Showing papers on "Mathematical model published in 2007"

Efficient collocational approach for parametric uncertainty analysis

Abstract: A numerical algorithm for effective incorporation of parametric uncertainty into mathematical models is presented. The uncertain parameters are modeled as random variables, and the governing equations are treated as stochastic. The solutions, or quantities of interests, are expressed as convergent series of orthogonal polynomial expansions in terms of the input random parameters. A high-order stochastic collocation method is employed to solve the solution statistics, and more importantly, to reconstruct the polynomial expansion. While retaining the high accuracy by polynomial expansion, the resulting “pseudo-spectral” type algorithm is straightforward to implement as it requires only repetitive deterministic simulations. An estimate on error bounded is presented, along with numerical examples for problems with relatively complicated forms of governing equations.

Online Prediction Under Model Uncertainty via Dynamic Model Averaging: Application to a Cold Rolling Mill

Abstract: We consider the problem of online prediction when it is uncertain what the best prediction model to use is. We develop a method called Dynamic Model Averaging (DMA) in which a state space model for the parameters of each model is combined with a Markov chain model for the correct model. This allows the "correct" model to vary over time. The state space and Markov chain models are both specified in terms of forgetting, leading to a highly parsimonious representation. As a special case, when the model and parameters do not change, DMA is a recursive implementation of standard Bayesian model averaging, which we call recursive model averaging. The method is applied to the problem of predicting the output strip thickness for a cold rolling mill, where the output is measured with a time delay. We found that when only a small number of physically motivated models were considered and one was clearly best, the method quickly converged to the best model, and the cost of model uncertainty was small; indeed DMA performed slightly better than the best physical model. When model uncertainty and the number of models considered were large, our method ensured that the penalty for model uncertainty was small. At the beginning of the process, when control is most difficult, we found that DMA over a large model space led to better predictions than the single best performing physically motivated model. We also applied the method to several simulated examples, and found that it recovered both constant and time-varying regression parameters and model specifications quite well.

Error modelling, calibration and analysis of an AM–CW terrestrial laser scanner system

Abstract: A rigorous method for terrestrial laser scanner self-calibration using a network of signalised points is presented. Exterior orientation, object point co-ordinates and additional parameters are estimated simultaneously by free network adjustment. Spherical co-ordinate observation equations are augmented with a set of additional parameters that model systematic errors in range, horizontal direction and elevation angle. The error models include both physically interpretable and empirically identified components. Though the focus is on one particular make and model of AM–CW scanner system, the Faro 880, the mathematical models are formulated in a general framework so their application to other instruments only requires selection of an appropriate set of additional parameters. Results from controlled testing show that significant improvement is achieved by using the proposed model in terms of both reducing the magnitude of observational residuals as well as the three-dimensional positioning accuracy of signalised points. Ten self-calibration datasets captured over the course of 13 months are used to examine short- and long-term additional parameter stability via standard hypothesis testing techniques. Detailed investigations into correlation mechanisms between model parameters accompany the self-calibration solution analyses. Other contributions include an observation model for incorporation of integrated inclinometer observations into the self-calibration solution and an effective a priori outlier removal method. The benefit of the former is demonstrated to be reduced correlation between exterior orientation and additional parameters, even if inclinometer precision is low. The latter is arrived at by detailed analysis of the influence of incidence angle on range.

Biology by numbers: mathematical modelling in developmental biology.

Abstract: In recent years, mathematical modelling of developmental processes has earned new respect. Not only have mathematical models been used to validate hypotheses made from experimental data, but designing and testing these models has led to testable experimental predictions. There are now impressive cases in which mathematical models have provided fresh insight into biological systems, by suggesting, for example, how connections between local interactions among system components relate to their wider biological effects. By examining three developmental processes and corresponding mathematical models, this Review addresses the potential of mathematical modelling to help understand development.

