Добірка наукової літератури з теми "Dirac measures estimation"

The scope of this paper is to offer an overview of the main results obtained by the authors in recent literature in connection with the introduction of a Delta formalism, á la Dirac-Schwartz, for random generalized functions (distributions) associated with random closed sets, having an integer Hausdorff dimension n lower than the full dimension d of the environment space Rd. A concept of absolute continuity of random closed sets arises in a natural way in terms of the absolute continuity of suitable mean content measures, with respect to the usual Lebesgue measure on Rd. Correspondingly mean geometric densities are introduced with respect to the usual Lebesgue measure; approximating sequences are provided, that are of interest for the estimation of mean geometric densities of lower dimensional random sets such as fbre processes, surface processes, etc. Many models in material science and in biomedicine include time evolution of random closed sets, describing birthand-growth processes; the Delta formalism provides a natural framework for deriving evolution equations for mean densities at all (integer) Hausdorff dimensions, in terms of the relevant kinetic parameters.

We propose and analyze a reliable and efficienta posteriorierror estimator for a control-constrained linear-quadratic optimal control problem involving Dirac measures; the control variable corresponds to the amplitude of forces modeled as point sources. The proposeda posteriorierror estimator is defined as the sum of two contributions, which are associated with the state and adjoint equations. The estimator associated with the state equation is based on Muckenhoupt weighted Sobolev spaces, while the one associated with the adjoint is in the maximum norm and allows for unbounded right hand sides. The analysis is valid for two and three-dimensional domains. On the basis of the deviseda posteriorierror estimator, we design a simple adaptive strategy that yields optimal rates of convergence for the numerical examples that we perform.

The double-indicator technique with intracarotid bolus injection is useful for the estimation of transfer rates across the human blood-brain barrier. A method using intravenous tracer injection is developed whereby the input is measured at a peripheral artery and the output is measured at the jugular vein. To correct for differences in the brain input of test and reference substances, a five-parameter Dirac impulse response for passage through the cerebrovascular bed is computed from the input and output of the reference substance. This response is then combined with a capillary model of the brain. This is then convoluted with the arterial input curve of the test substance to yield a theoretical test output curve, which is compared with the actual test output curve. On the basis of these two curves and an appropriate mathematical model for the brain, estimates of blood-brain barrier permeability are obtained. In the present study, the techniques are compared in 13 patients in whom alternating intracarotid and intravenous bolus injections were given. For D-glucose, the two techniques yielded similar results. This was also the case for L-phenylalanine, provided that the erythrocyte compartment was taken into account. Data obtained after intravenous injection of leucine and water yielded similar results compared with previous intracarotid data.

We present a study of multicore fiber (MCF) crosstalk using the coupled mode theory. We derived a general closed-form simulation formula for the crosstalk of MCF under random perturbations, which includes both the average crosstalk and the crosstalk statistical distribution. From this general formula, we further derived simple analytical expressions for the average crosstalk under the assumption of exponential distribution of fiber segment lengths. We show that the analytical expressions approximate very well the results for other distributions, such as Dirac and Gaussian, and thus they can be used as a general analytical approach for estimating the average crosstalk. Results from numerical simulations of average crosstalk are shown to be in full correspondence with analytic results. We also performed numerical simulations of crosstalk statistical distributions generated from our general closed-form simulation formula and find that these agree well with the χ2-distribution function with four degrees of freedom. Finally, we conducted crosstalk measurements under different bending deployment conditions, and the measured crosstalk distributions and average crosstalk are found to be in agreement with the modeling results.

We have developed a method to obtain a misfit function for robust waveform inversion. In this method, called adaptive traveltime inversion (ATI), a matching filter that matches predicted data to measured data is computed. If the velocity model is relatively accurate, the resulting matching filter is close to a Dirac delta function. Its traveltime shift, which characterizes the defocusing of the matching filter, is computed by minimization of the crosscorrelation between a penalty function such as [Formula: see text] and the matching filter. ATI is constructed by minimization of the least-squares errors of the calculated traveltime shift. Further analysis indicates that the resulting traveltime shift corresponds to a first-order moment, the mean value of the resulting matching filter distribution. We extend ATI to a more general misfit function formula by computing different order moment of the resulting matching filter distribution. Choosing the penalty function in adaptive waveform inversion (AWI) as [Formula: see text], the misfit function of AWI is the second-order moment, the variance of the resulting matching filter distribution with zero mean. Because our ATI method is based on a global comparison using deconvolution, such as AWI, it can resolve the “cycle skipping” issue. We evaluate our ATI misfit function and compare it with state-of-the-art options such as least-squares inversion (L2 norm), wave-equation traveltime inversion, and AWI using schematic examples before moving to more complex examples, such as the Marmousi model. For the Marmousi model, starting with a 1D [Formula: see text] model, with data without low frequencies (no energy below 3 Hz), a meaningful estimation of the P-wave velocity model is recovered. Our ATI misfit function (first-order moment) indicates comparable performance with the AWI misfit function (the second-order moment). We also include a real data example from the Gulf of Mexico to demonstrate the effectiveness of our method.

