The aim of this work is to test the Levemberg Marquardt and BFGS Broyden Fletcher Goldfarb Shanno algorithms, implemented by the matlab functions lsqnonlin and fminunc of the Optimization Toolbox, for modeling the kinetic terms occurring in chemical processes of adsorption.
We are interested in tests with noisy data that are obtained by adding Gaussian random noise to the solution of a model with known parameters. While both methods are very precise with noiseless data, by adding noise the quality of the results is greatly worsened. The semi-convergent behaviour of the relative error curves is observed for both methods. Therefore a stopping criterion, based on the Discrepancy Principle is proposed and tested.
Great improvement is obtained for both methods, making it possible to compute stable solutions also for noisy data. An important topic in many engineering applications is that of estimating parameters of differential models from partial and possibly noisy measurements. For example the removal of pollutants from surface water and groundwater requires the optimization of partial differential models where the dispersion, mass transfer and reaction terms are estimated from data in column reactor experiments [ 123 ].
Hence the final problem consists in the minimization of the distance between the data and the computed approximation F q measured in a norm dependent on the model of the data noise. In Sect.
The optimization methods implemented by the functions fminunc and lsqnolin are outlined in Sect. Finally in Sect. In this section we define the stopping rule applied to the iterative methods used by fminunc and lsqnolin functions to solve the test problem obtained by the model described in Sect. We start by a brief outline of the iterative numerical methods tested in the numerical experiments. In the first experiment we compare the results obtained by the Levemberg Marquardt and BFGS methods without data noise.
Test with noiseless measurements. Levemberg Marquardt LM. The present work reports tests of the Levemberg Marquardt and BFGS algorithms for modeling the kinetic terms, occurring in chemical processes of adsorption, in the presence of noisy data.
Conference paper First Online: 02 April Download conference paper PDF. We describe here the differential problem state equation used as a test problem. It consists of a system of two time dependent differential equations representing the dynamic evolution of the liquid and solid phases of Polyphenolic compounds [ 4 ].
The first method, Levemberg Marquardt, is specific of the non linear Least squares minimization while the second method, BFGS Broyden Fletcher Goldfarb Shanno quasi Newton method, is applied to more general nonlinear minimization problems.
See [ 56 ] for details.Kinetic models contain unknown parameters that are estimated by optimizing the fit to experimental data.
This task can be computationally challenging due to the presence of local optima and ill-conditioning. While a variety of optimization methods have been suggested to surmount these issues, it is difficult to choose the best one for a given problem a priori.
A systematic comparison of parameter estimation methods for problems with tens to hundreds of optimization variables is currently missing, and smaller studies provided contradictory findings. We use a collection of benchmarks to evaluate the performance of two families of optimization methods: i multi-starts of deterministic local searches and ii stochastic global optimization metaheuristics; the latter may be combined with deterministic local searches, leading to hybrid methods.
A fair comparison is ensured through a collaborative evaluation and a consideration of multiple performance metrics. We discuss possible evaluation criteria to assess the trade-off between computational efficiency and robustness.
Our results show that, thanks to recent advances in the calculation of parametric sensitivities, a multi-start of gradient-based local methods is often a successful strategy, but a better performance can be obtained with a hybrid metaheuristic. The best performer combines a global scatter search metaheuristic with an interior point local method, provided with gradients estimated with adjoint-based sensitivities.
We provide an implementation of this method to render it available to the scientific community. Supplementary data are available at Bioinformatics online. Mechanistic kinetic models provide a basis to answering biological questions via mathematical analysis.
Dynamical systems theory can be used to interrogate these kinetic models, enabling a more systematic analysis, explanation and understanding of complex biochemical pathways. Ultimately, the goal is the model-based prediction of cellular functions under new experimental conditions Almquist et al. During the last decade, many efforts have been devoted to developing increasingly detailed and, therefore, larger systems biology models Karr et al.
Such models are often formulated as nonlinear ordinary differential equations ODEs with unknown parameters. As it is impossible to measure all parameters directly, parameter estimation i. The unknown parameters are typically estimated by solving a mathematical optimization problem which minimizes the mismatch between model predictions and measured data Ashyraliyev et al.