Resolving the biodiversity paradox

Abstract: The paradox of biodiversity involves three elements, (i) mathematical models predict that species must differ in specific ways in order to coexist as stable ecological communities, (ii) such differences are difficult to identify, yet (iii) there is widespread evidence of stability in natural communities. Debate has centred on two views. The first explanation involves tradeoffs along a small number of axes, including ‘colonization-competition’, resource competition (light, water, nitrogen for plants, including the ‘successional niche’), and life history (e.g. high-light growth vs. low-light survival and few large vs. many small seeds). The second view is neutrality, which assumes that species differences do not contribute to dynamics. Clark et al. (2004) presented a third explanation, that coexistence is inherently high dimensional, but still depends on species differences. We demonstrate that neither traditional low-dimensional tradeoffs nor neutrality can resolve the biodiversity paradox, in part by showing that they do not properly interpret stochasticity in statistical and in theoretical models. Unless sample sizes are small, traditional data modelling assures that species will appear different in a few dimensions, but those differences will rarely predict coexistence when parameter estimates are plugged into theoretical models. Contrary to standard interpretations, neutral models do not imply functional equivalence, but rather subsume species differences in stochastic terms. New hierarchical modelling techniques for inference reveal high-dimensional differences among species that can be quantified with random individual and temporal effects (RITES), i.e. process-level variation that results from many causes. We show that this variation is large, and that it stands in for species differences along unobserved dimensions that do contribute to diversity. High dimensional coexistence contrasts with the classical notions of tradeoffs along a few axes, which are often not found in data, and with ‘neutral models’, which mask, rather than eliminate, tradeoffs in stochastic terms. This mechanism can explain coexistence of species that would not occur with simple, low-dimensional tradeoff scenarios.

Parametric reduced-order models for probabilistic analysis of unsteady aerodynamic applications

Abstract: DOI: 102514/135850 We address the problem of propagating input uncertainties through a computational fluid dynamics model Methods such as Monte Carlo simulation can require many thousands (or more) of computational fluid dynamics solves, rendering them prohibitively expensive for practical applications This expense can be overcome with reduced-order models that preserve the essential flow dynamics The specific contributions of this paper are as follows: first, to derive a linearized computational fluid dynamics model that permits the effects of geometry variations to be represented with an explicit affine function; second, to propose an adaptive sampling method to derive a reduced basis that is effective over the joint probability density of the geometry input parameters The method is applied to derive efficient reduced models for probabilistic analysis of a two-dimensional problem governedbythelinearized EulerequationsReduced-order modelsthatachieve 3-orders-of-magnitude reduction in the number of states are shown to accurately reproduce computational fluid dynamics Monte Carlo simulation results at a fraction of the computational cost

Stochastic delay propagation in railway networks and phase-type distributions

Abstract: It is difficult to analyse stochastic models for the propagation of delays in railway networks. It seems that the choice is between very global mathematical (queueing) models at one extreme and simulation models at the other. In this paper we discuss a (fairly general but not too detailed) model for delay propagation and show that in a world of so-called phase-type distributions it is possible to derive secondary delay distributions from primary delay distributions. We shall explain why phase-type distributions and the delay propagation model are suited for each other and show by an example that it is possible to develop algorithms that analyse such networks.

Modeling and Dynamic Simulations of Doubly Fed Adjustable-Speed Pumped Storage Units

Abstract: This paper derives the mathematical models of doubly fed adjustable-speed pumped storage units (DFASPSUs) to be utilized in the power system analysis. It adopts the improved induction machine model with ac excitation on the wire-wound rotor as well as the field-oriented control theory for the study and analysis of DFASPSU models. First, the rotor excitation voltage is separated into two components, i.e., the q-axis voltage and the d-axis voltage, so as to control the output active and reactive powers, respectively. Next, a dynamic model of DFASPSU is derived using the swing equation of rotary electric machine and the equation of motion of rotor rotating mass. Finally, computer simulation is carried out to compare its results with practical examples in terms of dynamic characteristics for verifying the correctness of the proposed models.

A simplified approach to quantifying predictive and parametric uncertainty in artificial neural network hydrologic models

Abstract: [1] One of the principal sources of uncertainty in hydrological models is the absence of understanding of the complex physical processes of the hydrological cycle within the system. This leads to uncertainty in input selection and consequently its associated parameters, and hence evaluation of uncertainty in a model becomes important. While there has been considerable interest in developing methods for uncertainty analysis of artificial neural network (ANN) models, most of the methods are relatively complex and/or require assumption about the prior distribution of the uncertain parameters. This paper presents an effective and simple way to perform uncertainty analysis for ANN-based hydrologic model. The method is based on the concept of bootstrap technique and is demonstrated through a case study of the Kolar River basin located in India. The method effectively quantifies uncertainty in the model output and the parameters arising from variation in input data used for calibration. In the current study, the uncertainty due to model architecture and the input vector are not directly considered; they have been minimized during the model calibration. The results from the case study suggest that the sampling variability of the training patterns as well as the initial guess of the parameters of ANN do not have significant impact on the model performance. However, despite good generalization properties for the models developed in this study, most of them fail to capture the hydrograph peak flow characteristics. The proposed method of uncertainty analysis is very efficient, can be easily applied to an ANN-based hydrologic model, and clearly illustrates the strong and weak points of the ANN model developed.