Introduction. The article is devoted to substantiation and comprehension of the concept of investments in modern economic conditions. Modern research in world economics does not have a clearly defined concept of investment, because this concept should take into account not only its types, but also the depth of understanding of this process, its dynamism in the modern world. The mechanism of the investment process is to be clear and transparent for everyone who chooses any direc-tion for investment. The purpose of the article is to analyze the theoretical and methodological aspects of investment areas, determine the nature, structure of the functioning mechanism of domestic government bonds, determine the effectiveness of investment areas. Results. The usual areas of investment have ceased to provide a stable return for Ukrainian investors. Therefore, the issue of finding modern investment trends that ensure the most efficient use of investment capital becomes acute. It is im-portant to pay attention to the movement of both international capital and resident capital, because its speed creates a posi-tive investment climate inside the country. The investment process helps to ensure the proper functioning of all economic institutions. Therefore, it is important to consider current investment trends from the average investor, to analyze all possi-ble risks and warnings about a particular investment direction. One of the effective measures to eliminate the crisis in Ukraine is the issue of domestic government bonds. This approach allows attracting additional funds, which are used by the state to cover the budget deficit. In today's reality, this process is a means of lending that is more efficient than loans from other countries. In the modern economy, the development of investment areas is the driving force, so we believe that the topic of investment is relevant, because such measures are developed by businesses, which, in turn, develops economic sectors and the economy as a whole. Conclusions. The article reveals the essence of the investment process. All modern investment directions are defined, their components are considered and characterized. It is justified that the need to invest only in clear areas of investment. The estimation of all offered directions of capital transfer is considered and carried out. Key words: investments, directions of investment, domestic government bonds, investment risks, investment capital, capital movement, non-resident investors, capital attraction, investment asset.

In two dimensions, we analyze a hybridized discontinuous Galerkin (HDG) method for the Navier-Stokes equations with Dirac measures. The approximate velocity field obtained by the HDG method is shown to be pointwise divergence-free and $H$(div)-conforming. Under a smallness assumption on the continuous and discrete solutions, a posteriori error estimator, that provides an upper bound for the $L^2$-norm in the velocity, is proposed in the convex domain. In the polygonal domain, reliable and efficient a posteriori error estimator for the $W^{1,q}$-seminorm in the velocity and $L^q$-norm in the pressure is also proved. Finally, a Banach's fixed point iteration method and an adaptive HDG algorithm are introduced to solve the discrete system and show the performance of the obtained a posteriori error estimators.

