Computational complexity of numerical solutions of initial value problems for differential algebraic equations (abstract only)
Pages 21 - 22
Abstract
We investigate the cost of solving initial value problems for differential algebraic equations depending on the number of digits of accuracy requested. A recent result showed that the cost of solving initial value problems (IVP) for ordinary differential equations (ODE) is polynomial in the number of digits of accuracy. This improves on the classical result of information-based complexity, which predicts exponential cost. The new theory is based on more realistic assumptions.
The algorithm analyzed in this thesis is based on a previously published Taylor series method for solving a general class of differential algebraic equations. We consider DAE of constant index to which the method applies. The DAE is allowed to be of arbitrary index, fully implicit and have derivatives of order higher than one.
Similarly, by considering a realistic model, we show that the cost of computing the solution of IVP for DAE with the algorithm adopted and by using automatic differentiation is polynomial in the number of digits of accuracy. We also show that non-adaptation is more expensive than adaptation, giving thus a theoretical justification of the success of adaptivity in practice.
A particular case frequently arising in practical applications, the index-1 DAE, is treated separately, in more depth. On the other hand, an analysis of the higher-index DAE is significantly more complicated and applies to a wider class of problems. In both cases, continuous output is also given.
These results apply to many important problems arising in practice. We present an interesting theoretical application to polynomial system solving.
Index Terms
- Computational complexity of numerical solutions of initial value problems for differential algebraic equations (abstract only)
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Copyright © 2008 Author.
Publisher
Association for Computing Machinery
New York, NY, United States
Publication History
Published: 25 July 2008
Published in SIGSAM-CCA Volume 42, Issue 1-2
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