168 results found
Fu J, Faust JMM, Chachuat B, et al., 2015, Local optimization of dynamic programs with guaranteed satisfaction of path constraints, Automatica, Vol: 62, Pages: 184-192, ISSN: 1873-2836
An algorithm is proposed for locating a feasible point satisfying the KKT conditions to a specified tolerance of feasible inequality-path-constrained dynamic programs (PCDP) within a finite number of iterations. The algorithm is based on iteratively approximating the PCDP by restricting the right-hand side of the path constraints and enforcing the path constraints at finitely many time points. The main contribution of this article is an adaptation of the semi-infinite program (SIP) algorithm proposed in Mitsos (2011) to PCDP. It is proved that the algorithm terminates finitely with a guaranteed feasible point which satisfies the first-order KKT conditions of the PCDP to a specified tolerance. The main assumptions are: (i) availability of a nonlinear program (NLP) local solver that generates a KKT point of the constructed approximation to PCDP at each iteration if this problem is indeed feasible; (ii) existence of a Slater point of the PCDP that also satisfies the first-order KKT conditions of the PCDP to a specified tolerance; (iii) all KKT multipliers are nonnegative and uniformly bounded with respect to all iterations. The performance of the algorithm is analyzed through two numerical case studies.
Houska B, Villanueva ME, Chachuat B, 2015, Stable set-valued integration of nonlinear dynamic systems using affine set-parameterizations, Siam Journal of Numerical Analysis, Vol: 53, Pages: 2307-2328, ISSN: 0036-1429
Many set-valued integration algorithms for parametric ordinary differential equations (ODEs) implement a combination of Taylor series expansion with either interval arithmetic or Taylor model arithmetic. Due to the wrapping effect, the diameter of the solution-set enclosures computed with these algorithms typically diverges to infinity on finite integration horizons, even though the ODE trajectories themselves may be asymptotically stable. This paper starts by describing a new discretized set-valued integration algorithm that uses a predictor-validation approach to propagate generic affine set-parameterizations, whose images are guaranteed to enclose the ODE solution set. Sufficient conditions are then derived for this algorithm to be locally asymptotically stable, in the sense that the computed enclosures are guaranteed to remain stable on infinite time horizons when applied to a dynamic system in the neighborhood of a locally asymptotically stable periodic orbit (or equilibrium point). The key requirement here is quadratic Hausdorff convergence of function extensions in the chosen affine set-parameterization, which is proved to be the case, for instance, for Taylor models with ellipsoidal remainders. These stability properties are illustrated with the case study of a cubic oscillator system.
Rayjaguru J, Villanueva ME, Houska B, et al., 2015, Continuous-time enclosures for uncertain implicit differential equations, 9th IFAC Symposium on Advanced Control of Chemical Processes, Publisher: Elsevier, Pages: 94-99, ISSN: 1474-6670
The computation of enclosures for the reachable set of uncertain dynamic systems is a crucial component in a wide variety of applications, from global and robust dynamic optimization to safety verification and fault detection. Even though many systems in engineering are best modeled as implicit differential equations (IDEs) and differential algebraic equations (DAEs), methods for the construction of enclosures for these are not as well developed as they are for ordinary differential equations (ODEs). In this paper, we propose a continuous-time approach for the guaranteed over approximations of the reachable set for quasilinear IDEs. This approach builds on novel high-order inclusion techniques for the solution set of algebraic equations and state-of-the-art techniques for bounding the solution of nonlinear ODEs.We show how this approach can be used to bound the reachable set of uncertain semi-explicit DAEs by bounding the underlying IDEs. We demonstrate this approach on two case studies, a double pendulum where it proves superior with delayed break-down times compared to other methods, and anaerobic digestion of microalgae which has nine differential and two algebraic states.
Puchongkawarin C, Fitzgerald S, Chachuat B, 2015, Plant-wide optimization of a full-scale activated sludge plant with anaerobic sludge treatment, 9th IFAC Symposium on Advanced Control of Chemical Processes, Publisher: Elsevier, Pages: 1234-1239, ISSN: 1474-6670
© 2015, (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. This paper presents the application of a plant-wide model-based methodology to wastewater treatment plants. The focus is on a tertiary activated sludge plant with anaerobic sludge treatment, owned and operated by Sydney Water. A dynamic plant-wide model is first developed and calibrated using historical data. A scenario-based optimization procedure is then applied for computing the effect of key discharge constraints on the minimal net power consumption, via the repeated solution of a dynamic optimization problem. The results show a potential for reduction of the energy consumption by about 20%, through operational changes only, without compromising the current effluent quality. It is also found that nitrate (and hence total nitrogen) discharge could be reduced from its current level around 22 mg(N)/L to less than 15 with no increase in net power consumption, and could be further reduced to <10 mg(N)/L subject to a 15% increase in net power consumption upon diverting part of the primary sludge to the secondary treatment stage. This improved understanding of the relationship between nutrient removal and energy use will feed into discussions with environmental regulators regarding nutrient discharge licensing.