Cette thèse de doctorat se propose de définir et d'expliciter les outils pour la reconstruction de courbes par des méthodes variationnelles sans-grille dans une perspective d'applications en imagerie biomédicale, et plus précisément en microscopie de fluorescence.Plusieurs algorithmes numériques convexes ou non-convexes ont déjà été développés au sein de l'équipe de recherche Morpheme qui a accueilli cette thèse, dans le cadre de l'optimisation non convexe parcimonieuse. Ces techniques ont été développées dans le cas discret, sur grille, ce qui signifie que l'image super-résolue est définie sur une grille plus fine que les images observées. A contrario, dans cette thèse de doctorat, nous développons des méthodes sans grille, en travaillant sur l'espace des mesures bornées. Ce choix apporte plusieurs avantages : la source n'est pas limitée par une grille fine et peut être localisée continûment, des bornes quantitatives sur l'écart entre la reconstruction et la source peuvent être énoncées, et ces approches permettent d'encoder une information structurelle ou géométrique dans le problème variationnel, de sorte à reconstruire une source d'une certaine forme, telle qu'un point ou une courbe.Dans un premier temps, nous rappellerons l'état de l'art des méthodes sans-grille, qui porte sur la reconstruction de sources ponctuelles. Nous illustrerons en particulier son intérêt pratique en microscopie de fluorescence, notamment face à des modèles classiques : les reconstructions proposées sont aussi convaincantes que celles produites par des modèles d'apprentissage profond, et ce, sans nécessiter d'ensemble d'apprentissage ou d'hyperparamètres à régler.Puis, nous introduirons une méthode sans-grille dans le cadre de la microscopie de fluorescence par fluctuation. Cette dernière repose sur l'indépendance des fluctuations de luminosité des fluorophores, ce qui permet en pratique de faire l'économie de matériel d'acquisition spécifique et coûteux. Les limites de ce modèle, formulé pour la reconstruction de sources ponctuelles, nous conduirons à nous intéresser sur la formulation d'une nouvelle méthode sans-grille, cette fois-ci pour la reconstruction de sources courbes.Nous nous pencherons sur la conception d'une méthode sans-grille, qui répond à la problématique de fluorophores le long de filaments qui forment de courbes dans les images, et qui n'est à notre connaissance pas traitée dans la littérature. En particulier, nous définirons l'espace d'optimisation impliqué, à savoir l'espace des mesures vectorielles à divergence finie, aussi appelé espace des charges. Nous poursuivrons en explicitant la manière de définir des mesures portées par des courbes, et en écrivant une nouvelle fonctionnelle appelée CROC (pour Curves Represented On Charges). Nous finirons en montrant qu'il existe une solution de cette énergie qui s'écrit comme une combinaison linéaire finie de mesures portées par des courbes.Finalement, nous étudierons la mise en œuvre numérique de la minimisation de cette énergie, à l'aide d'une adaptation d'algorithme glouton répandu dans la littérature. Nous discuterons des propriétés de notre algorithme, appelé Charge Sliding Frank-Wolfe, des difficultés rencontrées dans la définition du modèle d'observation, dans le choix de la paramétrisation discrète des courbes, etc.Le propos sera clôturé avec une conclusion qui rappellera les principaux points de chaque chapitre, ainsi qu'une ouverture, récapitulant les pistes de recherche possibles que nous avons dégagées au fil des chapitres
This PhD thesis aims to define and explain the tools for curve reconstruction using off-the-grid variational methods, as part of a range of applications in biomedical imaging, and more specifically in fluorescence microscopy.Several convex or non-convex numerical algorithms have already been developed within the Morpheme research team, where this thesis was conducted, in the context of non-convex sparse optimisation. These techniques were developed in the discrete case, on a grid, which means that the super-resolved image is defined on a finer grid than the observed images. In contrast, in this PhD thesis, we are developing off-the-grid also called gridless methods, while working on the space of bounded measures. This choice brings several advantages: the source is not limited by a fine grid and can be located continuously, quantitative bounds on the discrepancy between the reconstruction and the source can be written, and these approaches allow encoding structural/geometric information in the variational problem, to recover a source of a certain shape, such as a point or a curve.We will begin by reviewing the state of the art in off-the-grid methods, focusing on the reconstruction of point sources. In particular, we will illustrate its practical interest in fluorescence microscopy, especially compared with classical models: the reconstructions proposed are as convincing as those produced by deep learning models, and this without requiring a learning set or hyperparameters tuning.We will then introduce a gridless method for fluctuation-based fluorescence microscopy. This method exploits the independence of fluctuations in fluoroform luminosity, which means that specific and expensive acquisition equipment is not required. The limitations of this model, formulated for the reconstruction of point sources, will lead us to define a new off-the-grid method, this time for the reconstruction of curve.We will further study the design of an off-the-grid method, addressing the struggle of fluoroform along filaments that form curves in the images; a problem which is not taken care of in the literature. In particular, we will define the involved optimisation space, namely the space of vector measures with finite divergence, also known as the space of charges. We will further explain how to define curve measures, and we will write a new functional called CROC (for Curves Represented On Charges). We will finish by showing that there is a solution to this energy that can be written as a finite linear combination of measurements carried by curves.Finally, we will study the numerical implementation of this problem, using an adaptation of the greedy algorithm widely used in the literature. We will discuss the properties of our algorithm, called Charge Sliding Frank-Wolfe, and the difficulties encountered in the definition of the observation model, in choosing the discrete parametrisation of the curves, and so on.The thesis will be concluded with a reminder of the main points encountered in each chapter, and an opening section summing up the possible avenues of research that we have identified over the course of the chapters

