## Reliable Modeling and Optimization for Chemical Engineering Applications: Interval Analysis Approach. Reliable Computing

Citations: | 5 - 0 self |

### BibTeX

@MISC{Lin_reliablemodeling,

author = {Youdong Lin and C. Ryan Gwaltney and Mark A. Stadtherr},

title = {Reliable Modeling and Optimization for Chemical Engineering Applications: Interval Analysis Approach. Reliable Computing},

year = {}

}

### OpenURL

### Abstract

Abstract. In many applications of interest in chemical engineering it is necessary to deal with nonlinear models of complex physical phenomena, on scales ranging from the macroscopic to the molecular. Frequently these are problems that require solving a nonlinear equation system and/or finding the global optimum of a nonconvex function. Thus, the reliability with which these computations can be done is often an important issue. Interval analysis provides tools with which these reliability issues can be addressed, allowing such problems to be solved with complete certainty. This paper will focus on three types of applications: 1) Parameter estimation in the modeling of phase equilibrium, including the implications of using locally vs. globally optimal parameters in subsequent computations; 2) Nonlinear dynamics, in particular the location of equilibrium states and bifurcations of equilibria in ecosystem models used to assess the risk associated with the introduction of new chemicals into the environment; 3) Molecular modeling, with focus on transition state analysis of the diffusion of a sorbate molecule in a zeolite. 1.

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6 |
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Citation Context |

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Citation Context ...l minimum of the function f(x, y) = exp(sin(50x)) + sin(60 exp(y)) + sin(70 sin(x)) + sin(sin(80y)) − sin(10(x + y)) + (x 2 + y 2 )/4 is sought, where x ∈ [−1, 1] and y ∈ [−1, 1]. On the unit square (=-=[0, 1]-=-× [0, 1]) alone, the function has 667 local minima, as well as many other stationary points. This global optimization problem was solved successfully, with more than 10 digits of precision, in only 0.... |

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1 |
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Citation Context ...ndence should be addressed. E-mail: markst@nd.edu c○ 2005 Kluwer Academic Publishers. Printed in the Netherlands.2 from activity coefficient models [37, 46, 49], cubic equation-of-state (EOS) models =-=[6, 21, 22, 48]-=- and statistical associating fluid theory [54], calculation of critical points from cubic EOS models [47], location of azeotropes [34] and reactive azeotropes [35], computation of solidfluid equilibri... |

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Citation Context ...n the impact on one species, but rather on the larger impacts on the food chain and ecosystem. Of course, ecological modeling is just one part of a much larger suite of tools, including toxicological =-=[8, 23]-=-, hydrological and microbiological studies, that must be used in addressing this issue. Food chain models are often simple, but display rich mathematical behavior, with varying numbers and stability o... |

1 |
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Citation Context ...critical points from cubic EOS models [47], location of azeotropes [34] and reactive azeotropes [35], computation of solidfluid equilibrium [44, 55], parameter estimation using standard least squares =-=[9]-=- and error-in-variables (EIV) [10, 12, 11], and calculation of adsorption in nanoscale pores from a density function theory model [36]. In each case, the interval approach provides a mathematical and ... |

1 |
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Citation Context ...dels [47], location of azeotropes [34] and reactive azeotropes [35], computation of solidfluid equilibrium [44, 55], parameter estimation using standard least squares [9] and error-in-variables (EIV) =-=[10, 12, 11]-=-, and calculation of adsorption in nanoscale pores from a density function theory model [36]. In each case, the interval approach provides a mathematical and computational guarantee either that all so... |

1 |
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Citation Context ...dels [47], location of azeotropes [34] and reactive azeotropes [35], computation of solidfluid equilibrium [44, 55], parameter estimation using standard least squares [9] and error-in-variables (EIV) =-=[10, 12, 11]-=-, and calculation of adsorption in nanoscale pores from a density function theory model [36]. In each case, the interval approach provides a mathematical and computational guarantee either that all so... |

1 |
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Citation Context ...dels [47], location of azeotropes [34] and reactive azeotropes [35], computation of solidfluid equilibrium [44, 55], parameter estimation using standard least squares [9] and error-in-variables (EIV) =-=[10, 12, 11]-=-, and calculation of adsorption in nanoscale pores from a density function theory model [36]. In each case, the interval approach provides a mathematical and computational guarantee either that all so... |

1 |
Stadtherr: 2002c, ‘New Interval Methodologies for Reliable Chemical Process Modeling
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(Show Context)
Citation Context ...e approach [28]. Recently, a hybrid preconditioning approach (HP/RP), which combines a simple pivoting preconditioner with the standard inversemidpoint scheme, has been described by Gau and Stadtherr =-=[13]-=- and shown to achieve substantially more efficient computational performance than the inverse-midpoint preconditioner alone, in some cases by multiple orders of magnitude. However, it still cannot yie... |

