講演抄録/キーワード |
講演名 |
2020-10-23 13:15
[招待講演]混合行列を係数とする微分代数方程式の指数減少法 岩田 覚・○大城泰平(東大)・高松瑞代(中大) COMP2020-12 |
抄録 |
(和) |
(まだ登録されていません) |
(英) |
Differential-algebraic equations (DAEs) are widely used for the modeling of dynamical systems. The difficulty in numerically solving a DAE is measured by its differentiation index. For highly accurate simulation of dynamical systems, it is important to convert high-index
DAEs into low-index DAEs. Most of the existing simulation software packages for dynamical systems are equipped with an index-reduction algorithm given by Mattsson and Söderlind. Unfortunately, this algorithm fails if there are numerical cancellations.
These numerical cancellations are often caused by accurate constants in structural equations. Distinguishing those accurate constants from generic parameters that represent physical quantities, Murota and Iri introduced the notion of a mixed matrix as a mathematical tool for faithful model description in a structural approach to systems analysis. For DAEs described with the use of mixed matrices, efficient algorithms to compute the index have been developed by exploiting matroid theory.
In this talk, we present an index-reduction algorithm for linear DAEs whose coefficient matrices are mixed matrices, i.e., linear DAEs containing physical quantities as parameters. Our algorithm detects numerical cancellations between accurate constants and transforms a DAE into an equivalent DAE to which Mattsson–Söderlind’s index-reduction algorithm is applicable. Our algorithm is based on the combinatorial relaxation approach, which is a framework to solve a linear algebraic problem by iteratively relaxing it into an efficiently solvable combinatorial optimization problem. The algorithm does not rely on symbolic manipulations but on fast combinatorial algorithms on graphs and matroids. Our algorithm is proved to work for any linear DAEs whose coefficient matrices are mixed matrices.
Furthermore, we provide an improved algorithm under an assumption based on dimensional analysis of dynamical systems. Through numerical experiments, it is confirmed that our algorithms run sufficiently fast for large-scale DAEs and output DAEs such that physical meanings of coefficients are easy to interpret. Our algorithms can also be applied
to nonlinear DAEs by regarding nonlinear terms as parameters. |
キーワード |
(和) |
微分代数方程式 / 指数減少 / 組合せ緩和 / マトロイド理論 / 組合せ的行列理論 / 組合せ的科学計算 / / |
(英) |
Differential-algebraic equations / index reduction / combinatorial relaxation / matroid theory / combinatorial matrix theory / combinatorial scientific computing / / |
文献情報 |
信学技報, vol. 120, no. 209, COMP2020-12, pp. 9-9, 2020年10月. |
資料番号 |
COMP2020-12 |
発行日 |
2020-10-16 (COMP) |
ISSN |
Online edition: ISSN 2432-6380 |
著作権に ついて |
技術研究報告に掲載された論文の著作権は電子情報通信学会に帰属します.(許諾番号:10GA0019/12GB0052/13GB0056/17GB0034/18GB0034) |
PDFダウンロード |
COMP2020-12 |
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