Index Reduction for Differential-Algebraic Equations with Mixed Matrices

Differential-algebraic equations (DAEs) are widely used for modeling of dynamical systems. The difficulty in solving numerically 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 existing simulation software packages for dynamical systems are equipped with an index-reduction algorithm given by Mattsson and S\"{o}derlind. 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 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. This paper presents 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--S\"{o}derlind'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. Furthermore, we provide an improved algorithm under an assumption based on dimensional analysis of dynamical systems.Comment: A preliminary version of this paper is to appear in Proceedings of the Eighth SIAM Workshop on Combinatorial Scientific Computing, Bergen, Norway, June 201

Optimizing Train Stopping Patterns for Congestion Management

In this paper, we optimize train stopping patterns during morning rush hour in Japan. Since trains are extremely crowded, we need to determine stopping patterns based not only on travel time but also on congestion rates of trains. We exploit a Wardrop equilibrium model to compute passenger flows subject to congestion phenomena and present an efficient local search algorithm to optimize stopping patterns which iteratively computes a Wardrop equilibrium. We apply our algorithm to railway lines in Tokyo including Keio Line with six types of trains and succeed in relaxing congestion with a small effect on travel time

Hybrid analysis of nonlinear time-varying circuits providing DAEs with index at most one.

Abstract Commercial packages for transient circuit simulation are often based on the modified nodal analysis (MNA) which allows an automatic setup of model equations and requires a nearly minimal number of variables. However, it may lead to differential-algebraic equations (DAEs) with higher index. Here, we present a hybrid analysis for nonlinear time-varying circuits leading to DAEs with index at most one. This hybrid analysis is based merely on the network topology, which possibly leads to an automatic setup of the hybrid equations from netlists. Moreover, we prove that the minimum index of the DAE arising from the hybrid analysis never exceeds the index from MNA. As a positive side effect, the number of equations from the hybrid analysis is always no greater than that one from MNA. This suggests that the hybrid analysis is superior to MNA in numerical accuracy and computational effort

Optimizing train stopping patterns for congestion management

In this paper, we optimize train stopping patterns during the morning rush hour in Japan. Since trains are extremely crowded, we need to determine stopping patterns based not only on travel time but also on congestion rates of trains. We exploit a Wardrop equilibrium model to compute passenger flows subject to congestion phenomena and present an efficient local search algorithm to optimize stopping patterns which iteratively computes a Wardrop equilibrium. The framework of the proposed algorithm is extended to solve the problem of optimizing the number of services for each train type. We apply our algorithms to railway lines in Tokyo including the Keio Line with six types of trains and demonstrate that we succeed in relaxing congestion