Chachuat B, Houska B, Paulen R, et al., 2015, Set-theoretic approaches in analysis, estimation and control of nonlinear systems, 9th IFAC Symposium on Advanced Control of Chemical Processes, Publisher: Elsevier, Pages: 981-995, ISSN: 1474-6670
This paper gives an overview of recent developments in set-theoretic methods for nonlinear systems, with a particular focus on the activities in our own research group. Central to these approaches is the ability to compute tight enclosures of the range of multivariate systems, e.g. using ellipsoidal calculus or higher-order inclusion techniques based on multivariate polynomials, as well as the ability to propagate these enclosures to enclose the trajectories of parametric or uncertain differential equations. We illustrate these developments with a range of applications, including the reach ability analysis of nonlinear dynamic systems; the determination of all equilibrium points and bifurcations in a given state-space domain; and the solution of set-membership parameter estimation problems. We close the paper with a discussion about on-going research in tube-based methods for robust model predictive control.
Puchongkawarin C, Menichini C, Laso-Rubido C, et al., 2015, Model-based methodology for plant-wide analysis of wastewater treatment plants: Industrial case study, Water Practice and Technology, Vol: 10, Pages: 517-526, ISSN: 1751-231X
This paper presents the application of a model-based methodology for improved understanding of the tight interplay between effluent quality, energy use, and fugitive emissions in wastewater treatment plants. Dynamic models are developed and calibrated in an objective to predict the performance of a conventional activated sludge plant owned and operated by Sydney Water, Australia. A scenario-based approach is applied to quantify the effect of key operating variables on the effluent quality, energy use, and fugitive emissions. Operational strategies that enable a reduction in aeration energy by 10-20% or a reduction of total nitrogen discharge down to 3 mg L<sup>-1</sup> are identified. These results are also compared to an upgraded plant with reverse osmosis in terms of energy consumption and greenhouse gas emissions. This improved understanding of the relationship between nutrient removal, energy use, and emissions will feed into discussions with environmental regulators regarding nutrient discharge licensing.
Paulen R, Villanueva ME, Chachuat B, 2015, Guaranteed parameter estimation of non-linear dynamic systems using high-order bounding techniques with domain and CPU-time reduction strategies, IMA Journal of Mathematical Control and Information, Vol: 33, Pages: 563-587, ISSN: 0265-0754
This paper is concerned with guaranteed parameter estimation of non-linear dynamic systems in a context of bounded measurement error. The problem consists of finding - or approximating as closely as possible - the set of all possible parameter values such that the predicted values of certain outputs match their corresponding measurements within prescribed error bounds. A set-inversion algorithm is applied, whereby the parameter set is successively partitioned into smaller boxes and exclusion tests are performed to eliminate some of these boxes, until a given threshold on the approximation level is met. Such exclusion tests rely on the ability to bound the solution set of the dynamic system for a finite parameter subset, and the tightness of these bounds is therefore paramount; equally important in practice is the time required to compute the bounds, thereby defining a trade-off. In this paper, we investigate such a trade-off by comparing various bounding techniques based on Taylor models with either interval or ellipsoidal bounds as their remainder terms. We also investigate the use of optimization-based domain reduction techniques in order to enhance the convergence speed of the set-inversion algorithm, and we implement simple strategies that avoid recomputing Taylor models or reduce their expansion orders wherever possible. Case studies of various complexities are presented, which show that these improvements using Taylor-based bounding techniques can significantly reduce the computational burden, both in terms of iteration count and CPU time.