On the mathematical theory of vehicular traffic flow II: Discrete velocity kinetic models

Abstract: This paper deals with the modelling of vehicular traffic flow by methods of the discrete mathematical kinetic theory. The discretization is developed in the velocity space by a grid adapted to the local density. The discretization overcomes, at least in part, some technical difficulties related to the selection of the correct representation scale, while the adaptative grid allows an improved description of various phenomena related to vehicular traffic flow. Specific models are proposed and a qualitative and computational analysis is developed to show the properties of the model and their ability to describe real flow conditions. A critical analysis, proposed in the last part of the paper, outlines suitable research perspectives.

Approximation errors and truncation of computational domains with application to geophysical tomography

Abstract: Numerical realization of mathematical models always induces errors to the computational models, thus affecting both predictive simulations and related inversion results. Especially, inverse problems are typically sensitive to modeling and measurement errors, and therefore the accuracy of the numerical model is a crucial issue in inverse computations. For instance, in problems related to partial differential equation models, the implementation of a numerical model with high accuracy necessitates the use of fine discretization and realistic boundary conditions. However, in some cases realistic boundary conditions can be posed only for very large or even unbounded computational domains. Fine discretization and large domains lead to very high-dimensional models that may be of prohibitive computational cost. Therefore, it is often necessary in practice to use coarser discretization and smaller computational domains with more or less incorrect boundary conditions in order to decrease the dimensionality of the model. In this paper we apply the recently proposed approximation error approach to the problem of incorrectly posed boundary conditions. As a specific computational example we consider the imaging of conductivity distribution of soil using electrical resistance tomography. We show that the approximation error approach can also be applied to domain truncation problems and that it allows one to use significantly smaller scale forward models in the inversion.

Nonlinear plasma dynamics in capacitive radio frequency discharges

Abstract: The excitation of harmonics in the current of capacitive radio frequency (RF) discharges is a frequently observed phenomenon. The effect is of interest for several reasons. It forms, for instance, the basis of a successful diagnostic concept for technical plasmas, and it is intimately connected to the process of electron heating in capacitive discharges. Recently, mathematical models were proposed which interpret the phenomenon as the self-excitation of the plasma series resonance by the nonlinearity of the boundary sheath. These models are surprisingly successful but suffer from the limitation that they analyse the plasma dynamics in terms of global equations with concentrated parameters. They are unable to account for the complex multi-mode current waveforms seen in experiments and also cannot resolve the electromagnetic effects which dominate contemporary/large area, high frequency processing discharges. This paper aims to correct the deficiency by presenting a spatially resolved, fully electromagnetic model of the nonlinear RF dynamics of a bounded plasma. The model holds for arbitrary plasma reactor geometries and external RF excitations and makes no assumptions on the homogeneity of the plasma or the characteristics of the boundary sheath. A functional analytic (Hilbert space) formulation of the model is given which allows for an exact solution in terms of an infinite power/Fourier series. For the case of an idealized cylindrical reactor, the model is also explicitly evaluated. The calculated RF current wave forms exhibit the complex multi-mode structure of the currents observed in experiments and follow the same scaling laws. It is concluded that the presented model is capable of describing the nonlinear dynamics of capacitive RF discharges of all sizes and that it provides a significant improvement over both the established nonlinear global models and linear models with spatial resolution.

Novel Anchor System for CFRP Rod: Finite-Element and Mathematical Models

Abstract: An innovative wedge-anchor system with a longitudinal circular profile has been designed for gripping carbon-fiber-reinforced-polymer rods. The anchor system, consisting of a barrel, a number of wedges, and a sleeve, is capable of carrying the ultimate tensile strength of the rod. Three-dimensional (3D) and axisymmetric finite-element models have been developed and applied in order to determine the stresses inside the anchor system. Using the 3D model, the overall performance of the anchor represented by the tensile load-displacement relationship compared favorably with the experimental results. A comparison study between the 3D and axisymmetric models revealed that the axisymmetric model underestimated the contact pressure on the rod surface. Using the force-fitting principle of thick cylinders, a mathematical model was also developed, and under specific assumptions, the mathematical solution compared favorably with finite-element model results.

