Projected explicit and implicit Taylor series methods for DAEs
The recently developed new algorithm for computing consistent initial values and Taylor coefficients for DAEs using projector-based constrained optimization opens new possibilities to apply Taylor series integration methods. In this paper, we show how corresponding projected explicit and implicit Ta...
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| Published in | Numerical algorithms Vol. 88; no. 2; pp. 615 - 646 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.10.2021
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1017-1398 1572-9265 1572-9265 |
| DOI | 10.1007/s11075-020-01051-z |
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| Abstract | The recently developed new algorithm for computing consistent initial values and Taylor coefficients for DAEs using projector-based constrained optimization opens new possibilities to apply Taylor series integration methods. In this paper, we show how corresponding projected explicit and implicit Taylor series methods can be adapted to DAEs of arbitrary index. Owing to our formulation as a projected optimization problem constrained by the derivative array, no explicit description of the inherent dynamics is necessary, and various Taylor integration schemes can be defined in a general framework. In particular, we address higher-order Padé methods that stand out due to their stability. We further discuss several aspects of our prototype implemented in Python using Automatic Differentiation. The methods have been successfully tested on examples arising from multibody systems simulation and a higher-index DAE benchmark arising from servo-constraint problems. |
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| AbstractList | The recently developed new algorithm for computing consistent initial values and Taylor coefficients for DAEs using projector-based constrained optimization opens new possibilities to apply Taylor series integration methods. In this paper, we show how corresponding projected explicit and implicit Taylor series methods can be adapted to DAEs of arbitrary index. Owing to our formulation as a projected optimization problem constrained by the derivative array, no explicit description of the inherent dynamics is necessary, and various Taylor integration schemes can be defined in a general framework. In particular, we address higher-order Padé methods that stand out due to their stability. We further discuss several aspects of our prototype implemented in Python using Automatic Differentiation. The methods have been successfully tested on examples arising from multibody systems simulation and a higher-index DAE benchmark arising from servo-constraint problems. |
| Author | Estévez Schwarz, Diana Lamour, René |
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| Cites_doi | 10.1007/s002110050165 10.1007/978-3-030-53905-4_14 10.1142/6746 10.1007/s11075-017-0379-9 10.1007/3-540-62598-4_85 10.1016/j.cam.2013.09.018 10.1007/978-3-030-53905-4_1 10.1016/j.amc.2004.02.015 10.5194/ms-4-113-2013 10.1016/j.jocs.2011.10.007 10.1023/A:1019150322187 10.4171/017 10.1007/s11075-016-0107-x 10.1016/S0898-1221(97)00117-X 10.1080/10556788.2018.1428605 10.1007/978-3-642-05221-7 10.1007/978-3-642-27555-5 10.1002/cta.168 10.1137/0906024 10.1007/s11075-009-9304-1 10.1016/j.cam.2019.112486 |
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| Keywords | Consistent initial value Integration Projector-based analysis DAE Index Taylor series methods Derivative array Differential-algebraic equation Nonlinear constrained optimization Automatic differentiation |