1 |
Stadtherr: 2004, ‘Reliable Computation of Equilibrium States and Bifurcations in Nonlinear Dynamics
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Citation Context ...pacts of an ecotoxin that might not otherwise be apparent. The interval methodology has been applied successfully to several other ecological models by Gwaltney et al. [18] and Gwaltney and Stadtherr =-=[17]-=-. We anticipate that this methodology will also be useful for computing equilibrium states and bifurcations of equilibria in a wide variety of other problems in engineering and science in which nonlin... |

1 |
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Citation Context ...ndence should be addressed. E-mail: markst@nd.edu c○ 2005 Kluwer Academic Publishers. Printed in the Netherlands.2 from activity coefficient models [37, 46, 49], cubic equation-of-state (EOS) models =-=[6, 21, 22, 48]-=- and statistical associating fluid theory [54], calculation of critical points from cubic EOS models [47], location of azeotropes [34] and reactive azeotropes [35], computation of solidfluid equilibri... |

1 |
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Citation Context ...n the impact on one species, but rather on the larger impacts on the food chain and ecosystem. Of course, ecological modeling is just one part of a much larger suite of tools, including toxicological =-=[8, 23]-=-, hydrological and microbiological studies, that must be used in addressing this issue. Food chain models are often simple, but display rich mathematical behavior, with varying numbers and stability o... |

1 |
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Citation Context ...d (xi, yi, zi), i = 1, . . . , N are the known Cartesian coordinates of the N = 192 oxygen atoms. The silicon atoms, being recessed within the SiO4 tetrahedra, are neglected in the potential function =-=[29]-=-. Therefore, the total potential energy, V, of a single sorbate molecule in the absence of neighboring sorbate molecules is represented by a sum over all lattice oxygens, N∑ V = Vi. (21) i=1 The inter... |

1 |
Stadtherr: 2004a, ‘Advances in Interval Methods for Deterministic Global Optimization in Chemical Engineering
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(Show Context)
Citation Context ...e, in some cases by multiple orders of magnitude. However, it still cannot yield the tightest enclosure of the solution set, which, as noted above, is in general an NP-hard problem. Lin and Stadtherr =-=[31, 33]-=- have recently suggested a strategy (LISS LP) based on linear programming (LP) for solving the linear interval system, Eq. (1), arising in the context of intervalNewton methods. Using this approach, e... |

1 |
Stadtherr: 2004b, ‘Locating Stationary Points of Sorbate-Zeolite Potential Energy Surfaces Using Interval Analysis
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(Show Context)
Citation Context ...nce of the interval-Newton method, the mathematical properties of the Lennard-Jones potential and its first- and second-order derivatives can be exploited, as described in detail by Lin and Stadtherr =-=[32]-=-. 5.2. Results and Discussion The interval-Newton methodology described above (LISS LP) is now applied to find the stationary points of the potential energy surface V for the case of xenon as a sorbat... |

1 |
Stadtherr: 2004c, ‘LP Strategy for Interval-Newton Method in Deterministic Global
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(Show Context)
Citation Context ...e, in some cases by multiple orders of magnitude. However, it still cannot yield the tightest enclosure of the solution set, which, as noted above, is in general an NP-hard problem. Lin and Stadtherr =-=[31, 33]-=- have recently suggested a strategy (LISS LP) based on linear programming (LP) for solving the linear interval system, Eq. (1), arising in the context of intervalNewton methods. Using this approach, e... |

1 |
Stadtherr: 1998, ‘Reliable Computation of Homogeneous Azeotropes
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(Show Context)
Citation Context ...els [37, 46, 49], cubic equation-of-state (EOS) models [6, 21, 22, 48] and statistical associating fluid theory [54], calculation of critical points from cubic EOS models [47], location of azeotropes =-=[34]-=- and reactive azeotropes [35], computation of solidfluid equilibrium [44, 55], parameter estimation using standard least squares [9] and error-in-variables (EIV) [10, 12, 11], and calculation of adsor... |

1 |
Stadtherr: 2000, ‘Reliable Computation of Reactive Azeotropes
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(Show Context)
Citation Context ...ion-of-state (EOS) models [6, 21, 22, 48] and statistical associating fluid theory [54], calculation of critical points from cubic EOS models [47], location of azeotropes [34] and reactive azeotropes =-=[35]-=-, computation of solidfluid equilibrium [44, 55], parameter estimation using standard least squares [9] and error-in-variables (EIV) [10, 12, 11], and calculation of adsorption in nanoscale pores from... |

1 |
Stadtherr: 2001, ‘Reliable Density-Functional-Theory Calculations of Adsorption in Nanoporous Materials
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(Show Context)
Citation Context ...ibrium [44, 55], parameter estimation using standard least squares [9] and error-in-variables (EIV) [10, 12, 11], and calculation of adsorption in nanoscale pores from a density function theory model =-=[36]-=-. In each case, the interval approach provides a mathematical and computational guarantee either that all solutions have been located in a nonlinear equation solving problem or that the global optimum... |

1 |
Gumel: 2003, ‘Dynamical and Numerical Analysis of a Generalized Foodchain Model
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Citation Context ... used in addressing this issue. Food chain models are often simple, but display rich mathematical behavior, with varying numbers and stability of equilibria that depend on the model parameters (e.g., =-=[16, 38]-=-). Therefore, bifurcation analysis is quite useful in characterizing the mathematical behavior of predator/prey systems, as it allows for the concise representation of model behavior over a wide range... |

