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| Presentation | 2020-10-23 13:15 [Invited Talk] Index reduction for differential-algebraic equations with mixed matrices Satoru Iwata, Taihei Oki (The Univ. of Tokyo), Mizuyo Takamatsu (Chuo Univ.)  COMP2020-12 |
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| Abstract | (in Japanese) | (See Japanese page) |
| (in English) | 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. |
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| Keyword | (in Japanese) | (See Japanese page) |
| (in English) | Differential-algebraic equations / index reduction / combinatorial relaxation / matroid theory / combinatorial matrix theory / combinatorial scientific computing / / | |
| Reference Info. | IEICE Tech. Rep., vol. 120, no. 209, COMP2020-12, pp. 9-9, Oct. 2020. | |
| Paper # | COMP2020-12 | |
| Date of Issue | 2020-10-16 (COMP) | |
| ISSN | Online edition: ISSN 2432-6380 | |
| Copyright and reproduction |
All rights are reserved and no part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Notwithstanding, instructors are permitted to photocopy isolated articles for noncommercial classroom use without fee. (License No.: 10GA0019/12GB0052/13GB0056/17GB0034/18GB0034) | |
| Download PDF | COMP2020-12 |
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| Conference Information | |
| Committee | COMP |
| Conference Date | 2020-10-23 - 2020-10-23 |
| Place (in Japanese) | (See Japanese page) |
| Place (in English) | Osaka Univ. |
| Topics (in Japanese) | (See Japanese page) |
| Topics (in English) | |
| Paper Information | |
| Registration To | COMP |
| Conference Code | 2020-10-COMP |
| Language | Japanese |
| Title (in Japanese) | (See Japanese page) |
| Sub Title (in Japanese) | (See Japanese page) |
| Title (in English) | Index reduction for differential-algebraic equations with mixed matrices |
| Sub Title (in English) | |
| Keyword(1) | Differential-algebraic equations |
| Keyword(2) | index reduction |
| Keyword(3) | combinatorial relaxation |
| Keyword(4) | matroid theory |
| Keyword(5) | combinatorial matrix theory |
| Keyword(6) | combinatorial scientific computing |
| Keyword(7) | |
| Keyword(8) | |
| 1st Author's Name | Satoru Iwata |
| 1st Author's Affiliation | The University of Tokyo (The Univ. of Tokyo) |
| 2nd Author's Name | Taihei Oki |
| 2nd Author's Affiliation | The University of Tokyo (The Univ. of Tokyo) |
| 3rd Author's Name | Mizuyo Takamatsu |
| 3rd Author's Affiliation | Chuo University (Chuo Univ.) |
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| Speaker | Author-2 |
| Date Time | 2020-10-23 13:15:00 |
| Presentation Time | 60 minutes |
| Registration for | COMP |
| Paper # | COMP2020-12 |
| Volume (vol) | vol.120 |
| Number (no) | no.209 |
| Page | p.9 |
| #Pages | 1 |
| Date of Issue | 2020-10-16 (COMP) |
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