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| References | Scott, J.R., Solving, ODE: Initial Value Problems with Implicit Taylor Series Methods. Technical report NASA/TM-2000-209400 (2000) Estévez Schwarz, D., Lamour, R.: A projector based decoupling of DAEs obtained from the derivative array. In: Progress in Differential-Algebraic Equations II, Differential-Algebraic Equations Forum (DAE-F) (2020) PryceJDSolving high-index DAEs by Taylor seriesNumer. Algorithms1998191–4195211166829910.1023/A:1019150322187 PryceJDNedialkovNSTanGLiXHow AD can help solve differential-algebraic equationsOptim. Methods Softw.2018334–6729749385321110.1080/10556788.2018.1428605 Estévez Schwarz, D., Lamour, R.: InitDAE: Computation of consistent values, index determination and diagnosis of singularities of DAEs using automatic differentiation in Python. Journal of Computational and Applied Mathematics. https://doi.org/10.1016/j.cam.2019.112486 (2019) OttoSSeifriedRApplications of Differential-Algebraic Equations: Examples and Benchmarks, chapter Open-loop Control of Underactuated Mechanical Systems Using Servo-constraints: Analysis and Some Examples. Differential-Algebraic Equations Forum2019ChamSpringer07218212 SeifriedRBlajerWAnalysis of servo-constraint problems for underactuated multibody systemsMech. Sci.2013411312910.5194/ms-4-113-2013 CampbellSLGearCWThe index of general nonlinear DAEsNumer. Math.1995722173196136225910.1007/s002110050165 AbramowitzMStegunIAHandbook of Mathematical Functions with Formulas Graphs and Mathematical Tables1972New YorkDover0543.33001 Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Springer (1996) WalterSFLehmannLAlgorithmic differentiation in Python with AlgoPyJ. Comput. Sci.20134533434410.1016/j.jocs.2011.10.007 Estévez Schwarz, D., Lamour, R., März, R.: Singularities of the Robotic Arm DAE. Progress in Differential-Algebraic Equations II, Differential-Algebraic Equations Forum (DAE-F) (2020) RiazaRDifferential-Algebraic Systems. Analytical Aspects and Circuit Applications2008HackensackWorld Scientific10.1142/6746 Estévez SchwarzDLamourRA new approach for computing consistent initial values and Taylor coefficients for DAEs using projector-based constrained optimizationNumer. Algorithms2018782355377380335010.1007/s11075-017-0379-9 Estévez SchwarzDA step-by-step approach to compute a consistent initialization for the MNAInt. J. Circuit Theory Appl.20023011610.1002/cta.168 BarrioRPerformance of the Taylor series method for ODEs/DAEsAppl. Math. Comput.2005163252554521218081067.65063 BrenanKECampbellSLPetzoldLRNumerical Solution of Initial-Value Problems in Differential-Algebraic Equations. Unabridged, corr. republ. Classics in Applied Mathematics1996PhiladelphiaSIAM Society for Industrial and Applied Mathematics1414 Estévez SchwarzDConsistent initialization for DAEs in Hessenberg formNumer. Algorithms2009524629648256371810.1007/s11075-009-9304-1 Estévez Schwarz, D., Lamour, R.: InitDAE’s documentation. Available from: https://www.mathematik.hu-berlin.de/~lamour/software/python/InitDAE/html Estévez SchwarzDLamourRConsistent initialization for higher-index DAEs using a projector based minimum-norm specification. Technical Report 12016Humboldt-Universität zu BerlinInstitut für Mathematik Kirlinger, G., Corliss, G.F.: On implicit Taylor series methods for stiff ODEs. In: Computer Arithmetic and Enclosure Methods. Proceedings of the 3rd International IMACS-GAMM Symposium on Computer Arithmetic and Scientific Computing (SCAN-91), Oldenburg, Germany, 1-4 October 1991, pp 371–379, Amsterdam (1992) DeuflhardPBornemannFNumerical mathematics 2. Ordinary differential equations. (Numerische Mathematik 2 Gewöhnliche Differentialgleichungen.) 4th revised and augmented ed.2013Berlinde Gruyter Studium1273.65001 Estévez SchwarzDLamourRA new projector based decoupling of linear DAEs for monitoring singularitiesNumer. Algorithms2016732535565354909110.1007/s11075-016-0107-x Mazzia, F., Magherini, C.: Test set for initial value problems, release 2.4. Technical report, Department of Mathematics, University of Bari and INdAM, Research Unit of Bari, February 2008. Available from: http://pitagora.dm.uniba.it/~testset AkishinPGPuzyninIVVinitskySIA hybrid numerical method for analysis of dynamics of the classical Hamiltonian systemsComput. Math. Appl.1997342-44573147875210.1016/S0898-1221(97)00117-X Corliss, G.F., Griewank, A., Henneberger, P., Kirlinger, G., Potra, F.A., Stetter, H.J.: High-order stiff ODE solvers via automatic differentiation and rational prediction. In: Vulkov, L., Waśniewski, J., Yalamov, P. (eds.) Numerical Analysis and Its Applications. WNAA 1996. Lecture Notes in Computer Science, vol. 1196, pp 114–124 (1997) Estévez SchwarzDLamourRProjector based integration of DAEs with the Taylor series method using automatic