Tesis sobre el tema "Equations aux dérivées partielles stochastiques singulières"
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Popier, Alexandre François Roland. "Equations différentielles stochastiques rétrogrades avec condition finale singulière". Aix-Marseille 1, 2004. http://www.theses.fr/2004AIX11037.
Texto completoDiop, Mamadou Abdoul. "Equations aux dérivées partielles stochastiques et homogénéisation". Aix-Marseille 1, 2003. http://www.theses.fr/2003AIX11017.
Texto completoThis thesis is devoted to some problems connected to the theory of homogenization of random parabolic operators with large potential. It is assumed that the said operators have a periodic spatial microstructure whose characteristics are rapidly oscillating stationary random process in time. Two different cases of non diffusive scaling are addressed. Namely, the case when the oscillation in time is faster than that in spatial variables and the opposite case when the time oscillation is slower than that the spatial one. It is shown that in the former case, under certain mixing conditions,the corresponding Cauchy problem admits homogenization and its solution converges in probability to a solution of a deterministic semilinear operator. In the latter case the limit equation is a stochastic partial differential equation. Here a solution of the original Cauchy problem converges in law in the energy functional space, while con vergence in probability does not takes place. The thesis consists of an introduction and three different parts. In the introduction we give an elementary presentation of the basic ideas in the homogenization theory. The first chapter, deals with the results contained in this thesis. In the second chapter the operators with Markov driving processes are considered. In the second part the operators with non Markov coefficients are investigated
Hsu, Yueh-Sheng. "On the random Schrödinger operators in the continuous setting". Electronic Thesis or Diss., Université Paris sciences et lettres, 2024. http://www.theses.fr/2024UPSLD009.
Texto completoThis thesis studies the random Schrödinger operators in continuous setting, particularly those with Gaussian white noise potential. The definition of such differential operators is generally non-trivial and necessitates renormalization in dimensions d ≥ 2. We first present a general framework to translate the problem of operator construction into stochastic PDEs. This approach enables us to define the operator at stake and establishes its self-adjointness, as well as to investigate its spectrum.Subsequently, we proceed to study the continuous Anderson Hamiltonian under two distinct spatial settings: first on a bounded box with side length L with zero Dirichlet boundary condition for dimensions d ≤ 3, and second on the full Euclidean space Rd, for d ∈ {2, 3}. In the former case, the operator admits eigenvalues λn,L, for which we identify the almost sure asymptotic as L → ∞. This asymptotic aligns with previous findings in the literature for dimension 1 and 2, while our result in dimension 3 is new. In the latter case, we propose a new construction technique employing the solution theory to the associated parabolic equation which allows to prove self-adjointness and show that the spectrum equals to R almost surely. This approach reconfirms the recently established result in dimension 2, but our construction seems to be more elementary; for dimension 3, our result is new.Lastly, we present an ongoing project addressing the case where a uniform magnetic field is applied to the system : this leads to the study of Landau Hamiltonian perturbed by the white noise potential. Our objective is to define the operator on full space R² without resorting to sophisticated renormalization theory. However, the unboundedness of white noise on R² poses additional technical challenges. To overcome this, the usage of Faris-Lavine theorem is discussed
Debbi, Latifa. "Equations aux dérivées partielles déterministes et stochastiques avec opérateurs fractionnaires". Nancy 1, 2006. http://www.theses.fr/2006NAN10046.
Texto completoThis thesis treats application of fractional calculus in stochastic analysis. In the first part, the definition of the the multidimensional Riesz-Feller fractional differential operator is extended to higher order. The operator obtained generalizes several known fractional differential and pseudodifferential operators. High order fractional Fokker-Plank equations are studied in both the probabilistic and the quasiprobabilistic approaches. In particular, the solutions are represented via stable Lévy processes and generalization of Airy's function. In the second part, onedimensional stochastic fractional partial differential equations perturbed by space-time white noise are considered. The existence and the uniqueness of field solutions and of L2solutions are proved under different Lipschtz conditions. Spatial and temporal Hölder exponents of the field solutions are obtained. Further, equivalence between several definitions of L2solutions is proven. In particular, Fourier transform is used to give meaning to some stochastic fractional partial differential equations
Zhang, Jing. "Les équations aux dérivées partielles stochastiques avec obstacle". Thesis, Evry-Val d'Essonne, 2012. http://www.theses.fr/2012EVRY0020/document.
