## Robust Process Simulation Using Interval Methods (1996)

Venue: | Comput. Chem. Eng |

Citations: | 31 - 19 self |

### BibTeX

@ARTICLE{Stadtherr96robustprocess,

author = {Mark A. Stadtherr},

title = {Robust Process Simulation Using Interval Methods},

journal = {Comput. Chem. Eng},

year = {1996},

volume = {20},

pages = {187--199}

}

### OpenURL

### Abstract

Ideally, for the needs of robust process simulation, one would like a nonlinear equation solving technique that can find any and all roots to a problem, and do so with mathematical certainty. In general, currently used techniques do not provide such rigorous guarantees. One approach to providing such assurances can be found in the use of interval analysis, in particular the use of interval Newton methods combined with generalized bisection. However, these methods have generally been regarded as extremely inefficient. Motivated by recent progress in interval analysis, as well as continuing advances in computer speed and the availability of parallel computing, we consider here the feasibility of using an interval Newton/generalized bisection algorithm on process simulation problems. An algorithm designed for parallel computing on an MIMD machine is described, and results of tests on several problems are reported. Experiments indicate that the interval Newton/generalized bisection method works quite well on relatively small problems, providing a powerful method for finding all solutions to a problem. For larger problems, the method performs inconsistently with regard to efficiency, at least when reasonable initial bounds are not provided.

### Citations

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Citation Context ...(k) ) is a suitable interval extension of the real Jacobian J(x) off(x) over the current box X (k) , and x (k) is a point in the interior of X (k) , usually taken to be the midpoint. It can be shown (=-=Moore, 1966-=-) that any root x * ∈ X (k) of f(x) is also contained in N (k) . This suggests the iteration X (k+1) = X (k) ∩ N (k) . (3) The various interval Newton methods differ in how they determine N (k) from E... |

506 |
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(Show Context)
Citation Context ... of using such techniques for process simulation problems has been dismissed forthwith, on the assumption that they would be extremely inefficient. However, recent advances in interval methods (e.g., =-=Neumaier, 1990-=-; Kearfott and Novoa, 1990), together with the continuing rapid advance in computer speed and the availability of parallel computing, may make such methods viable, as originally suggested by Schnepper... |

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Citation Context ...find, with mathematical certainty, any and all solutions to a system of nonlinear equations lying within the variable bounds. The techniques of interval analysis provide just such a class of methods (=-=Kearfott, 1990-=-a), namely interval Newton methods combined with generalized bisection. In general, the thought of using such techniques for process simulation problems has been dismissed forthwith, on the assumption... |

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(Show Context)
Citation Context ...d bisection method can find with mathematical certainty any and all such solutions to a specified tolerance, or can determine with mathematical certainty that there are no solutions in the given box (=-=Kearfott, 1987-=-a,1989,1990a). The technique used here for computing N (k) is the preconditioned Gauss-Seidel-like technique developed by Hansen and Sengupta (1981) and Hansen and Greenburg (1983). Eq. (2) is first p... |

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Citation Context ...echniques for process simulation problems has been dismissed forthwith, on the assumption that they would be extremely inefficient. However, recent advances in interval methods (e.g., Neumaier, 1990; =-=Kearfott and Novoa, 1990-=-), together with the continuing rapid advance in computer speed and the availability of parallel computing, may make such methods viable, as originally suggested by Schnepper and Stadtherr (1990). In ... |

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Citation Context ...ess is poor or if singular points are encountered. To improve convergence in these circumstances, various methods have been used. These include trust-region techniques such as Powell’s dogleg method (=-=Powell, 1970-=-; Chen and Stadtherr, 1981), homotopy-based methods (e.g., Wayburn and Seader, 1987; Kuno and Seader, 1988), and techniques based on iterative mathematical programming, as reviewed recently by Bullard... |

26 |
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Citation Context ...d bisection method can find with mathematical certainty any and all such solutions to a specified tolerance, or can determine with mathematical certainty that there are no solutions in the given box (=-=Kearfott, 1987-=-a,1989,1990a). The technique used here for computing N (k) is the preconditioned Gauss-Seidel-like technique developed by Hansen and Sengupta (1981) and Hansen and Greenburg (1983). Eq. (2) is first p... |

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(Show Context)
Citation Context ...find, with mathematical certainty, any and all solutions to a system of nonlinear equations lying within the variable bounds. The techniques of interval analysis provide just such a class of methods (=-=Kearfott, 1990-=-a), namely interval Newton methods combined with generalized bisection. In general, the thought of using such techniques for process simulation problems has been dismissed forthwith, on the assumption... |