differentiationJ. Comput. Appl Math.20142626272316230410.1016/j.cam.2013.09.018 LamourRMärzRTischendorfCDifferential-Algebraic Equations: A Projector Based Analysis. Differential-Algebraic Equations Forum, vol. 12013BerlinSpringer10.1007/978-3-642-27555-5 KunkelPMehrmannVDifferential-Algebraic Equations - Analysis and Numerical Solution2006ZürichEMS Publishing House10.4171/017 CampbellSLThe numerical solution of higher index linear time varying singular systems of differential equationsSIAM J. Sci. Stat. Comput.1985633434877940910.1137/0906024 Dimova, S.N., Hristov, I.G., Hristova, R.D., Puzynin, I.V., Puzynina, T.P., Sharipov, Z.A., Shegunov, N.G., Tukhliev, Z.K.: Combined explicit-implicit Taylor Series Methods. In: Proceedings of the VIII International Conference “Distributed Computing and Grid-technologies in Science and Education” (GRID 2018), Dubna, Moscow region, Russia, September 10 -14 (2018) D Estévez Schwarz (1051_CR16) 2018; 78 1051_CR21 1051_CR20 SL Campbell (1051_CR6) 1995; 72 R Seifried (1051_CR30) 2013; 4 1051_CR24 M Abramowitz (1051_CR1) 1972 D Estévez Schwarz (1051_CR13) 2014; 262 JD Pryce (1051_CR27) 2018; 33 D Estévez Schwarz (1051_CR11) 2009; 52 1051_CR29 SF Walter (1051_CR31) 2013; 4 R Barrio (1051_CR3) 2005; 163 R Riaza (1051_CR28) 2008 JD Pryce (1051_CR26) 1998; 19 D Estévez Schwarz (1051_CR10) 2002; 30 R Lamour (1051_CR23) 2013 P Kunkel (1051_CR22) 2006 S Otto (1051_CR25) 2019 P Deuflhard (1051_CR8) 2013 D Estévez Schwarz (1051_CR14) 2016; 73 1051_CR12 1051_CR19 1051_CR18 1051_CR17 PG Akishin (1051_CR2) 1997; 34 1051_CR7 1051_CR9 KE Brenan (1051_CR4) 1996 SL Campbell (1051_CR5) 1985; 6 D Estévez Schwarz (1051_CR15) 2016 |
| References_xml | – reference: Mazzia, F., Magherini, C.: Test set for initial value problems, release 2.4. Technical report, Department of Mathematics, University of Bari and INdAM, Research Unit of Bari, February 2008. Available from: http://pitagora.dm.uniba.it/~testset – reference: CampbellSLGearCWThe index of general nonlinear DAEsNumer. Math.1995722173196136225910.1007/s002110050165 – reference: Estévez SchwarzDA step-by-step approach to compute a consistent initialization for the MNAInt. J. Circuit Theory Appl.20023011610.1002/cta.168 – reference: Estévez SchwarzDLamourRConsistent initialization for higher-index DAEs using a projector based minimum-norm specification. Technical Report 12016Humboldt-Universität zu BerlinInstitut für Mathematik – reference: Estévez SchwarzDLamourRA new approach for computing consistent initial values and Taylor coefficients for DAEs using projector-based constrained optimizationNumer. Algorithms2018782355377380335010.1007/s11075-017-0379-9 – reference: AbramowitzMStegunIAHandbook of Mathematical Functions with Formulas Graphs and Mathematical Tables1972New YorkDover0543.33001 – reference: Kirlinger, G., Corliss, G.F.: On implicit Taylor series methods for stiff ODEs. In: Computer Arithmetic and Enclosure Methods. Proceedings of the 3rd International IMACS-GAMM Symposium on Computer Arithmetic and Scientific Computing (SCAN-91), Oldenburg, Germany, 1-4 October 1991, pp 371–379, Amsterdam (1992) – reference: Estévez SchwarzDConsistent initialization for DAEs in Hessenberg formNumer. Algorithms2009524629648256371810.1007/s11075-009-9304-1 – reference: RiazaRDifferential-Algebraic Systems. Analytical Aspects and Circuit Applications2008HackensackWorld Scientific10.1142/6746 – reference: OttoSSeifriedRApplications of Differential-Algebraic Equations: Examples and Benchmarks, chapter Open-loop Control of Underactuated Mechanical Systems Using Servo-constraints: Analysis and Some Examples. Differential-Algebraic Equations Forum2019ChamSpringer07218212 – reference: Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Springer (1996) – reference: Estévez Schwarz, D., Lamour, R.: InitDAE’s documentation. Available from: https://www.mathematik.hu-berlin.de/~lamour/software/python/InitDAE/html/ – reference: Estévez Schwarz, D., Lamour, R., März, R.: Singularities of the Robotic Arm DAE. Progress in Differential-Algebraic Equations II, Differential-Algebraic Equations Forum (DAE-F) (2020) – reference: AkishinPGPuzyninIVVinitskySIA hybrid numerical method for analysis of dynamics of the