A graph-theoretic approach to sparse matrix inversion for implicit differential algebraic equations

    Research output: Contribution to journalArticle

    Abstract

    In this paper, we propose an efficient numerical scheme to compute sparse matrix inversions for Implicit Differential Algebraic Equations of large-scale nonlinear mechanical systems. We first formulate mechanical systems with constraints by Dirac structures and associated Lagrangian systems. Second, we show how to allocate <i>input-output relations</i> to the variables in kinematical and dynamical relations appearing in DAEs by introducing an oriented bipartite graph. Then, we also show that the matrix inversion of Jacobian matrix associated to the kinematical and dynamical relations can be carried out by using the input-output relations and we explain solvability of the sparse Jacobian matrix inversion by using the bipartite graph. Finally, we propose an efficient symbolic generation algorithm to compute the sparse matrix inversion of the Jacobian matrix, and we demonstrate the validity in numerical efficiency by an example of the stanford manipulator.

    Original languageEnglish
    Pages (from-to)243-250
    Number of pages8
    JournalMechanical Sciences
    Volume4
    Issue number1
    DOIs
    Publication statusPublished - 2013

    Fingerprint

    Jacobian matrices
    Differential equations
    Manipulators

    ASJC Scopus subject areas

    • Fluid Flow and Transfer Processes
    • Civil and Structural Engineering
    • Control and Systems Engineering
    • Industrial and Manufacturing Engineering
    • Mechanical Engineering
    • Mechanics of Materials

    Cite this

    A graph-theoretic approach to sparse matrix inversion for implicit differential algebraic equations. / Yoshimura, Hiroaki.

    In: Mechanical Sciences, Vol. 4, No. 1, 2013, p. 243-250.

    Research output: Contribution to journalArticle

    @article{f0b7a6a357ff4e5e9763103448e73499,
    title = "A graph-theoretic approach to sparse matrix inversion for implicit differential algebraic equations",
    abstract = "In this paper, we propose an efficient numerical scheme to compute sparse matrix inversions for Implicit Differential Algebraic Equations of large-scale nonlinear mechanical systems. We first formulate mechanical systems with constraints by Dirac structures and associated Lagrangian systems. Second, we show how to allocate input-output relations to the variables in kinematical and dynamical relations appearing in DAEs by introducing an oriented bipartite graph. Then, we also show that the matrix inversion of Jacobian matrix associated to the kinematical and dynamical relations can be carried out by using the input-output relations and we explain solvability of the sparse Jacobian matrix inversion by using the bipartite graph. Finally, we propose an efficient symbolic generation algorithm to compute the sparse matrix inversion of the Jacobian matrix, and we demonstrate the validity in numerical efficiency by an example of the stanford manipulator.",
    author = "Hiroaki Yoshimura",
    year = "2013",
    doi = "10.5194/ms-4-243-2013",
    language = "English",
    volume = "4",
    pages = "243--250",
    journal = "Mechanical Sciences",
    issn = "2191-9151",
    publisher = "Copernicus GmbH",
    number = "1",

    }

    TY - JOUR

    T1 - A graph-theoretic approach to sparse matrix inversion for implicit differential algebraic equations

    AU - Yoshimura, Hiroaki

    PY - 2013

    Y1 - 2013

    N2 - In this paper, we propose an efficient numerical scheme to compute sparse matrix inversions for Implicit Differential Algebraic Equations of large-scale nonlinear mechanical systems. We first formulate mechanical systems with constraints by Dirac structures and associated Lagrangian systems. Second, we show how to allocate input-output relations to the variables in kinematical and dynamical relations appearing in DAEs by introducing an oriented bipartite graph. Then, we also show that the matrix inversion of Jacobian matrix associated to the kinematical and dynamical relations can be carried out by using the input-output relations and we explain solvability of the sparse Jacobian matrix inversion by using the bipartite graph. Finally, we propose an efficient symbolic generation algorithm to compute the sparse matrix inversion of the Jacobian matrix, and we demonstrate the validity in numerical efficiency by an example of the stanford manipulator.

    AB - In this paper, we propose an efficient numerical scheme to compute sparse matrix inversions for Implicit Differential Algebraic Equations of large-scale nonlinear mechanical systems. We first formulate mechanical systems with constraints by Dirac structures and associated Lagrangian systems. Second, we show how to allocate input-output relations to the variables in kinematical and dynamical relations appearing in DAEs by introducing an oriented bipartite graph. Then, we also show that the matrix inversion of Jacobian matrix associated to the kinematical and dynamical relations can be carried out by using the input-output relations and we explain solvability of the sparse Jacobian matrix inversion by using the bipartite graph. Finally, we propose an efficient symbolic generation algorithm to compute the sparse matrix inversion of the Jacobian matrix, and we demonstrate the validity in numerical efficiency by an example of the stanford manipulator.

    UR - http://www.scopus.com/inward/record.url?scp=84897762409&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=84897762409&partnerID=8YFLogxK

    U2 - 10.5194/ms-4-243-2013

    DO - 10.5194/ms-4-243-2013

    M3 - Article

    VL - 4

    SP - 243

    EP - 250

    JO - Mechanical Sciences

    JF - Mechanical Sciences

    SN - 2191-9151

    IS - 1

    ER -