Homotopy parameter bounding in increasing the robustness of homotopy continuation methods in multiplicity studies

Malinen, Ilkka; Tanskanen, Juha

doi:10.1016/j.compchemeng.2010.03.013

Cited by 37 publications

(30 citation statements)

References 49 publications

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“…In this study, it will be shown that the target of finding all the feasible solutions for the investigated CSTR systems can be reached with the Newton homotopy-based solving method if it is equipped with the features of homotopy parameter and variables bounding presented in Malinen and Tanskanen [30][31]. The solving strategy of the present work extends the applicability of the method presented in Malinen et al [32] in the determination of multiple steady states of chemical engineering systems.…”

Section: Introductionmentioning

confidence: 51%

“…In order to improve the capabilities of homotopy methods to approach solutions existing on separate homotopy path branches, Malinen and Tanskanen [31] proposed the concept of homotopy parameter bounding. However, as was concluded in Malinen [49], when used separately, both the concept of homotopy parameter bounding and bounding with respect to problem variables are incapable of finding all the solutions if the solutions exist on isolated branches.…”

Section: Characteristics Of Some Present Solving Methodsmentioning

confidence: 99%

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A New Homotopy-Based Strategy for the Robust Determination of All the Feasible Solutions for CSTR Systems

Malinen¹,

Kangas

Ahola

et al. 2016

Period. Polytech. Chem. Eng.

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Section: Introductionmentioning

confidence: 51%

Section: Characteristics Of Some Present Solving Methodsmentioning

confidence: 99%

A New Homotopy-Based Strategy for the Robust Determination of All the Feasible Solutions for CSTR Systems

Malinen¹,

Kangas

Ahola

et al. 2016

Period. Polytech. Chem. Eng.

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“…Mature and robust software implementations for solving polynomial systems are, for example, Bertini [11,12], and PHCpack [86,87]. Homotopy methods with problem-specific homotopy maps were successfully applied to large-scale industrial problems with transcendental equations [21,51,85]. This approach with problem-specific homotopy maps is also suitable for component-based (also referred to as component-oriented) modeling of large, complex, and heterogeneous technical systems [70;72,: Once the models of the devices (components) are implemented and put into a library, practically no understanding of probability-one homotopy method is required from the end-user.…”

Section: General-purpose Methods For Finding Multiple Solutionsmentioning

confidence: 99%

A robust approach for finding all well-separated solutions of sparse systems of nonlinear equations

2016

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Tearing is a long-established decomposition technique, widely adapted across many engineering fields. It reduces the task of solving a large and sparse nonlinear system of equations to that of solving a sequence of low-dimensional ones. The most serious weakness of this approach is well-known: It may suffer from severe numerical instability. The present paper resolves this flaw for the first time. The new approach requires reasonable bound constraints on the variables. The worst-case time complexity of the algorithm is exponential in the size of the largest subproblem of the decomposed system. Although there is no theoretical guarantee that all solutions will be found in the general case, increasing the so-called sample size parameter of the method improves robustness. This is demonstrated on two particularly challenging problems. Our first example is the steady-state simulation a challenging distillation column, belonging to an infamous class of problems where tearing often fails due to numerical instability. This column has three solutions, one of which is missed using tearing, but even with problem-specific methods that are not based on tearing. The other example is the Stewart-Gough platform with 40 real solutions, an extensively studied benchmark in the field of numerical algebraic geometry. For both examples, all solutions are found with a fairly small amount of sampling.

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“…Input multiplicity occurs when two or more sets of input variables produce the same output conditions, while output multiplicity occurs when one set of input variables results in two or more independent sets of output variables [23]; Fig. 1 illustrates these types of multiplicity.…”

Section: Introductionmentioning

confidence: 99%

“…According to literature, reactive separation systems can exhibit two types of multiplicity: input and output multiplicity [23]. Input multiplicity occurs when two or more sets of input variables produce the same output conditions, while output multiplicity occurs when one set of input variables results in two or more independent sets of output variables [23]; Fig.…”

Section: Introductionmentioning

confidence: 99%

Process Intensification

VandenBussche¹

2007

Micro Instrumentation

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a b s t r a c tA systematic study of the operating conditions and parameter sensibility of multiple steady states in a reactive distillation column for production of fuel ethers is presented. The multiplicity analysis is performed using bifurcation diagrams for several scenarios using the process simulator Aspen One Aspen Plus. Our results indicate that the main variables that affect both the presence and behavior of multiple solutions in the reactive distillation column are the reflux ratio, the reboiler duty and the thermal condition in the feed stream. In particular, we have determined that the vapor-liquid stream in the feed has a major influence on the occurrence of this phenomenon and this parameter especially establishes the type of multiplicity that can occur in the reactive distillation column for production of fuel ethers.

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Homotopy parameter bounding in increasing the robustness of homotopy continuation methods in multiplicity studies

Cited by 37 publications

References 49 publications

A New Homotopy-Based Strategy for the Robust Determination of All the Feasible Solutions for CSTR Systems

A New Homotopy-Based Strategy for the Robust Determination of All the Feasible Solutions for CSTR Systems

A robust approach for finding all well-separated solutions of sparse systems of nonlinear equations

Process Intensification

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