Parameter estimation for dynamical systems is an inverse problem Villaverde and Banga, that exhibits many possible challenges and pitfalls, mostly associated with ill-conditioning and non-convexity Schittkowski, These properties, which are in general only known a posterioriinfluence the performance of optimization methods. Even if we restrict our attention to a specific class of problems within the same field e.
Hence, methods need to be benchmarked for a representative collection of problems of interest in order to reach meaningful conclusions. In this study, we consider the class of medium to large scale kinetic models. These models pose several challenges, such as computational complexity, and an assessment of the performance of optimization methods is particularly important Babtie and Stumpf, ; Degasperi et al.
The calibration of large-scale kinetic models usually requires the optimization of a multi-modal objective function Chen et al. Local optimization methods, such as Levenberg-Marquardt or Gauss-Newton Schittkowski,which converge to local optima, will only find a global optimum for appropriate starting points.
Convergence to a suboptimal solution is an estimation artifact that can lead to wrong conclusions: we might think that the mechanism considered is not suitable to explain the data, while the real reason might be that the method failed to locate the global optimum Chachuat et al.
In order to avoid suboptimal solutions, many studies have recommended the use of global optimization techniques Ashyraliyev et al. One of the earliest and simplest global optimization methods is the multi-start, which consists of launching many local searches from different initial points in parameter space, assuming that one of them will be inside the basin of attraction of the global solution. The comparison of global optimization methods was the topic of several research papers.
Interestingly, the evaluation results led to apparently contradictory conclusions, advocating the use of either multi-start local optimization Hross and Hasenauer, ; Raue et al.
These contradictions cannot be explained by the no-free lunch theorems for optimization Wolpert and Macready,since i the problems analyzed possessed relatively similar characteristics i. Hence, we suggest two alternative explanations: i the comparisons were carried out by researchers who had substantially more experience with the tuning of one type of the considered methods, and ii the performance metrics differed and might even have been biased towards particular approaches.
To circumvent these issues, we established an intensive collaboration between experienced users of multi-start local optimization the HZM group and metaheuristics the CSIC group. Through a joint development of performance metrics and evaluation procedures we attempted to ensure a fair comparison of different approaches. In this study, we present the results of this collaboration: the development of a performance metric suited for the comparison of different methods, and the evaluation of the state-of-the-art in parameter estimation methodologies.
Based on these results we provide guidelines for their application to large kinetic models in systems biology.
To this end, we use seven previously published estimation problems to benchmark a number of optimization methods.Mathematical modeling is a key process to describe the behavior of biological networks. One of the most difficult challenges is to build models that allow quantitative predictions of the cells' states along time. Recently, this issue started to be tackled through novel in silico approaches, such as the reconstruction of dynamic models, the use of phenotype prediction methods, and pathway design via efficient strain optimization algorithms.
The use of dynamic models, which include detailed kinetic information of the biological systems, potentially increases the scope of the applications and the accuracy of the phenotype predictions. New efforts in metabolic engineering aim at bridging the gap between this approach and other different paradigms of mathematical modeling, as constraint-based approaches.
These strategies take advantage of the best features of each method, and deal with the most remarkable limitation—the lack of available experimental information—which affects the accuracy and feasibility of solutions. Parameter estimation helps to solve this problem, but adding more computational cost to the overall process. Moreover, the existing approaches include limitations such as their scalability, flexibility, convergence time of the simulations, among others.
The aim is to establish a trade-off between the size of the model and the level of accuracy of the solutions. In this work, we review the state of the art of dynamic modeling and related methods used for metabolic engineering applications, including approaches based on hybrid modeling.
We describe approaches developed to undertake issues regarding the mathematical formulation and the underlying optimization algorithms, and that address the phenotype prediction by including available kinetic rate laws of metabolic processes.
Then, we discuss how these have been used and combined as the basis to build computational strain optimization methods for metabolic engineering purposes, how they lead to bi-level schemes that can be used in the industry, including a consideration of their limitations. Systems biology and bioinformatics tools help to analyze relevant data and properties e.
This has stimulated the interest to build genome-scale networks, allowing to perform in silico simulations of complex biological systems, and to understand the way metabolic flux distributions change within a particular biological network for predicting cellular phenotypes McCloskey et al.
