Reduced-Order Modeling (ROM) for Simulation and Optimization PDF
Preview Reduced-Order Modeling (ROM) for Simulation and Optimization
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 ©SpringerInternationalPublishingAG,partofSpringerNature2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbytheregisteredcompanySpringerInternationalPublishingAG partofSpringerNature Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland 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:[email protected] P.Benner e-mail:[email protected] ©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
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