Nikolaou A, Chachuat B, 2015, 427331 Scaling-up microalgae production systems: Inferring biomass productivity in raceway ponds using numerical simulation, Pages: 453-456
Villanueva ME, Pereira HSA, Rajyaguru J, et al., 2015, Bifurcation and stability analysis of nonlinear dynamic systems using complete search, Pages: 627-628
Bernardi A, Nikolaou A, Meneghesso A, et al., 2015, Using Fluorescence Measurements to Model Key Phenomena in Microalgae Photosynthetic Mechanisms, ICHEAP12: 12TH INTERNATIONAL CONFERENCE ON CHEMICAL & PROCESS ENGINEERING, Vol: 43, Pages: 217-222, ISSN: 2283-9216
Bernardi A, Nikolaou A, Meneghesso A, et al., 2015, A Framework for the Dynamic Modelling of PI Curves in Microalgae, 12TH INTERNATIONAL SYMPOSIUM ON PROCESS SYSTEMS ENGINEERING AND 25TH EUROPEAN SYMPOSIUM ON COMPUTER AIDED PROCESS ENGINEERING, PT C, Vol: 37, Pages: 2483-2488, ISSN: 1570-7946
Villanueva ME, Rajyaguru J, Houska B, et al., 2015, Ellipsoidal Arithmetic for Multivariate Systems, 12TH INTERNATIONAL SYMPOSIUM ON PROCESS SYSTEMS ENGINEERING (PSE) AND 25TH EUROPEAN SYMPOSIUM ON COMPUTER AIDED PROCESS ENGINEERING (ESCAPE), PT A, Vol: 37, Pages: 767-772, ISSN: 1570-7946
Sun M, Chachuat B, Pistikopoulos EN, 2015, Design of multiparametric NCO tracking controllers for linear dynamic systems, 12TH INTERNATIONAL SYMPOSIUM ON PROCESS SYSTEMS ENGINEERING (PSE) AND 25TH EUROPEAN SYMPOSIUM ON COMPUTER AIDED PROCESS ENGINEERING (ESCAPE), PT B, Vol: 37, Pages: 1511-1516, ISSN: 1570-7946
Nikolaou A, Bernardi A, Meneghesso A, et al., 2014, A model of chlorophyll fluorescence in microalgae integrating photoproduction, photoinhibition and photoregulation, Journal of Biotechnology, Vol: 194, Pages: 91-99, ISSN: 1873-4863
This paper presents a mathematical model capable of quantitative prediction of the state of the photosynthetic apparatus of microalgae in terms of their open, closed and damaged reaction centers under variable light conditions. This model combines the processes of photoproduction and photoinhibition in the Han model with a novel mathematical representation of photoprotective mechanisms, including qE-quenching and qI-quenching. For calibration and validation purposes, the model can be used to simulate fluorescence fluxes, such as those measured in PAM fluorometry, as well as classical fluorescence indexes. A calibration is carried out for the microalga Nannochloropsis gaditana, whereby 9 out of the 13 model parameters are estimated with good statistical significance using the realized, minimal and maximal fluorescence fluxes measured from a typical PAM protocol. The model is further validated by considering a more challenging PAM protocol alternating periods of intense light and dark, showing a good ability to provide quantitative predictions of the fluorescence fluxes even though it was calibrated for a different and somewhat simpler PAM protocol. A promising application of the model is for the prediction of PI-response curves based on PAM fluorometry, together with the long-term prospect of combining it with hydrodynamic and light attenuation models for high-fidelity simulation and optimization of full-scale microalgae production systems.
Villanueva ME, Houska B, Chachuat B, 2014, Unified framework for the propagation of continuous-time enclosures for parametric nonlinear ODEs, Journal of Global Optimization, Vol: 62, Pages: 575-613, ISSN: 1573-2916
This paper presents a framework for constructing and analyzing enclosures ofthe reachable set of nonlinear ordinary differential equations using continuous-time setpropagation methods. The focus is on convex enclosures that can be characterized in terms of their support functions. A generalized differential inequality is introduced, whose solutionsdescribe such support functions for a convex enclosure of the reachable set under mild conditions. It is shown that existing continuous-time bounding methods that are based on standard differential inequalities or ellipsoidal set propagation techniques can be recovered as special cases of this generalized differential inequality. A way of extending this approach for the construction of nonconvex enclosures is also described, which relies on Taylor modelswith convex remainder bounds. This unifying framework provides a means for analyzing the convergence properties of continuous-time enclosure methods. The enclosure techniques and convergence results are illustrated with numerical case studies throughout the paper, including a six-state dynamic model of anaerobic digestion.
