Spelling suggestions: "subject:"modulation functions""
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Modulating FunctionBased Method for Parameter and Source Estimation of Partial Differential EquationsAsiri, Sharefa M. 08 October 2017 (has links)
Partial Differential Equations (PDEs) are commonly used to model complex systems that arise for example in biology, engineering, chemistry, and elsewhere. The parameters (or coefficients) and the source of PDE models are often unknown and are estimated from available measurements. Despite its importance, solving the estimation problem is mathematically and numerically challenging and especially when the measurements are corrupted by noise, which is often the case. Various methods have been proposed to solve estimation problems in PDEs which can be classified into optimization methods and recursive methods. The optimization methods are usually heavy computationally, especially when the number of unknowns is large. In addition, they are sensitive to the initial guess and stop condition, and they suffer from the lack of robustness to noise. Recursive methods, such as observerbased approaches, are limited by their dependence on some structural properties such as observability and identifiability which might be lost when approximating the PDE numerically. Moreover, most of these methods provide asymptotic estimates which might not be useful for control applications for example. An alternative nonasymptotic approach with less computational burden has been proposed in engineering fields based on the socalled modulating functions. In this dissertation, we propose to mathematically and numerically analyze the modulating functions based approaches. We also propose to extend these approaches to different situations. The contributions of this thesis are as follows. (i) Provide a mathematical analysis of the modulating functionbased method (MFBM) which includes: its wellposedness, statistical properties, and estimation errors. (ii) Provide a numerical analysis of the MFBM through some estimation problems, and study the sensitivity of the method to the modulating functions' parameters. (iii) Propose an effective algorithm for selecting the method's design parameters. (iv) Develop a twodimensional MFBM to estimate spacetime dependent unknowns which is illustrated in estimating the source term in the damped wave equation describing the physiological characterization of brain activity. (v) Introduce a moving horizon strategy in the MFBM for online estimation and examine its effectiveness on estimating the source term of a first order hyperbolic equation which describes the heat transfer in distributed solar collector systems.

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Methods and Algorithms for Solving Inverse Problems for Fractional AdvectionDispersion EquationsAldoghaither, Abeer 12 November 2015 (has links)
Fractional calculus has been introduced as an e cient tool for modeling physical phenomena, thanks to its memory and hereditary properties. For example, fractional models have been successfully used to describe anomalous di↵usion processes such as contaminant transport in soil, oil flow in porous media, and groundwater flow. These models capture important features of particle transport such as particles with velocity variations and longrest periods.
Mathematical modeling of physical phenomena requires the identification of pa rameters and variables from available measurements. This is referred to as an inverse problem.
In this work, we are interested in studying theoretically and numerically inverse problems for space Fractional AdvectionDispersion Equation (FADE), which is used to model solute transport in porous media. Identifying parameters for such an equa tion is important to understand how chemical or biological contaminants are trans ported throughout surface aquifer systems. For instance, an estimate of the di↵eren tiation order in groundwater contaminant transport model can provide information about soil properties, such as the heterogeneity of the medium.
Our main contribution is to propose a novel e cient algorithm based on modulating functions to estimate the coe cients and the di↵erentiation order for space FADE,
which can be extended to general fractional Partial Di↵erential Equation (PDE). We also show how the method can be applied to the source inverse problem.
This work is divided into two parts: In part I, the proposed method is described and studied through an extensive numerical analysis. The local convergence of the proposed twostage algorithm is proven for 1D space FADE. The properties of this method are studied along with its limitations. Then, the algorithm is generalized to the 2D FADE.
In part II, we analyze direct and inverse source problems for a space FADE. The problem consists of recovering the source term using final observations. An analytic solution for the nonhomogeneous case is derived and existence and uniqueness of the solution are established. In addition, the uniqueness and stability of the inverse problem is studied. Moreover, the modulating functionsbased method is used to solve the problem and it is compared to a standard Tikhonobased optimization technique.

