On solving large sparse nonlinear equation systems

Chen, Hern-Shann; Stadtherr, Mark A.

doi:10.1016/0098-1354(84)80010-1

Cited by 28 publications

(11 citation statements)

References 24 publications

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“…Typically, the Harwell subroutine MA28 (Duff and Reid, 1979) or something similar is used. Some alternative routines (Stadtherr and Wood, 1984a, b; Chen and Stadtherr, 1984) have also been proposed, one of which, LUlSOL, is now used in an industrial EB simulator, and significantly outperforms MA28. It is also worth noting that LUlSOL was recently rated in an independent study (Kaijaluoto et al, 1989) as the best available for EB flowsheeting on conventional machines.…”

Section: Aiche Journalmentioning

confidence: 98%

Vector processing strategies for chemical process flowsheeting

Vegeais

Stadtherr

1990

AIChE Journal

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The potential for effective vectorization of the sequential-modular, simultaneous-modular, and equation-based approaches to process flowsheeting is assessed. The last appears to be the most promising, but the required sparse matrix techniques currently used do not vectorize well. Different approaches to vectorizing the sparse matrix calculation are considered. With a good matrix reordering scheme, the most promising sparse matrix strategy for the vectorization of flowsheeting problems is the frontal method, a technique currently used primarily on finite-element problems.

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Section: Aiche Journalmentioning

confidence: 98%

Vector processing strategies for chemical process flowsheeting

Vegeais

Stadtherr

1990

AIChE Journal

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“…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 discuss some of its components in more detail.…”

Section: Methodsmentioning

confidence: 99%

Robust process simulation using interval methods

Schnepper

Stadtherr

1996

Computers & Chemical Engineering

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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.

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“…This type of approach achieves superlinear convergence only when they are well within the neighbourhood of the solution. Since these methods tend to fail if the initial guess is not sufficiently close to the root or if singular points are encountered, some modifications have been developed to avoid singularities or to enhance solution efficiency, such as Powell's dog leg method (Powell 1970), steepest descent techniques (Chen andStadtherr 1984, Duff et al 1987) and the monomial method (Burns and Locascio 1991). However, they cannot provide guarantees for convergence.…”

Section: Introductionmentioning

confidence: 97%

Finding all solutions of systems of nonlinear equations with free variables

Tsai

Lin

2007

Engineering Optimization

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Systems of nonlinear equations often represent mathematical models in engineering design. This study proposes a novel method for finding all solutions of systems of nonlinear equations with free variables. The original problem is first transformed into a global optimization problem whose multiple global minima with a zero objective value correspond to all solutions of the original problem. Then, by using variable substitution on free variables and applying convexification strategies and piecewise linearization techniques on nonconvex functions, the transformed optimization problem is reformulated as a convex mixed-integer program solvable to reach an approximately global optimum. An algorithm is developed to find all solutions of the reformulated problem. Numerical examples in real applications are presented to demonstrate the usefulness of the proposed method in engineering design.

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On solving large sparse nonlinear equation systems

Cited by 28 publications

References 24 publications

Vector processing strategies for chemical process flowsheeting

Vector processing strategies for chemical process flowsheeting

Robust process simulation using interval methods

Finding all solutions of systems of nonlinear equations with free variables

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