Houska B, Chachuat B, 2014, Branch-and-Lift Algorithm for Deterministic Global Optimization in Nonlinear Optimal Control, Journal of Optimization Theory and Applications, Vol: 162, Pages: 208-248, ISSN: 0022-3239
This paper presents a branch-and-lift algorithm for solving optimal control problems with smooth nonlinear dynamics and potentially nonconvex objective and constraint functionals to guaranteed global optimality. This algorithm features a direct sequential method and builds upon a generic, spatial branch-and-bound algorithm. A new operation, called lifting, is introduced, which refines the control parameterization via a Gram-Schmidt orthogonalization process, while simultaneously eliminating control subregions that are either infeasible or that provably cannot contain any global optima. Conditions are given under which the image of the control parameterization error in the state space contracts exponentially as the parameterization order is increased, thereby making the lifting operation efficient. A computational technique based on ellipsoidal calculus is also developed that satisfies these conditions. The practical applicability of branch-and-lift is illustrated in a numerical example. © 2013 Springer Science+Business Media New York.
Gros S, Chachuat B, 2014, Optimization-based load reduction during rapid shutdown of multi-megawatt wind turbine generators, Wind Energy, Vol: 17, Pages: 1055-1075, ISSN: 1095-4244
This paper describes an optimization-based approach to reducing extreme structural loads during rapid or emergency shutdown of multi-megawatt wind turbine generators. The load reduction problem is cast into an optimal control formulation, and a simple, low-order model is developed in order for this optimization problem to be tractable in reasonable time using state-of-the-art numerical methods. To handle the variations in wind speed and turbulence inherent to wind turbine operation as well as the presence of model mismatch, a real-time optimization strategy based on fast sensitivity updates is also considered, whose online computational burden is limited to the repeated solution of quadratic programs that are designed offline. The low-order model and both the open-loop and closed-loop optimal control strategies are validated against a high-fidelity model in the simulation environment Bladed™ for an industrial 3 MW wind turbine. Under favorable shutdown scenarios, i.e. when the wind turbine is operating properly and the actuators and sensors are not faulty, large reductions of the first compressive peak and subsequent compressive/tensile peaks of the tower load pattern are obtained at various above-rated wind speeds compared with normal pitch control shutdown. Extension to more challenging shutdown scenarios are also discussed.
Khor CS, Chachuat B, Shah N, 2014, Optimization of water network synthesis for single-site and continuous processes: milestones, challenges, and future directions, Industrial & Engineering Chemistry Research, Vol: 53, Pages: 10257-10275, ISSN: 1520-5045
Increasing water demand in the process and its allied industries coupled with global water stress and scarcity has underlined the importance of water as a crucial resource and elevated a need for widespread adoption of water reuse and recycle. This paper provides a state-of-the-art review of the area of water network synthesis focusing on single-site and continuous process problems since its inception in the 1980s. The survey centers around model-based optimization or mathematical programming methods for water network synthesis and covers key findings from the water pinch analysis technique, which are often essential in enhancing model formulations. Major modeling and computational challenges are discussed that explore the issues of nonconvexity, nonlinearity, and uncertainty inherent in water network synthesis problems. The review concludes by providing a perspective of future research directions to be tackled to address the challenges highlighted.
Khor CS, Chachuat B, Shah N, 2014, Fixed-flowrate total water network synthesis under uncertainty with risk management, Journal of Cleaner Production, Vol: 77, Pages: 79-93, ISSN: 0959-6526
This work addresses the problem of integrated water network synthesis under uncertainty with risk management. We consider a superstructure consisting of water sources, regenerators, and sinks that leads to a mixed-integer quadratically-constrained quadratic program (MIQCQP) for a fixed-flowrate total water network synthesis problem. Uncertainty in the problem is accounted for via a recourse-based two-stage stochastic programming formulation with discrete scenarios that gives rise to a multiscenario MIQCQP comprising network design in the first stage and its operation in the second stage acting as recourse. In addition, we extend the model to address risk management using the Conditional Value-at-Risk (CVaR) metric. Because a large number of scenarios is often required to capture the underlying uncertainty of the problem, causing the model to suffer from the curse of dimensionality, we propose a stepwise solution strategy to reduce the computational load. We illustrate this methodology on a case study inspired from the water network of a petroleum refinery in Malaysia. The presence of nonconvex bilinear terms necessitates the use of global optimization techniques for which we employ a new global MIQCQP solver, GAMS/GloMIQO and verify the solutions with BARON. Our computational results show that total water network synthesis under uncertainty with risk management problems can be solved to global optimality in reasonable time.