Texto completoThis thesis deals with quasilinear Stochastic Partial Differential Equations (in short SPDE). It is divided into two parts, the first part concerns the obstacle problem for quasilinear SPDE and the second part solves quasilinear SPDE driven by G-Brownian motion. In the first part we begin with the existence and uniqueness result for the obstacle problem of quasilinear stochastic partial differential equations (in short OSPDE). Our method is based on analytical technics coming from the parabolic potential theory. The solution is expressed as a pair (u, v) where u is a predictable continuous process which takes values in a proper Sobolev space and v is a random regular measure satisfying minimal Skohorod condition. Then we prove a maximum principle for a local solution of quasilinear stochastic partial differential equations with obstacle. The proofs are based on a version of Itô’s formula and estimates for the positive part of a local solution which is negative on the lateral boundary. The objective of the second part is to study the well-posedness of stochastic partial differential equations driven by G-Brownian motion in the framework of sublinear expectation spaces. One can also establish an Itô formula for the solution and a comparison theorem
Riviere, Olivier. "Equations différentielles stochastiques progressives rétrogrades couplées : équations aux dérivées partielles et discrétisation". Phd thesis, Université René Descartes - Paris V, 2005. http://tel.archives-ouvertes.fr/tel-00011231.
Texto completoRivière, Olivier. "Equations différentielles stochastiques progressives rétrogrades couplées : équations aux dérivées partielles et discrétisation". Paris 5, 2005. http://www.theses.fr/2005PA05S028.
Texto completoThis thesis deals with the forward backward stochastic differential equations, in particular those with a coefficient of progressive diffusion which depends on all unknowns of the problem. We propose an original way to get onto this subject, letting us to reobtain some classical results of existence and uniqueness in the spirit of Pardoux-Tang and Yong's results, and to find a probabilistic representation of a new class of parabolic PDE, in which derivation coefficient of order 2 depends on the gradient of the solution. We also propose an iterative discretization scheme. We prove its convergence and give an evaluation of the error on a particular example
Carrizo, Vergara Ricardo. "Développement de modèles géostatistiques à l’aide d’équations aux dérivées partielles stochastiques". Thesis, Paris Sciences et Lettres (ComUE), 2018. http://www.theses.fr/2018PSLEM062/document.
Texto completoThis dissertation presents theoretical advances in the application of the Stochastic Partial Differential Equation (SPDE) approach in Geostatistics. This recently developed approach consists in interpreting a regionalised data-set as a realisation of a Random Field satisfying a SPDE. Within the theoretical framework of Generalized Random Fields, the influence of a linear SPDE over the covariance structure of its potential solutions can be studied with a great generality. A criterion of existence and uniqueness of stationary solutions for a wide-class of linear SPDEs has been obtained, together with an expression for the related spectral measures. These results allow to develop spatio-temporal covariance models presenting non-trivial properties through the analysis of evolution equations presenting a fractional temporal derivative order. Suitable parametrizations of such models allow to control their separability, symmetry and separated space-time regularities. Results concerning stationary solutions for physically inspired SPDEs such as the Heat equation and the Wave equation are also presented. A method of non-conditional simulation adapted to these models is then studied. This method is based on the computation of an approximation of the Fourier Transform of the field, and it can be implemented efficiently thanks to the Fast Fourier Transform algorithm. The convergence of this method has been theoretically proven in suitable weak and strong senses. This method is applied to numerically solve the SPDEs studied in this work. Illustrations of models presenting non-trivial properties and related to physically driven equations are then given
Fedrizzi, Ennio. "Partial differential equations and noise". Paris 7, 2012. http://www.theses.fr/2012PA077176.