Moreover, mathematical modeling of cellular metabolism, studied under various environmental and genetic conditions, has started to reasonably support metabolic engineering ME tasks, such as design of desirable strains, by optimal selection of gene deletions or expression modulation for the overproduction of industrial compounds Stephanopoulos et al. Metabolic network modeling can be based on the knowledge of enzyme mechanisms and experimental data to build a representation of a dynamic system, able to describe changes on concentrations of metabolites over time by using systems of ordinary differential equations ODEs.
These ODEs contain initial values for metabolite concentrations, reaction rate equations and kinetic parameters. These representations were applied to model small-scale central metabolic pathways of well-known organisms, such as Saccharomyces cerevisiae Rizzi et al. However, dynamic representations for large-scale systems are not always possible due to the lack of experimental kinetic information to build proper reaction rate equations.
This framework, that tackles cell metabolism modeling tasks using a formulation based on the dynamics of metabolic processes, gives detailed and unique solutions in time for the transient and the equilibrium states, from any initial metabolite concentration condition. It is based on kinetic rate laws inferred from biochemical and mechanistic information, while the values of the final fluxes are obtained directly from the rate laws and the metabolite concentrations at equilibrium.
However, this type of modeling requires considerable amounts of data that are not always available, such as kinetic parameters or total enzyme concentrations Smallbone et al. Opposite to the dynamic case, an alternative is to restrict models to contain only reaction stoichiometry and reversibility, based on the assumption of steady-state operation, thus unable to express transient behaviors.
In this approach, models use formulations based on linear equation systems, which are typically underdetermined, i. This leads to the imposition of certain restrictions constraints and assumptions objective functions to be able to find an optimal solution Lewis et al.
This formulation is based on the stoichiometry, via constraint-based modeling CBMhelping to define limits on the behavior of a system depending on physical and chemical restrictions, such as fluxes, mass balance and thermodynamics. This approach yields solutions that might not be unique, represented as steady-state flux distributions, which are within the space of feasible solutions called the flux hypercone Wagner and Urbanczik, While CBM approaches do not include physiological knowledge about metabolite concentrations in time nor transient behavior, they remove the need for a detailed mechanistic knowledge, since only parameters for minimum and maximum flux bounds are required.
As a consequence, a solution space from a dynamic formulation is a subset of a constraint-based solution, since they use the same core constraints, knowing that the dynamic model adds other constraints from specific values of kinetic information Machado et al.
Mathematical modeling can be used to explain or to predict the behavior of a system. This work is mainly focused on modeling frameworks based on the knowledge of systems' dynamics to increase the accuracy of the predictions of strain optimization methods, considering the limitations that the approaches can present.The reaction of phenolphthalein with a base solution follows the 2 reaction sequence given below.
Reaction 1 is very fast and can be assumed to be instantaneous when the phenolphthalein Ph is added to a hydroxide solution. We are considering using phenolphthalein as an indicator to determine the residence time of several large CSTR reactors in a pilot plant. Please determine the kinetic parameters for reaction 2 with respect to phenolphthalein associated with the fading of phenolphthalein in sodium hydroxide solutions reaction orders, A1, A2, Ea1, and Ea2.
An engineer collected time-varying data for a non-isothermal run and needs you to estimate the kinetic parameters. The temperature ranged from about 60 - degF.
Nonlinear confidence intervals can also be visualized as a function of 2 parameters. In this case, both parameters are simultaneously varied to find the confidence region. The confidence interval is determined with an F-test that specifies an upper limit to the deviation from the optimal solution. For many problems, this creates a multi-dimensional nonlinear confidence region. In the case of 2 parameters, the nonlinear confidence region is a 2-dimensional space.
Below is an example that shows the confidence region for the dye fading experiment confidence region for forward and reverse activation energies. This plot demonstrates that the 2D confidence region is not necessarily symmetric. Design Optimization. Syllabus Book Schedule. Estimate Kinetic Parameters from Dynamic Data. Options: fixed fluid orange blue green pink cyan red violet.
View Edit History Print.The Seahawks' 24-10 win over Philadelphia last week was another reminder that they shouldn't be written off in their injury-weakened state. An improved offensive line, a running game that has shown signs of life and a defense that has still been good without Richard Sherman and Kam Chancellor will be enough on Sunday in Jacksonville.
The defense -- which leads the NFL in sacks, pass defense and scoring -- has its hands full with Wilson, Jimmy Graham and Doug Baldwin, but there are enough playmakers at all levels to limit the big plays that Wilson creates with his legs. However, the Jags' offense is going to have a hard time moving the ball consistently.