Une grande part des muqueuses à l’intérieur du corps humain sont recouvertes de cils qui, par leurs mouvements coordonnés, conduisent à une circulation de la couche de fluide nappant la muqueuse. Dans le cas de la paroi interne des bronches, ce processus permet l’évacuation des impuretés inspirées à l’extérieur de l’appareil respiratoire.Dans cette thèse, nous nous intéressons aux effets du ou des cils sur le fluide, en nous plaçant à l’échelle du cil, et on considère pour cela les équations de Stokes incompressible. Due à la finesse du cil, une simulation directe demanderait un raffinement important du maillage au voisinage du cil, pour un maillage qui évoluerait à chaque pas de temps. Cette approche étant trop onéreuse en terme de coûts de calculs, nous avons considéré l’asymptotique d’un diamètre du cil tendant vers 0 et d’une vitesse qui tend vers l’infini : le cil est modélisé par un Dirac linéique de forces en terme source. Nous avons montré qu’il était possible de remplacer ce Dirac linéique par une somme de Dirac ponctuels distribués le long du cil. Ainsi, nous nous sommes ramenés, par linéarité, à étudier le problème de Stokes avec en terme source une force ponctuelle. Si les calculs sont ainsi simplifiés (et leurs coûts réduits), le problème final est lui plus singulier, ce qui motive une analyse numérique fine et l’élaboration d’une nouvelle méthode de résolution.Nous avons d’abord étudié une version scalaire de ce problème : le problème de Poisson avec une masse de Dirac en second membre. La solution exacte étant singulière, la solution éléments finis est à définir avec précaution. La convergence de la méthode étant dégradée dans ce cas-là, par rapport à celle dans le cas régulier, nous nous sommes intéressés à des estimations locales. Nous avons démontré une convergence quasi-optimale en norme Hs (s ě 1) sur un sous-domaine qui exclut la singularité. Des résultats analogues ont été obtenus dans le cas du problème de Stokes.Pour palier les problèmes liés à une mauvais convergence sur l’ensemble du domaine, nous avons élaboré une méthode pour résoudre des problème elliptiques avec une masse de Dirac ou une force ponctuelle en terme source. Basée sur celle des éléments finis standard, elle s’appuie sur la connaissance explicite de la singularité de la solution exacte. Une fois données la position de chacun des cils et leur paramétrisation, notre méthode rend possible la simulation directe en 3d d’un très grand nombre de cils. Nous l’avons donc appliquée au cas du transport mucociliaire dans les poumons. Cet outil numérique nous donne accès à des informations que l’on ne peut avoir par l’expérience, et permet de simuler des cas pathologiques comme par exemple une distribution éparse des cils
Numerous mucous membranes inside the human body are covered with cilia which, by their coordinated movements, lead to a circulation of the layer of fluid coating the mucous membrane, which allows, for example, in the case of the internal wall of the bronchi, the evacuation of the impurities inspired outside the respiratory system.In this thesis, we integrate the effects of the cilia on the fluid, at the scale of the cilium. For this, we consider the incompressible Stokes equations. Due to the very small thickness of the cilia, the direct computation would request a time-varying mesh grading around the cilia. To avoid too prohibitive computational costs, we consider the asymptotic of a zero diameter cilium with an infinite velocity: the cilium is modelled by a lineic Dirac of force in source term. In order to ease the computations, the lineic Dirac of forces can be approached by a sum of punctual Dirac masses distributed along the cilium. Thus, by linearity, we have switched our initial problem with the Stokes problem with a punctual force in source term. Thus, we simplify the computations, but the final problem is more singular than the initial problem. The loss of regularity involves a deeper numerical analysis and the development of a new method to solve the problem.We have first studied a scalar version of this problem: Poisson problem with a Dirac right-hand side. The exact solution is singular, therefore the finite element solution has to be defined with caution. In this case, the convergence is not as good as in the regular case, and thus we focused on local error estimates. We have proved a quasi-optimal convergence in H1-norm (s ď 1) on a sub-domain which does not contain the singularity. Similar results have been shown for the Stokes problem too.In order to recover an optimal convergence on the whole domain, we have developped a numerical method to solve elliptic problems with a Dirac mass or a punctual force in source term. It is based on the standard finite element method and the explicit knowl- edge of the singularity of the exact solution. Given the positions of the cilia and their parametrisations, this method permits to compute in 3d a very high number of cilia. We have applied this to the study of the mucociliary transport in the lung. This numerical tool gives us information we do not have with the experimentations and pathologies can be computed and studied by this way, like for example a small number of cilia