Texto completoIn this work we present examples of the effects of noise on the solution of a partial differential equation in three different settings. We first consider random initial conditions for two nonlinear dispersive partial differential equations, the nonlinear Schrodinger equation and the Korteweg - de Vries equation, and analyze their effects on some special solutions, the soliton solutions. The second case considered is a linear PDE, the wave equation, with random initial conditions. We show that special random initial conditions allow to I substantially decrease the computational and data storage costs of an algorithm to solve the inverse problem based on the boundary measurements of the solution of this equation. Finally, the third example considered is that of the linear transport equation with a singular drift term, where we will show that the addition of a multiplicative noise term forbids the blow up of solutions, under very weak hypothesis for which we have finite-time blow up of solutions in the deterministic case
Piozin, Lambert. "Quelques résultats sur les équations rétrogrades et équations aux dérivées partielles stochastiques avec singularités". Thesis, Le Mans, 2015. http://www.theses.fr/2015LEMA1004/document.
Texto completoThis thesis is devoted to the study of some problems in the field of backward stochastic differential equations (BSDE), and their applications to partial differential equations.In the first chapter, we introduce the notion of backward doubly stochastic differential equations (BDSDE) with singular terminal condition. A first work consists to study the case of BDSDE with monotone generator. We then obtain existing result by an approximating scheme built considering a truncation of the terminal condition. The last part of this chapter aim to establish the link with stochastic partial differential equations, using a weak solution approach developed by Bally, Matoussi in 2001.The second chapter is devoted to the BSDEs with singular terminal conditions and jumps. As in the previous chapter the tricky part will be to prove continuity in T. We formulate sufficient conditions on the jumps in order to obtain it. A section is then dedicated to establish a link between a minimal solution of our BSDE and partial integro-differential equations.The last chapter is dedicated to doubly reflected second order backward stochastic differential equations (2DRBSDE). We have been looking to establish existence and uniqueness for such equations. In order to obtain this, we had to focus first on the upper reflection problem for 2BSDEs. We combined then these results to those already existing to give a well-posedness context to 2DRBSDE. Uniqueness is established as a straight consequence of a representation property. Existence is obtained using shifted spaces, and regular conditional probability distributions. A last part is then consecrated to the link with some Dynkin games and Israeli options
Wang, Hao. "Equations différentielles stochastiques rétrogrades réfléchies et applications au problème d'investissement réversible et aux équations aux dérivées partielles". Le Mans, 2009. http://cyberdoc.univ-lemans.fr/theses/2009/2009LEMA1013.pdf.
Texto completoThe main objective of the thesis is to study the existence and uniqueness of solutions of reflected backward stochastic differential equations and to relate this notion to the study of the problems such as the reversible investment or so-called optimal switching problem, the mixed zero-sum stochastic differential games and the probabilistic interpretation of the weak solution of partial differential equations, either in viscosity sense or in Sobolev space under different framework
Rouis, Moeiz. "Equations aux dérivées partielles en finance : problèmes inverses et calibration de modèle". Phd thesis, Ecole Polytechnique X, 2007. http://pastel.archives-ouvertes.fr/pastel-00003888.
Texto completoFurlan, Marco. "Structures contrôlées pour les équations aux dérivées partielles". Thesis, Paris Sciences et Lettres (ComUE), 2018. http://www.theses.fr/2018PSLED008/document.
Texto completoThe thesis project has various possible directions: a) Improve the understanding of the relations between the theory of Regularity Structures developed by M.Hairer and the method of Paracontrolled Distributions developed by Gubinelli, Imkeller and Perkowski, and eventually to provide a synthesis. This is highly speculative and at the moment there are no clear path towards this long term goal. b) Use the theory of Paracontrolled Distributions to study different types of PDEs: transport equations and general hyperbolic evolution equation, dispersive equations, systems of conservation laws. These PDEs are not in the domain of the current methods which were developed mainly to handle parabolic semilinear evolution equations. c) Once a theory of transport equation driven by rough signals have been established it will become possible to tackle the phenomena of regularization by transport noise which for the moment has been studied only in the context of transport equations driven by Brownian motion, using standard tools of stochastic analysis. d) Renormalization group (RG) techniques and multi-scale expansions have already been used both to tackle PDE problems and to define Euclidean Quantum Field Theories. Paracontrolled Distributions theory can be understood as a kind of mul- tiscale analysis of non-linear functionals and it would be interesting to explore the interplay of paradifferential techniques with more standard techniques like cluster expansions and RG methods
El, Dika Khaled. "Comportement qualitatif des solutions de l'équation de Benjamin-Bona-Mahony déterministe et stochastique". Paris 11, 2004. http://www.theses.fr/2004PA112194.