Semi-Analytic Modelling of Subsidence

Abstract: This paper presents a forward model for subsidence prediction caused by extraction of hydrocarbons. The model uses combinations of analytic solutions to the visco-elastic equations, which approximate the boundary conditions. There are only a few unknown parameters to be estimated, and, consequently, calculations are very fast. The semi-analytic model is applicable to a uniform and layer-cake stratigraphy, with visco-elastic parameters changing per layer, and an arbitrary depletion pattern. By its capabilities to handle a multi-layered visco-elastic subsurface, the semi-analytic model fills the gap between the analytic single-layered elastic models available to date and the more elaborate numerical, e.g. finite element, models. © 2006 Springer Science+Business Media, LLC.

Study on mathematical model for VHF partial discharge of typical insulated defects in GIS

Abstract: Very high frequency (VHF) measurement is a valid technique for partial discharge (PD) detection, and PD mathematical models are very important to study PD denoising and waveshape pattern recognition etc. However, these mathematical models based on the traditional PD measurement have been established according to RC or LCR resistance detection so that they are not suitable for VHF measurements. PD signals are detected based on four typical physical models of defects in gas insulated substation (GIS) using VHF method. The VHF PD mathematical models are established, and the principles and methods of establishment are summarized. Furthermore, the parameters of the mathematical models are given and the validity of PD mathematical model is proved by fitting error, power spectrum and time-frequency analysis. Results show that the VHF PD mathematical models can facilitate theoretical simulation of different insulated defects by using the values of given parameters. In the end, applications of the VHF PD mathematical models and complex wavelet transform for extracting the relevant signals from a white noise background are illustrated

The effect of interpolation methods in surface definition: an experimental study

Abstract: Mappings of the earth surface and their representation in 3D (three-dimensional) models are commonly used in most recent research. Modeling research, which starts with classical surveying methods, acquires new dimensions matching the modern technologies. 3D models of any object or earth surface can be used in much visual and scientific research. A digital model of the landscape is an important part within creation of geo-information systems used in the public administration and in the commercial sphere. It is an important tool in applications such as geomorphology, hydrology, geology, cartography, ecology, mining etc. Values of volume in terrains that do not have regular geometric structure can be obtained more accurately by using 3D models of surfaces with respect to developing technology. Basic data of 3D models must indicate 3D coordinates of the surveyed object in the reference frame. Distribution and intensity of points are important factors in modeling earth surfaces. A minimum number of points is desired in defining an object in 3D. Interpolation methods employing different mathematical models are used to obtain 3D models of terrain surfaces. In this study, the effect of interpolation methods in defining a terrain surface is investigated. For this purpose, a uniform surface, hill-shaped artificial object with a known volume is employed. The 3D surface and volume are calculated by using 12 different interpolation methods. Point distribution, point intensity and accuracy of point measurements are not considered. The same data set was used for all the interpolation methods. The interpolation methods are compared and evaluated based on the results. Copyright © 2007 John Wiley & Sons, Ltd.

On mathematical modeling of circadian rhythms, performance, and alertness.

Abstract: Mathematical models of neurobehavioral performance and alertness have both basic science and practical applications. These models can be especially useful in predicting the effect of different sleep-wake schedules on human neurobehavioral objective performance and subjective alertness under many conditions. Several relevant models currently exist in the literature. In principle, the development and refinement of any mathematical model should be based on an explicit modeling methodology, such as the Box modeling paradigm, that formally defines the model structure and calculates the set of parameters. While most mathematical models of neurobehavioral performance and alertness include homeostatic, circadian, and sleep inertia components and their interactions, there may be fundamental differences in the equations included in these models. In part, these may be due to differences in the assumptions of the underlying physiology. Because the choice of model equations can have a dramatic influence on the results, it is necessary to consider these differences in assumptions when examining the results from a model and when comparing results across models. This article presents principles of mathematical modeling and examples of how such procedures can be applied to the development and refinement of mathematical models of neurobehavioral performance and alertness. This article also presents several methods of testing and comparing these models, suggests different uses of the models, and discusses problems with current models.

The leader-follower location model

Abstract: This paper summarizes some results for the leader–follower location model on networks in several scenarios Discretization results are considered and differences derived from the inelastic and elastic demand assumptions, as well as from the customer’s choice rule, are emphasized Finally, some issues for future lines of investigation are suggested