Papers
Topics
Authors
Recent
Search
2000 character limit reached

On the computation of accurate initial conditions for linear higher-index differential-algebraic equations and its application in initial value solvers

Published 25 Oct 2024 in math.NA and cs.NA | (2410.19585v2)

Abstract: In contrast to regular ordinary differential equations, the problem of accurately setting initial conditions just emerges in the context of differential-algebraic equations where the dynamic degree of freedom of the system is smaller than the absolute dimension of the described process, and the actual lower-dimensional configuration space of the system is deeply implicit. For linear higher-index differential-algebraic equations, we develop an appropriate numerical method based on properties of canonical subspaces and on the so-called geometric reduction. Taking into account the fact that higher-index differential-algebraic equations lead to ill-posed problems in naturally given norms, we modify this approach to serve as transfer conditions from one time-window to the next in a time stepping procedure and combine it with window-wise overdetermined least-squares collocation to construct the first fully numerical solvers for higher-index initial-value problems.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (63)
  1. Theorie der linearen Operatoren im Hilbert-Raum. Verlag Harry Deutsch, 1977.
  2. E.L. Albasiny. A subroutine for solving a system of differential equations in Chebyshev series. In B. Childs, M. Scott, J.W. Daniel, E. Denman, and P. Nelson, editors, Codes for Boundary-Value problems in Ordinary Differential Equations, volume 76 of Lecture Notes in Computer Science, pages 280–286, Berlin, Heidelberg, New York, 1979. Springer-Verlag.
  3. Lapack Users’ Guide. Third Edition. SIAM, 1999.
  4. Richard Baltensperger. Improving the accuracy of the matrix differentiation method for arbitrary colloation points. Applied Numer Math, 33:143–149, 2000.
  5. Spectral differencing with a twist. SIAM J Sci Comput, 24(5):1465–1487, 2003.
  6. Barycentric Lagrange interpolation. SIAM Review, 46(3):501–517, 2004.
  7. Christian H. Bischof. Incremental condition estimation. SIAM J Matrix Anal Appl, 11(2):312–322, 1990.
  8. Åke Björck. Numerical Methods in Matrix Computations, volume 59 of Texts in Applied Mathematics. Springer, 2015.
  9. Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, volume 14 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics, 1996.
  10. Numerical computation of an analytic singular value decomposition of a matrix valued function. Numer Math, 60(1):1–39, 1991.
  11. S.L. Campbell and E. Moore. Constraint preserving integrators for general nonlinear higher index DAEs. Numer Math, 69:383–399, 1995.
  12. Stephen L. Campbell. A general form for solvable linear time varying singular systems of differential equations. SIAM J Math Anal, 18(4):1101–1115, 1987.
  13. On rank-revealing factorizations. SIAM J Matrix Anal Appl, 15(2):592–622, 1994.
  14. V.F. Chistyakov. On the solution of singular linear systems of ordinary differential equations (in Russioan). In Yu.E. Botarintsev, editor, Vyroshdennye sistemy obyknovennykh differentsial’nykh uravnenij, pages 37–65. Nauka, Sibirskoje otfelenie, 1982.
  15. universal circuitfor studying and generating chaos. I: Routes to chaos. IEEE Trans. Circuit Theory, 40:732–744, 1993.
  16. LINPACK Users’ Guide. SIAM, 1979.
  17. GNU Octave version 6.3.0 manual: a high-level interactive language for numerical computations, 2021.
  18. InitDAE: Computation of consistent values, index determination and diagnosis of singularities of DAEs using automatic differentiation in Python. J Comput Appl Math, 387:112486, 2021.
  19. Diana Estévez Schwarz and René Lamour. Projected explicit and implicit Taylor series methods for DAEs. Numerical Algorithms, 88:615–646, 2021.
  20. M. Galassi et al. GNU Scientific Library Reference Manual. 3rd edition, 2009.
  21. Bengt Fornberg. A practical guide to pseudospectral methods. Cambridge University Press, 1994.
  22. Reliable Calculation of Numerical Rank, Null Space Bases, Pseudoinverse Solutions and Basic Solutions using SuiteSparseQR. ACM Trans on Math Software, 40(1):7:1–7:23, 2013.
  23. Gene H. Golub and Charles F. Van Loan. Matrix Computations. The Johns Hopkins University Press, 2 edition, 1989.
  24. Differential-Algebraic Equations and Their Numerical Treatment, volume 88 of Teubner-Texte zur Mathematik. BSB B.G. Teubner Verlagsgesellschaft, 1986.
  25. Eigen v3.4.0. http://eigen.tuxfamily.org, 2021.
  26. E. Hairer and G. Wanner. Solving ordinary differential equations II. Springer-Verlag, 2nd edition, 1996.
  27. A reliable direct numerical treatment of differential-algebraic equations by overdetermined collocation: An operator approach. J Comput Appl Math, 387:112520, 2021.
  28. Convergence analysis of least-squares collocation metods for nonlinear higher-index differential-algebraic equations. J Comp Appl Math, 387:112514, 2021.
  29. Least-squares collocation for higher-index DAEs: Global approach and attempts towards a time-stepping version. In Timo Reis, Sara Grundel, and Sebastian Schöps, editors, Progress in Differential_Algebraic Equations II, Differential-Algebraic Equation Forum, pages 91–135. Springer-Verlag, 2021.
  30. Towards a reliable implementation of least-squares collocation for higher index differential-algebraic equations—Part 1: basics and ansatz function choices. Numerical Algorithms, 89:931–963, 2022.