Nikolaou A, Bernardi A, Bezzo F, et al., 2014, Dynamic Model of Photoproduction, Photoregulation and Photoinhibition in Microalgae using Chlorophyll Fluorescence., IFAC WC, Publisher: Elsevier
Villanueva ME, Houska B, Chachuat B, 2014, On the Stability of Set-Valued Integration for Parametric Nonlinear ODEs, 24TH EUROPEAN SYMPOSIUM ON COMPUTER AIDED PROCESS ENGINEERING, PTS A AND B, Vol: 33, Pages: 595-600, ISSN: 1570-7946
Houska B, Villanueva ME, Chachuat B, 2013, A validated Integration algorithm for nonlinear ODEs using Taylor models and ellipsoidal calculus, 52nd IEEE Annual Conference on Decision and Control (CDC), Publisher: IEEE, Pages: 484-489, ISSN: 0191-2216
his paper presents a novel algorithm for bounding the reachable set of parametric nonlinear differential equations. This algorithm is based on a first-discretize-then-bound approach to enclose the reachable set via propagation of a Taylor model with ellipsoidal remainder, and it accounts for truncation errors that are inherent to the discretization. In contrast to existing algorithms that proceed in two phases-an a priori enclosure phase, followed by a tightening phase-the proposed algorithm first predicts a continuous-time enclosure and then seeks a maximal step-size for which validity of the predicted enclosure can be established. It is shown that this reversed approach leads to a natural step-size control mechanism, which no longer relies on the availability of an a priori enclosure. Also described in the paper is an open-source implementation of the algorithm in ACADO Toolkit. A simple numerical case study is presented to illustrate the performance and stability of the algorithm.
Hartmann P, Nikolaou A, Chachuat B, et al., 2013, A dynamic model coupling photoacclimation and photoinhibition in microalgae, ECC 2013, Publisher: IEEE, Pages: 4178-4183
Microalgae are often considered a promising alternative for production of renewable energy, particularly as a potential producer of biodiesel. In order to improve large-scale, industrial culturing systems, the development of mathematical models that are capable of predicting photosynthetic productivity under dynamic conditions is crucial. Especially important are the processes of growth inhibition due to excess light, known as photoinhibition, and of adjustment of the light harvesting capacity to photon flux, known as photoacclimation. In this paper, we develop a dynamic model that accounts for the processes of photoinhibition, photoacclimation and growth in microalgae, thereby spanning multiple time scales. The properties of the model are investigated under quasi steady-state conditions and the model is validated against several experimental data sets from the literature. We also discuss how the model can provide new insights into the mechanisms underlying photoacclimation. © 2013 EUCA.
Paulen R, Villanueva M, Fikar M, et al., 2013, Guaranteed parameter estimation in nonlinear dynamic systems using improved bounding techniques, 2013 European Control Conference (ECC), Publisher: IEEE, Pages: 4514-4519
This paper is concerned with guaranteed parameter estimation in nonlinear dynamic systems in a context of bounded measurement error. The problem consists of finding - or approximating as closely as possible - the set of all possible parameter values such that the predicted outputs match the corresponding measurements within prescribed error bounds. An exhaustive search procedure is applied, whereby the parameter set is successively partitioned into smaller boxes and exclusion tests are performed to eliminate some of these boxes, until a prespecified threshold on the approximation level is met. Exclusion tests rely on the ability to bound the solution set of the dynamic system for a given parameter subset and the tightness of these bounds is therefore paramount. Equally important is the time required to compute the bounds, thereby defining a trade-off. It is the objective of this paper to investigate this trade-off by comparing various bounding techniques based on interval arithmetic, Taylor model arithmetic and ellipsoidal calculus. When applied to a simple case study, ellipsoidal and Taylor model approaches are found to reduce the number of iterations significantly compared to interval analysis, yet the overall computational time is only reduced for tight approximation levels due to the computational overhead. © 2013 EUCA.
Paulen R, Villanueva M, Chachuat B, 2013, Optimization-based domain reduction in guaranteed parameter estimation of nonlinear dynamic systems, 9th IFAC Symposium on Nonlinear Control Systems, 2013, Publisher: International Federation of Automatic Control, Pages: 564-569, ISSN: 1474-6670
This paper is concerned with guaranteed parameter estimation in nonlinear dynamic systems in a context of bounded measurement error. The problem consists of finding-or approximating as closely as possible-the set of all possible parameter values such that the predicted outputs match the corresponding measurements within prescribed error bounds. An exhaustive search procedure is applied, whereby the parameter set is successively partitioned into smaller boxes and exclusion tests are performed to eliminate some of these boxes, until a prespecified threshold on the approximation level is met. In order to enhance the convergence of this procedure, we investigate the use of optimization-based domain reduction techniques for tightening the parameter boxes before partitioning. We construct such bound-reduction problems as linear programs from the polyhedral relaxation of Taylor models of the predicted outputs. When applied to a simple case study, the proposed approach is found to reduce the computational burden significantly, both in terms of CPU time and number of iterations. © IFAC.
