Polynomial cost for solving IVP for high-index DAE

Corless, Robert M.; Ilie, Silvana

doi:10.1007/s10543-008-0163-2

Cited by 13 publications

(8 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [10], the Σ-method is used to prove that the computational complexity of solving the initial value problem for a DAE of arbitrarily high index is polynomial in the number of bits of accuracy needed. The Σ-method also guides one to carry out a DDs-based index reduction procedure and to design software code for automating this procedure [31,32,34,35,44,47].…”

mentioning

confidence: 99%

Conversion methods for improving structural analysis of differential-algebraic equation systems

2017

View full text Add to dashboard Cite

Systems of differential-algebraic equations (DAEs) arise in many areas including chemical engineering, electrical circuit simulation, and robotics. Such systems are routinely generated by simulation and modeling environments, like MapleSim, Matlab/Simulink, and those based on the Modelica language. Before a simulation starts and a numerical solution method is applied, some kind of structural analysis (SA) is performed to determine the structure and the index of a DAE system. Structural analysis methods serve as a necessary preprocessing stage, and among them, Pantelides's graph-theory-based algorithm is widely used in industry. Recently, Pryce's Σ-method is becoming increasingly popular, owing to its straightforward approach and capability of analyzing high-order systems. Both methods are equivalent in the sense that (a) when one succeeds, producing a nonsingular Jacobian, the other also succeeds, and that (b) the two give the same structural index in the case of either success or failure. When SA succeeds, the structural results can be used to perform an index reduction process, or to devise a stage-by-stage solution scheme for computing derivatives or Taylor coefficients up to some order.Although such a success occurs on fairly many problems of interest, SA can fail on some simple, solvable DAEs with an identically singular Jacobian, and give incorrect structural information that usually includes the index. In this thesis, we focus on the iii Σ-method and investigate its failures. Aiming at making this SA more reliable, we develop two conversion methods for fixing SA failures. These methods reformulate a DAE on which the Σ-method fails into an equivalent problem on which SA is more likely to succeed with a nonsingular Jacobian. The implementation of our methods requires symbolic computations.We also combine our conversion methods with block triangularization of a DAE.Using a block triangular form of a Jacobian sparsity pattern, we identify which diagonal block(s) of the Jacobian is identically singular, and then perform a conversion on each singular block. This approach can reduce the computational cost and improve the efficiency of finding a suitable conversion for fixing SA's failures.

show abstract

mentioning

confidence: 99%

Conversion methods for improving structural analysis of differential-algebraic equation systems

2017

View full text Add to dashboard Cite

show abstract

“…But, in step (C.1.c), the number of stages is restricted to k = 0, 1. The step control is given by (8) with β = 0.9, ρ = 1, κ = 1/8, andỹ n+1 is obtained to order 7 by the step control predictorỹ…”

Section: The Solver Dp(87)daementioning

confidence: 99%

“…The global error of a numerical solution at final time t N = t f is the uniform vector norm y N − y(t f ) ∞ of the difference between the numerical solution y N and a reference solution found by DP (8,7)13M at tolerance 10 −12 or in [15], depending on the problem.…”

Section: Nomenclaturementioning

confidence: 99%

See 1 more Smart Citation

One-step 9-stage Hermite–Birkhoff–Taylor DAE solver of order 10

Nguyen‐Ba

Han

Yagoub

et al. 2009

J. Appl. Math. Comput.

View full text Add to dashboard Cite

The HBT(10)9 method for ODEs is expanded into HBT(10)9DAE for solving nonstiff and moderately stiff systems of fully implicit differential algebraic equations (DAEs) of arbitrarily high fixed index. A scheme to generate first-order derivatives at off-step points is combined with Pryce scheme which generates high order derivatives at step points. The stepsize is controlled by a local error estimator. HBT(10)9DAE uses only the first four derivatives of y instead of the first 10 required by Taylor's series method T10DAE of order 10. Dormand-Prince's DP(8,7)13M for ODEs is extended to DP(8,7)DAE for DAEs. HBT(10)9DAE wins over DP(8,7)DAE on several test problems on the basis of CPU time as a function of relative error at the end of the interval of integration. An index-5 problem is equally well solved by HBT(10)9DAE and T10DAE. On this problem, the error in the solution by DP(8,7)DAE increases as time increases.

show abstract

“…Our complexity analysis is based on a recent model of computation, which was introduced by Corless in [5] for solving ordinary differential equations and which was extended to solving differential algebraic equations in [6,14,15]. Under the assumptions of the new model, we show that the computational cost of extrapolation methods with different number sequences exhibit different behavior depending on the rate of growth of the sequence.…”

Section: Introductionmentioning

confidence: 99%

The Computational Complexity of Extrapolation Methods

2008

Self Cite

View full text Add to dashboard Cite

This paper analyzes the cost of extrapolation methods for non-stiff ordinary differential equations depending on the number of digits of accuracy requested. Extrapolation of the explicit midpoint rule is applied for various number sequences. We show that for sequences with arithmetic growth, the cost of the method is polynomial in the number of digits of accuracy, while for sequences of numbers with geometric growth, the cost is super-polynomial with respect to the same parameter. Mathematics Subject Classification (2000). 65L06, 68Q25.

show abstract

Polynomial cost for solving IVP for high-index DAE

Cited by 13 publications

References 28 publications

Conversion methods for improving structural analysis of differential-algebraic equation systems

Conversion methods for improving structural analysis of differential-algebraic equation systems

One-step 9-stage Hermite–Birkhoff–Taylor DAE solver of order 10

The Computational Complexity of Extrapolation Methods

Contact Info

Product

Resources

About