1 |
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Citation Context |

1 |
Stadtherr: 2003, ‘Phase Behavior and Reliable Computation of High-Pressure Solid-Fluid Equilibrium with
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Citation Context ...d statistical associating fluid theory [54], calculation of critical points from cubic EOS models [47], location of azeotropes [34] and reactive azeotropes [35], computation of solidfluid equilibrium =-=[44, 55]-=-, parameter estimation using standard least squares [9] and error-in-variables (EIV) [10, 12, 11], and calculation of adsorption in nanoscale pores from a density function theory model [36]. In each c... |

1 |
Westerberg: 2002, ‘Agent-based Strategies for Multiobjective Optimization
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Citation Context ...function N∏ 5∑ ( ) j5 f(x) = 100 cos(j + jxi) + 4425 1 N∑ (xi − x0,i) N 2 , (3) i=1 j=1 where xi ∈ [x0,i − 20, x0,i + 20] and x0,i = 3, i = 1, ..., N. This is used as a test problem by Siirola et al. =-=[45]-=-. There are 2048 local minima for the case N = 2 and on the order of a hundred million (10 8 ) local minima for the case N = 5. The problem also has multiple (N) global minimizer points. The problems ... |

1 |
Stadtherr: 2001, ‘Reliable Computation of Mixture Critical Points
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(Show Context)
Citation Context ...from activity coefficient models [37, 46, 49], cubic equation-of-state (EOS) models [6, 21, 22, 48] and statistical associating fluid theory [54], calculation of critical points from cubic EOS models =-=[47]-=-, location of azeotropes [34] and reactive azeotropes [35], computation of solidfluid equilibrium [44, 55], parameter estimation using standard least squares [9] and error-in-variables (EIV) [10, 12, ... |

1 |
Stadtherr: 2000, ‘Modeling and Design of an Environmentally Benign Reaction Process
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Citation Context ...ndence should be addressed. E-mail: markst@nd.edu c○ 2005 Kluwer Academic Publishers. Printed in the Netherlands.2 from activity coefficient models [37, 46, 49], cubic equation-of-state (EOS) models =-=[6, 21, 22, 48]-=- and statistical associating fluid theory [54], calculation of critical points from cubic EOS models [47], location of azeotropes [34] and reactive azeotropes [35], computation of solidfluid equilibri... |

1 |
Stadtherr: 2000, ‘Reliable Phase Stability Analysis for Excess Gibbs Energy Models
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Citation Context ...uid phase equilibrium ∗ Author to whom all correspondence should be addressed. E-mail: markst@nd.edu c○ 2005 Kluwer Academic Publishers. Printed in the Netherlands.2 from activity coefficient models =-=[37, 46, 49]-=-, cubic equation-of-state (EOS) models [6, 21, 22, 48] and statistical associating fluid theory [54], calculation of critical points from cubic EOS models [47], location of azeotropes [34] and reactiv... |

1 |
Stadtherr: 2005
- Ulas, Diwekar, et al.
(Show Context)
Citation Context ...pe compositions would be desirable. The difference between the use of the globally and locally optimal parameters can also have an effect on many other types of calculations. For example, Ulas et al. =-=[52]-=- demonstrate how batch distillation optimal control profiles are affected by using the globally optimal parameterTable III. Azeotrope prediction for benzene (1) – hexafluorobenzene (2) system. Data T... |

1 |
Floudas: 1999, ‘Locating All Transition States and Studying the Reaction Pathways of Potential Energy Surfaces
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(Show Context)
Citation Context ...tcoming, namely that they provide no guarantee that all local minima and first-order saddle points will actually be found. One approach to resolving this difficulty is given by Westerberg and Floudas =-=[53]-=-, who transform the equation-solving problem ∇V = 0 into an equivalent optimization problem that has global minimizers corresponding to the solutions of the equation system (i.e., the stationary point... |

1 |
Stadtherr: 2002, ‘Reliable Computation of Phase Stability and Equilibrium from the SAFT Equation of
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(Show Context)
Citation Context ... Kluwer Academic Publishers. Printed in the Netherlands.2 from activity coefficient models [37, 46, 49], cubic equation-of-state (EOS) models [6, 21, 22, 48] and statistical associating fluid theory =-=[54]-=-, calculation of critical points from cubic EOS models [47], location of azeotropes [34] and reactive azeotropes [35], computation of solidfluid equilibrium [44, 55], parameter estimation using standa... |

1 |
Stadtherr: 1998a, ‘Enhanced Interval Analysis for Phase Stability Cubic Equation of State Models
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(Show Context)
Citation Context ...n and nonlinear equation solving problems in chemical engineering, including computation of fluid phase equilibrium from activity coefficient models [35, 42, 45], cubic equation-of state (EOS) models =-=[5, 19, 20, 44]-=- and statistical associating fluid theory [50], calculation of critical points from cubic EOS models [43], location of azeotropes [32] and reactive azeotropes [33], computation of solid-fluid equilibr... |