Podmajersky M, Fikar M, Chachuat B, 2013, Measurement-based optimization of batch and repetitive processes using an integrated two-layer architecture, Journal of Process Control, Vol: 23, Pages: 943-955, ISSN: 1873-2771
This paper is concerned with optimal control of batch and repetitive processes in the presence of uncertainty. An integrated two-layer optimization strategy is proposed, whereby within-run corrections are performed using a neighboring-extremal update strategy and run-to-run corrections are based on a constraint-adaptation scheme. The latter is appealing since a feasible operating strategy is guaranteed upon convergence, and its combination with neighboring-extremal updates improves the reactivity and convergence speed. Moreover, these two layers are consistent in that they share the same objective function. The proposed optimization scheme is declined into two versions, namely an indirect version based on the Pontryagin maximum principle and a direct version that applies a control parameterization and nonlinear programming techniques. Although less rigorous, the latter approach can deal with singular extremals and path constraints as well as handle active-set changes more conveniently. Two case studies are considered. The indirect approach is demonstrated for a level-control problem in an experimental two-tank system, whereas the direct approach is illustrated in numerical simulation on a fed-batch reactor for acetoacetylation of pyrrole. The results confirm that faster adaptation is possible with the proposed integrated two-layer scheme compared to either constraint adaptation or neighboring-extremal update alone.
Villanueva M, Paulen R, Houska B, et al., 2013, Enclosing the Reachable Set of Parametric ODEs using Taylor Models and Ellipsoidal Calculus, 23 EUROPEAN SYMPOSIUM ON COMPUTER AIDED PROCESS ENGINEERING, Vol: 32, Pages: 979-984, ISSN: 1570-7946
One of the main bottlenecks of state of the art algorithm for global dynamic optimizationis the computation of enclosures of parametric differential equations (ODEs). Afterreviewing existing techniques based on Taylor model propagation and ellipsoidal calculusfor nonlinear dynamic processes, we introduce a novel algorithm for computing suchstate enclosures. Here, the bounding strategy employs a Taylor series with an ellipsoidalremainder bound. We analyze the convergence properties of the new ODE enclosure forsmall parameter intervals and provide conditions under which higher order convergencecan be proven. Moreover, we discuss implementation details and practical advantages byapplying the method to a numerical test example.
Rajyaguru J, Chachuat B, 2013, Taylor models in deterministic global optimization for large-scale systems with few degrees of freedom, Vol: 32, Pages: 973-978, ISSN: 1570-7946
Many process systems applications comprise large sets of nonlinear model equations, whose participating variables can be split naturally into independent and dependent variable subsets. This structure can be exploited for deterministic global optimization based on a sequential approach, which performs the optimization in the reduced space of independent variables by considering the model as implicit equations. This paper presents a new method for constructing Taylor model estimators of the implicit equation solutions in order to generate tighter lower bounds on the reduced-space optimization problem. The convergence properties of these estimators are analyzed through numerical examples, and the global optimization approach is demonstrated on a numerical case study featuring a discretized PDE system. © 2013 Elsevier B.V.
Nikolaou A, Hartmann P, Bernard O, et al., 2013, A dynamic model coupling photoacclimation and photoinhibition in microalgae, Pages: 351-352
Bompadre A, Mitsos A, Chachuat B, 2012, Convergence analysis of Taylor models and McCormick-Taylor models, Journal of Global Optimization, Vol: 57, Pages: 75-114, ISSN: 1573-2916
This article presents an analysis of the convergence order of Taylor models and McCormick-Taylor models, namely Taylor models with McCormick relaxations as the remainder bounder, for factorable functions. Building upon the analysis of McCormick relaxations by Bompadre and Mitsos (J Glob Optim 52(1):1–28, 2012), convergence bounds are established for the addition, multiplication and composition operations. It is proved that the convergence orders of both qth-order Taylor models and qth-order McCormick-Taylor models are at least q + 1, under relatively mild assumptions. Moreover, it is verified through simple numerical examples that these bounds are sharp. A consequence of this analysis is that, unlike McCormick relaxations over natural interval extensions, McCormick-Taylor models do not result in increased order of convergence over Taylor models in general. As demonstrated by the numerical case studies however, McCormick-Taylor models can provide tighter bounds or even result in a higher convergence rate.
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