  31. Canonical subspaces of linear time-varying differential-algebraic equations and their usefulness for formulating accurate initial conditions. DAE Panel, 1, 2023.
  32. Least-squares collocation for higher-index linear differential-algebraic equations: Estimating the instability threshold. Math Comp, 88:1647–1683, 2019.
  33. Least-squares collocation for linear higher-index differential-algebraic equations. J Comput Appl Math, 317:403–431, 2017.
  34. Richard Haverkamp. Approximationsfehler der Ableitungen von Interpolationspolynomen. J Approx Theory, 30:180–196, 1980.
  35. Richard Haverkamp. Zur Konvergenz der Ableitungen von Interpolationspolynomen. Computing, 32:343–355, 1984.
  36. N.S. Hoang. On node distributions for interpolation and spectral methods. Math Comp, 85(298):667–692, 2015.
  37. The MathWorks Inc. MATLAB version: 9.13.0 (R2022b). Natick, Massachusetts, United States, 2022.
  38. Intel Corp. The Intel oneAPI Math Kernel Library, 2023.
  39. L. Jansen. A dissection concept for DAEs, structural decouplings, unique solvability, convergence theory and half-explicit methods. PhD thesis, Humboldt-Universität, Berlin, 2014.
  40. A convergence analysis of regularization by discretization in preimage space. Math Comp, 81(280):2049–2069, 2012.
  41. Tosio Kato. Perturbation theory for nullity, deficiency and other quantities of linear operators. J Analyse Math, 6:261–322, 1958.
  42. A. Kielbasinski and H. Schwetlick. Numerische Lineare Algebra. VEB Deutscher Verlag der Wissenschaften, 1988.
  43. Smooth factorizations of matrix valued functions and their derivatives. Numer Math, 60(1):115–131, 1991.
  44. Differential-algebraic equations: Analysis and numerical solution. EMS Textbooks in Mathematics. European Mathematical Society Publishing House, 2006.
  45. A new software package for linear differential-algebraic equations. SIAM J Sci Comput, 18:115–138, 1997.
  46. Differential-Algebraic Equations: A Projector Based Analysis. Differential-Algebraic Equations Forum. Springer-Verlag, 2013.
  47. Boundary-value problems for differential-algebraic equations: A survey. In Achim Ilchmann and Timo Reis, editors, Surveys in Differential-Algebraic Equations III, Differential-Algebraic Equation Forum, pages 177–309. Springer, 2015.
  48. Approximation methods for the consistent initialization of differential-algebraic equations. SIAM J Numer Anal, 28(1):205–206, 1991.
  49. Index reduction in differential-algebraic equations using dummy derivatives. SIAM J Sci Comput, 14(3):677–692, 1993.
  50. A unified operator theory of generalized inverses. In M.Z. Nashed, editor, Generalized Inverses and Applications.Proceedings of an Advanced Seminar Sponsored by the Mathematics Research Center, The University of Wisconsin-Madison, October 8 - 1 0 , 197, pages 1–109. Academic Press, 1976.
  51. C.C. Pantelides. The consistent initialization of differential-algebraic systems. SIAM J Sci Stat Comput, 9:213–231, 1988.
  52. J.D. Pryce. Solving high-index DAEs by Taylor series. Numerical Algorithms, 19:195–211, 1998.
  53. Finite difference methods for time dependent, linear differential algebraic equations. Appl Math Lett, 7(2):29–34, 1994.
  54. Theoretical and numerical analysis of differential-algebraic equations. In P.G. Ciarlet and J.L. Lions, editors, Handbook of Numerical Analysis VIII, pages 183–540. Elsevier Science, 2002.
  55. Differential-algebraic equations of index 1 may have arbitrarily high structural index. SIAM J Sci Comput, 21(6):1987–1990, 2000.
  56. Ricardo Riaza. Differential-algebraic systems. Analytical aspects and circuit applications. World Scientific Publishing, 2008.
  57. Some new aspects of rational interpolation. Math Comp, 47(175):285–299, 1986.
  58. Using the Gnu Compiler collection. CreateSpace, Scotts Valley, 2009.
  59. J. Stoer and R. Bulirsch. Introduction to Numerical Analysis, volume 12 of Texts in Applied Mathematics. Springer-Verlag, New York, 1993.
  60. DAESA - Differential-algebraic structural analyzer, Version 1.0, 2014.
  61. Electronic circuit simulation with differential-algebraic equations. In MATHEON–Mathematics for Key Technologies, volume 1 of EMS Series in Industrial and Applied Mathematics, pages 233–242. European Mathmatical Society Publishing House Zürich, 2014.
  62. Explicit barycentric weights for polynomial interpolation in the roots or extrema of classical orthogonal polynomials. Math Comp, 83(290):2893–2914, 2014.
  63. AUGEM: Automatically Generate High Performance Dense Linear Algebra Kernels on x86 CPUs. In International Conference for High Performance Computing, Networking, Storage and Analysis (SC’13), Denver, November 2013.

Summary

No one has generated a summary of this paper yet.

Sign Up to Summarize

Paper to Video (Beta)

No one has generated a video about this paper yet.

Sign Up to Generate All Videos Subscribe on YouTube

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Sign Up to Generate

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Generate Now

Continue Learning

We haven't generated follow-up questions for this paper yet.

Generate Now

Authors (2)

  1. Michael Hanke 
  2. Roswitha März 

Collections

Sign up for free to add this paper to one or more collections.

Sign Up

Tweets

Sign up for free to view the 1 tweet with 0 likes about this paper.

Sign Up for Free