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Contributions to Datadriven and Fractionalorder Modelbased Approaches for Arterial Haemodynamics Characterization and Aortic Stiffness EstimationBahloul, Mohamed 26 April 2022 (has links)
Cardiovascular diseases (CVDs) remain the leading cause of death worldwide. Patients at risk of evolving CVDs are assessed by evaluating a risk factorbased score that incorporates different biomarkers ranging from age and sex to arterial stiffness (AS). AS depicts the rigidity of the arterial vessels and leads to an increase in the arterial pulse pressure, affecting the heart and vascular physiology. These facts have encouraged researchers to propose surrogate markers of cardiovascular risks and develop simple and noninvasive models to better understand cardiovascular system operations. This work thus fundamentally capitalizes on developing a novel class of lowdimensional physicsbased fractionalorder models of systemic arteries and exploring the feasibility of fractional differentiation order to portray the vascular stiffness. Fractionalorder modeling is a successful paradigm to integrate multiscale and interconnected mechanisms of the complex arterial system. However, this type of modeling alone often fails to efficiently integrate altered variabilities in vascular physiology from various sources of large datasets, multimodalities, and levels. In this regard, combining fractionalorderbased approaches with machine learning techniques presents a unique opportunity to develop a powerful prediction framework that reveals the correlation between intertwined vascular events.
This work is divided into three parts. The first part contributes to developing the fractionalorder lumped parametric model of the arterial system. First, we propose fractionalorder representations to model and characterize the complex and frequencydependent apparent arterial compliance. Second, we propose fractionalorder arterial Windkessel modeling the aortic input impedance and hemodynamic. Subsequently, the proposed models have been applied and validated using both human insilico healthy datasets and real vascular aging and hypertension.
The second part addresses the nonzero initial value problem for fractional differential equations (FDEs) and proposes an estimation technique for joint estimation of the input, parameters, and fractional differentiation order of noncommensurate FDEs. The performance of the proposed estimation techniques is illustrated on arterial and neurovascular hemodynamic response models.
The third part explores the feasibility of using machine learning algorithms to estimate the goldstandard measurement of AS, carotidtofemoral pulse wave velocity. Different modalities have been investigated to generate informative input features and reduce the dimensionality of the time series pulse waves.

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Identification et commande en ligne des robots avec utilisation de différentiateurs algébriques / Online identification and control of robots using algebraic differentiatorsGuo, Qi 17 December 2015 (has links)
Cette thèse traite de l'identification des paramètres dynamiques des robots, en s'appuyant sur les méthodes d'identification en robotique, qui utilisent le modèle dynamique inverse, ou le modèle de puissance, ou le modèle d'énergie du robot. Ce travail revisite le modèle d'énergie en exploitant le caractère intégral des fonctions modulatrices appliquées au modèle de puissance du robot. En outre, les procédures d'intégration sont analysées dans le domaine fréquentiel, et certains groupes de fonctions modulatrices sont sélectionnés afin d'offrir un bon comportement de filtre passebas. Ensuite, l'introduction d'un différentiateur algèbrique récemment développé est proposé, nommé différentiateurs de Jacobi. L'analyse est effectuée dans le domaine temporel, et dans le domaine fréquenciel, ce qui met en évidence la propriété de filtrage passe bande et permet de sélectionner les paramètres des différentiateurs. Puis, ces différentiateurs sont appliqués avec succès à l'identification de robot, ce qui prouve leur bonne performance. Les comparaisons entre les différents modèles d'identification, les différenciateurs, les techniques des moindres carrés sont présentées et des conclusions sont tirées dans le domaine de l'identification de robot. / This thesis discusses the identification issues of the robot dynamic parameters. Starting with the wellknown inverse dynamic identification model, power and energy identification models for robots, it extends the identification model from an energy point of view, by integrating modulating functions with robot power model. This new identification model avoids the computation of acceleration data. As well, the integration procedures are analyzed in frequency domain so that certain groups of modulating functions are selected in order to offer a good lowpass filtering property. Then, a recently developed high order algebraic differentiator is proposed and studied, named Jacobi differentiators. The analyses are done in both the time domain and in the frequency domain, which gives a clear clue about the differentiator filtering property and about how to select the differentiator parameters. Comparisons among different identification models, differentiators, least square techniques are presented and conclusions are drawn in the robot identification issues.

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