## Numerical Methods and Software for Sensitivity Analysis of Differential-Algebraic Systems (1995)

Venue: | Applied Numerical Mathematics |

Citations: | 23 - 6 self |

### BibTeX

@ARTICLE{Maly95numericalmethods,

author = {Timothy Maly and Linda R. Petzold},

title = {Numerical Methods and Software for Sensitivity Analysis of Differential-Algebraic Systems},

journal = {Applied Numerical Mathematics},

year = {1995},

volume = {20},

pages = {57--79}

}

### OpenURL

### Abstract

In this paper we present some new algorithms and software for sensitivity analysis of differential-algebraic equation (DAE) systems. The algorithms have several novel features which are described and analyzed. The codes, which are extensions of DASSL and DASPK, are easy to use, highly efficient, and well-suited for large-scale problems. 1 Introduction Many engineering and scientific problems are described by systems of differential -algebraic equations (DAEs). Parametric sensitivity analysis of the (DAE) 1 The work of this author was partially supported by DOE contract number DE-FG02-92ER25130 and by the Minnesota Supercomputer Institute and by the Army High Performance Computing Research Center under the auspices of the Department of the Army, Army Research Laboratory cooperative agreement number DAAH04-95-2-0003/contract number DAAH04-95-C-0008, the content of which does not necessarily reflect the position or the policy of the government, and no official endorsement should be infe...

### Citations

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(Show Context)
Citation Context ...ale of the problem is the relative size of s i to y. Recall that the directional difference increments y to obtain y+ ffi i s i . Using a rule of thumb which suggests perturbing half the digits of y (=-=[6]-=-), this would mean roughly that ffi i ks i k 2 = p ukyk 2 ; and hence ffi i = p u kyk 2 ks i k 2 : (8) Note also that units of s are the same as units of y divided by units of p. This yields a ffi i w... |

1398 | GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems - Saad, Schultz - 1986 |

985 |
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Citation Context ...Gamma1 ks2fi. Proof First note that k J(xs) \Gamma1 ( J(x) \Gamma J(xs))ksk J(xs) \Gamma1 kk J(x) \Gamma J(xs)k (12)sfifl 0 kx \Gamma xsksfifl 0 ffls1 2 : Then by the perturbation relation (3.1.20 in =-=[5]-=-),sJ(x) is nonsingular and ksJ(x) \Gamma1 ksksJ(xs) \Gamma1 k 1 \Gamma ksJ(xs) \Gamma1 ( J(x 0 ) \GammasJ(xs))k (13)s2fi:2 Lemma 2.2 Let the conditions of Lemma 2.1 hold and x i 2 N (xs; r). Define j ... |

157 | ADIFOR - Generating derivative codes from Fortran programs, Scienti¯c Programming 1
- Bischof, Carle, et al.
- 1992
(Show Context)
Citation Context ...can specify directly the residual of the sensitivity system at the same time as the residual of the original system. Eventually, we intend to incorporate the automatic differentiation software ADIFOR =-=[1]-=- for this purpose. Alternatively, we can approximate the right hand side of the sensitivity equations via a directional derivative finite difference approximation. As an example, define s i = dy dp i ... |

56 |
Using Krylov methods in the solution of large-scale di®erential-algebraic systems
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(Show Context)
Citation Context ... highly effective in DASPK. The new codes are easy to use, highly efficient, and well-suited for large-scale problems. We assume that the reader is already familiar with the codes DASSL [2] and DASPK =-=[3]-=- and the algorithms used in these codes, in order to concentrate on the extensions for sensitivity analysis. 2 Sensitivity Analysis To illustrate the basic approach, consider the general DAE system wi... |

24 |
Sensitivity analysis in chemical kinetics
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Citation Context ...: : 0 J 1 C C C C C C A (3) and J = 1 h @F @y 0 + @F @y , J i = @J @y s i + @J @p i . A number of codes for ODEs and DAEs solve the sensitivity system (1), or its special case for ODEs, directly (see =-=[4,11]-=-). If the partial derivative matrices are not available analytically, they are approximated by finite differences. The nonlinear system is usually solved by a staggered scheme, where the first block i... |

21 |
Sensitivity analysis of initial value problems with mixed ODEs and algebraic equations
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Citation Context ...: : 0 J 1 C C C C C C A (3) and J = 1 h @F @y 0 + @F @y , J i = @J @y s i + @J @p i . A number of codes for ODEs and DAEs solve the sensitivity system (1), or its special case for ODEs, directly (see =-=[4,11]-=-). If the partial derivative matrices are not available analytically, they are approximated by finite differences. The nonlinear system is usually solved by a staggered scheme, where the first block i... |

11 | The simultaneous solution and sensitivity analysis of systems described by ordinary di®erential equations - Leis, Kramer - 1988 |

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2 |
Parallel solution of large-scale differential-algebraic systems
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Citation Context ...s, which is most often the case, large errors may be introduced into the sensitivities. In addition, the staggered scheme for solving the nonlinear system is not advantageous for parallel computation =-=[10]-=-. To eliminate these problems, we focus on approximating the sensitivity system (1) directly, rather than via the matrices @F=@y, @F=@y 0 , and @F=@p. In the simplest case, the user can specify direct... |