.Hence,thelinearpartofthesystem(1)(H =0, N =0)
21 11 12 k
hasanindex-2structure,e.g.,see[20].ThestructureoftheQBdescriptorsystem(1)
occursafterthespacediscretizationofacontrolproblemwherethestatefunctions
aredescribedbytheincompressibleNavier–Stokesequations.
MORtechniquesforlinearsystemsarenowverywell-establishedandarewidely
appliedinnumerousapplications,e.g.,see[2,4,14,26].Severalofthosetechniques
havebeensuccessfullyextendedtospecialclassesofnonlinearODEsystems,namely
bilinearandQBsystems,see,e.g.,[6, 7, 9, 10, 12, 16, 18, 27].Thesetechniques
canbeclassifiedmainlyintotwocategories:trajectory-basedmethodsandsystem-
theoreticapproaches.Theprimaryideasoftrajectory-basedmethodsrelyonasetof
snapshotsofthestatesolutionsfortraininginputs,whichisthenusedtodetermine
aGalerkin projection,formoredetails,see,e.g.,[3, 15, 22, 24, 25].Ontheother
hand, in system-theoretic approaches, there are two prominent methods, the so-
called balanced truncated and moment-matching (interpolation) methods that are
widely used, see, e.g., [2]. The idea of balanced truncation is to find states which
are hard to control as well as hard to observe, and truncating such states gives us
areduced-order system.This method forQBsystemshas been recently studiedin
[12].Interpolation-basedmethodsaimatconstructingreduced-ordersystemswhich
approximatetheinput–outputbehaviorofthesystem.Withthisintent,suchaproblem
forQBsystemswasfirstconsideredin[18],whereaone-sidedprojectionmethodto
obtainaninterpolatingreduced-ordersystemisproposed.Lateron,asimilarproblem
wasaddressedin[8, 9]forsingle-inputsingle-output(SISO)QBsystems,wherea
two-sided projection method was proposed, ensuring a higher number of moment
tobematched.However,themainchallengesforthismethodareagoodselection
ofinterpolationpointsandtheapplicationtomulti-inputmulti-output(MIMO)QB
systems. To address these issues, an interpolation-based optimal model reduction
problem for QB systems was recently addressed in [13], where a reduced-order
system is constructed, aiming at minimizing a system norm of the error system,
whereatruncatedH -normissuggestedforthispurpose.
2