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Citation Context ...sed does not cover the entire feasible space, we cannot say with absolute certainty that this problem has only one solution. In fact, multiple solutions have been obtained for similar problems (e.g., =-=Jacobsen and Skogestad, 1991-=-). One thing that these results suggest is that the mathematical criterion used to select the variables to be bisected may be inadequate for process simulation problems. For instance, in Problem 5-1, ... |

6 |
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Citation Context ...of which is often suggested (e.g., Neumaier, 1990) as a good preconditioner. Other preconditioners for improving the performance of the Gauss-Seidel procedure are being developed 11s(Kearfott, 1990b; =-=Kearfott et al., 1991-=-a); however these are computationally more expensive than the inverse Jacobian, especially for large problems. INTBIS computes and stores the entire inverse Jacobian explicitly before proceeding with ... |

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6 |
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Citation Context ... w(Xj ) Fj′(X (k) ) , where Fj′(X (k) ) is the j-th column of the interval Jacobian F′(X (k) ). Then q is chosen so that s q = max j s j. Our computational experiments on process simulation problems (=-=Schnepper, 1992-=-) have shown the third strategy to be the most effective, and that is what is used here. Algorithm (serial) The basic IN/GB algorithm is outlined first in serial form. Its parallel implementation will... |

6 |
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Citation Context ...y of routines for computing interval extensions of these real functions was developed. INTBIS was then modified to use the sparse storage scheme used in SEQUEL-II. The efficient sparse solver LU1SOL (=-=Stadtherr and Wood, 1984-=-; Chen and Stadtherr, 1984; Kaijaluoto et al., 1989) was used in performing the preconditioning and in connection with the point-Newton iteration done in intervals having a positive root inclusion tes... |

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Citation Context ...e.g., Zitney and Stadtherr, 1988) involving truncation or reflection of the correction step. A more natural way of dealing with bounds is to use the iterative mathematical programming approach (e.g., =-=Bullard and Biegler, 1991-=-; Swaney and Wilhelm, 1990), in which case the bounds become an integral part of the problem. While a number of these techniques demonstrate excellent global convergence properties in practice, none o... |

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(Show Context)
Citation Context ... if singular points are encountered. To improve convergence in these circumstances, various methods have been used. These include trust-region techniques such as Powell’s dogleg method (Powell, 1970; =-=Chen and Stadtherr, 1981-=-), homotopy-based methods (e.g., Wayburn and Seader, 1987; Kuno and Seader, 1988), and techniques based on iterative mathematical programming, as reviewed recently by Bullard and Biegler (1991). An ad... |

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Citation Context ... various methods have been used. These include trust-region techniques such as Powell’s dogleg method (Powell, 1970; Chen and Stadtherr, 1981), homotopy-based methods (e.g., Wayburn and Seader, 1987; =-=Kuno and Seader, 1988-=-), and techniques based on iterative mathematical programming, as reviewed recently by Bullard and Biegler (1991). An additional difficulty is that in process simulation there are invariably upper and... |

3 |
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(Show Context)
Citation Context ...or use in process simulation problems. While for relatively small problems, IN/GB is usually efficient (Kearfott, 1987b; Kearfott and Novoa, 1990), for larger problems efficiency is less predictable (=-=Kearfott, 1989-=-; Kearfott and Novoa, 1990). In fact, for a properly implemented IN/GB method applied to an initial box with a finite number of roots, the only mode of failure is an excessive computational requiremen... |

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Citation Context ...of which is often suggested (e.g., Neumaier, 1990) as a good preconditioner. Other preconditioners for improving the performance of the Gauss-Seidel procedure are being developed 11s(Kearfott, 1990b; =-=Kearfott et al., 1991-=-a); however these are computationally more expensive than the inverse Jacobian, especially for large problems. INTBIS computes and stores the entire inverse Jacobian explicitly before proceeding with ... |

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Citation Context ... X + Y =[a + c, b + d] X − Y =[a − d, b − c] X Y = [min (ac, ad, bc, bd), max (ac, ad, bc, bd)] X / Y =[a,b] [1/d, 1/c], 0 ∉ [c,d]. For X / Y when 0 ∈ Y, an extended interval arithmetic is available (=-=Hansen, 1968-=-) which is useful in the execution of the interval Newton methods discussed below. The extended arithmetic produces the same set as defined by Eq. (1). The foregoing assumes that we are able to comput... |

1 |
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Citation Context ...hese real functions was developed. INTBIS was then modified to use the sparse storage scheme used in SEQUEL-II. The efficient sparse solver LU1SOL (Stadtherr and Wood, 1984; Chen and Stadtherr, 1984; =-=Kaijaluoto et al., 1989-=-) was used in performing the preconditioning and in connection with the point-Newton iteration done in intervals having a positive root inclusion test. Before summarizing the algorithm used, we discus... |

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