classical Hamiltonian systemsComput. Math. Appl.1997342-44573147875210.1016/S0898-1221(97)00117-X – reference: SeifriedRBlajerWAnalysis of servo-constraint problems for underactuated multibody systemsMech. Sci.2013411312910.5194/ms-4-113-2013 – reference: KunkelPMehrmannVDifferential-Algebraic Equations - Analysis and Numerical Solution2006ZürichEMS Publishing House10.4171/017 – reference: PryceJDSolving high-index DAEs by Taylor seriesNumer. Algorithms1998191–4195211166829910.1023/A:1019150322187 – reference: BrenanKECampbellSLPetzoldLRNumerical Solution of Initial-Value Problems in Differential-Algebraic Equations. Unabridged, corr. republ. Classics in Applied Mathematics1996PhiladelphiaSIAM Society for Industrial and Applied Mathematics1414 – reference: CampbellSLThe numerical solution of higher index linear time varying singular systems of differential equationsSIAM J. Sci. Stat. Comput.1985633434877940910.1137/0906024 – reference: PryceJDNedialkovNSTanGLiXHow AD can help solve differential-algebraic equationsOptim. Methods Softw.2018334–6729749385321110.1080/10556788.2018.1428605 – reference: Estévez Schwarz, D., Lamour, R.: A projector based decoupling of DAEs obtained from the derivative array. In: Progress in Differential-Algebraic Equations II, Differential-Algebraic Equations Forum (DAE-F) (2020) – reference: Corliss, G.F., Griewank, A., Henneberger, P., Kirlinger, G., Potra, F.A., Stetter, H.J.: High-order stiff ODE solvers via automatic differentiation and rational prediction. In: Vulkov, L., Waśniewski, J., Yalamov, P. (eds.) Numerical Analysis and Its Applications. WNAA 1996. Lecture Notes in Computer Science, vol. 1196, pp 114–124 (1997) – reference: Dimova, S.N., Hristov, I.G., Hristova, R.D., Puzynin, I.V., Puzynina, T.P., Sharipov, Z.A., Shegunov, N.G., Tukhliev, Z.K.: Combined explicit-implicit Taylor Series Methods. In: Proceedings of the VIII International Conference “Distributed Computing and Grid-technologies in Science and Education” (GRID 2018), Dubna, Moscow region, Russia, September 10 -14 (2018) – reference: Estévez Schwarz, D., Lamour, R.: InitDAE: Computation of consistent values, index determination and diagnosis of singularities of DAEs using automatic differentiation in Python. Journal of Computational and Applied Mathematics. https://doi.org/10.1016/j.cam.2019.112486 (2019) – reference: WalterSFLehmannLAlgorithmic differentiation in Python with AlgoPyJ. Comput. Sci.20134533434410.1016/j.jocs.2011.10.007 – reference: LamourRMärzRTischendorfCDifferential-Algebraic Equations: A Projector Based Analysis. Differential-Algebraic Equations Forum, vol. 12013BerlinSpringer10.1007/978-3-642-27555-5 – reference: DeuflhardPBornemannFNumerical mathematics 2. Ordinary differential equations. (Numerische Mathematik 2 Gewöhnliche Differentialgleichungen.) 4th revised and augmented ed.2013Berlinde Gruyter Studium1273.65001 – reference: Estévez SchwarzDLamourRA new projector based decoupling of linear DAEs for monitoring singularitiesNumer. Algorithms2016732535565354909110.1007/s11075-016-0107-x – reference: Estévez SchwarzDLamourRProjector based integration of DAEs with the Taylor series method using automatic differentiationJ. Comput. Appl Math.20142626272316230410.1016/j.cam.2013.09.018 – reference: BarrioRPerformance of the Taylor series method for ODEs/DAEsAppl. Math. Comput.2005163252554521218081067.65063 – reference: Scott, J.R., Solving, ODE: Initial Value Problems with Implicit Taylor Series Methods. Technical report NASA/TM-2000-209400 (2000) – volume: 72 start-page: 173 issue: 2 year: 1995 ident: 1051_CR6 publication-title: Numer. Math. doi: 10.1007/s002110050165 – ident: 1051_CR19 doi: 10.1007/978-3-030-53905-4_14 – volume-title: Differential-Algebraic Systems. 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| SubjectTerms | Algebra Algorithms Approximation Computer Science Constraints Linear algebra Methods Multibody systems Numeric Computing Numerical Analysis Optimization Original Paper Systems simulation Taylor series Theory of Computation |
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| Title | Projected explicit and implicit Taylor series methods for DAEs |
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