Leonard Fournette is still dealing with an ankle injury and has run for just 226 yards and one touchdown on 77 carries (2.Bayesian parameter estimation
The offensive line is banged up too. The biggest issue, though, are the receivers dealing with the half of the 'Legion of Boom' still standing. Rookie Keelan Colehas been up and down all season and has battled drops. Rookie Dede Westbrook is playing in just his fourth game. Marqise Lee is the Jaguars' top option, but he has battled rib and knee injuries over the past six weeks and leads the NFL with seven drops.
The Jaguars just don't have enough firepower to score a lot of points. Tough to do, given that the Jaguars own the No. Also not helping this pick: Earlier this season, Seattle was throttled by the Titans (before playing frantic catch-up football). I'm not confidentBlake Bortles and the Jags' receivers can capitalize on the Achilles' heel of Seattle right now: a banged-up secondary missing Richard Sherman and Kam Chancellor.
Offensively, maybe the Seahawks should come out in no-huddle, letting Russell Wilson play with a sense of urgency in quarter No. Wilson is tied with Eli Manning(2011) for the most fourth-quarter touchdown passes in a single season.
Blake Bortles played well against the Colts last week, and I think it will carry over here. It won't be as good, but good enough. The Jaguars win a low-scoring game. Schematically, these two defenses are mirror images of each other, but Jacksonville has the superior unit at this point.
In fact, the Jaguars are the most talented team in football on the defensive side. The advantage Seattle has in this matchup is the fact the Seahawks offense practices against this scheme every day.
Now, they have to travel all the way across the country, and Jacksonville is ready for a slugfest. This is a supreme test for both teams. Athlete of the Week Stats for Kids Youth ProgramsFuel Up To Play 60 Play 60 Tuesdays Jr. Watch Video Week 14: Seahawks at Jaguars Preview Posted Dec 8th, 2017 The Seattle Seahawks hit the road this weekend for a Week 14 matchup with the Jacksonville Jaguars.
Watch Video Seahawks Are "Fired Up For This Opportunity" At Jacksonville Posted Dec 8th, 2017 The Seattle Seahawks took advantage of being able to practice outside all week and are "fired up" for the upcoming game against Jacksonville on Sunday.There are also many products that offer online monitoring for your home. Indeed, nearly everything can be connected to the Internet these days, and Apple is even getting in on the action here with its HomeKit SDK for iOS.
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Parameter Estimation Algorithms for Kinetic Modeling from Noisy Data
All of them are free of charge. Definitely worth checking out. Coaches send game footage and roster information to Krossover. Coaches and players can sort through the plays, separating them by any number of categories.Look for products labeled "sodium-free" (less than 5 milligrams of sodium per serving), "very low sodium" (35 milligrams or less per serving), or "no salt added.
Watch for additives such as monosodium glutamate (MSG), baking soda, baking powder, and sodium benzoate, which are all sodium compounds. When a recipe calls for even a pinch of salt, replace it with another herb or spice. You can add rosemary, marjoram, thyme, tarragon, onion powder, garlic powder, pepper, nutmeg, cumin, curry powder, ginger, cilantro, bay leaf, oregano, dry mustard, or dill.
Or sprinkle in a commercial salt-free seasoning blend. You can use dozens of different ingredients to add flavor to your low-sodium meals. Here are just a few ideas for common foods:U.
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Start with these four tips: 1. Cook It Yourself Packaged foods often have a lot of salt in them, since salt is a preservative. Check on Salty Items Check every food label for sodium content. Don't Follow Directions When a recipe calls for even a pinch of salt, replace it with another herb or spice. Here are just a few ideas for common foods: Marinate chicken breasts or pork chops in lemon juice, orange juice, or wine.
Roll fish in sesame seeds before baking. Spice up beef with a mixture of onion, peppers, sage, and thyme. Simmer carrots in cinnamon and nutmeg. Sprinkle some dill and parsley onto potatoes before roasting.
Add a dash of chili powder to corn. Toss your pasta with fresh chopped garlic. WebMD Medical Reference Reviewed by Kathleen M.
Zelman, MPH, RD, LD on April 28, 2016 SOURCES: U. BMJ, published online April 20, 2007. Oklahoma Department of Human Services: "Tips for Managing Your Congestive Heart Failure.
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