Texto completoWe are interested in the qualitative behavior of solutions of the generalized BBM equation. Our main results :1- Asymptotic stability of the solitary waves of the gBBM eqaution in the energy space H^1. 2- Qualitative description of localized solutions for the gBBM equation traveling to the right. 3- Stability (and asymptotic stabiility) of the sum of N solitary waves for the generalized BBM equation, we prove also the existence and uniqueness of "N-solitary wave", i. E. Solutions behaving asymptotically as the the sum of N solitary waves of the gBBM eqaution. 4- We prove global existence for several stochastic BBM equation, including the case of the space derivative of a noise which is locally white in space and time
Touibi, Rim. "Sur le comportement qualitatif des solutions de certaines équations aux dérivées partielles stochastiques de type parabolique". Thesis, Université de Lorraine, 2018. http://www.theses.fr/2018LORR0263/document.
Texto completoThis thesis is concerned with stochastic partial differential equations of parabolic type. In the first part we prove new results regarding the existence and the uniqueness of global and local variational solutions to a Neumann initial-boundary value problem for a class of non-autonomous stochastic parabolic partial differential equations. The equations we consider are defined on unbounded open domains in Euclidean space satisfying certain geometric conditions, and are driven by a multiplicative noise derived from an infinite-dimensional fractional Wiener process characterized by a sequence of Hurst parameters H = (Hi) i ∈ N+ ⊂ (1/2,1). These parameters are in fact subject to further constraints that are intimately tied up with the nature of the nonlinearity in the stochastic term of the equations, and with the choice of the functional spaces in which the problem at hand is well-posed. Our method of proof rests on compactness arguments in an essential way. The second part is devoted to the study of the blowup behavior of solutions to semilinear stochastic partial differential equations with Dirichlet boundary conditions driven by a class of differential operators including (not necessarily symmetric) Lévy processes and diffusion processes, and perturbed by a mixture of Brownian and fractional Brownian motions. Our aim is to understand the influence of the stochastic part and that of the differential operator on the blowup behavior of the solutions. In particular we derive explicit expressions for an upper and a lower bound of the blowup time of the solution and provide a sufficient condition for the existence of global positive solutions. Furthermore, we give estimates of the probability of finite time blowup and for the tail probabilities of an upper bound for the blowup time of the solutions
Rainero, Sophie. "Sur les propriétés des solutions d'équations différentielles stochastiques rétrogrades à horizon aléatoire ou déterministe. Principes de grandes déviations et applications à des problèmes de perturbations singulières pour des équations aux dérivées partielles non linéaires". Paris 9, 2006. https://portail.bu.dauphine.fr/fileviewer/index.php?doc=2006PA090008.
Texto completoWe prove large deviations principles for solutions of forward-backward stochastic differential equations with determinist terminal time, and we give an application of these results to the theory of credit risk management. We also study the existence, uniqueness and stability of solutions of backward stochastic differential equations with random terminal time under new assumptions. We establish large deviations principles for the solutions of such equations, related to a family of Markov processes, the diffusion coefficient of which tends to zero. We deduce from these results some theorems of convergence of solutions of non linear partial differential equations, elliptic and parabolic, which extend Freidlin and Wentzell's
Touibi, Rim. "Sur le comportement qualitatif des solutions de certaines équations aux dérivées partielles stochastiques de type parabolique". Electronic Thesis or Diss., Université de Lorraine, 2018. http://www.theses.fr/2018LORR0263.
Texto completoThis thesis is concerned with stochastic partial differential equations of parabolic type. In the first part we prove new results regarding the existence and the uniqueness of global and local variational solutions to a Neumann initial-boundary value problem for a class of non-autonomous stochastic parabolic partial differential equations. The equations we consider are defined on unbounded open domains in Euclidean space satisfying certain geometric conditions, and are driven by a multiplicative noise derived from an infinite-dimensional fractional Wiener process characterized by a sequence of Hurst parameters H = (Hi) i ∈ N+ ⊂ (1/2,1). These parameters are in fact subject to further constraints that are intimately tied up with the nature of the nonlinearity in the stochastic term of the equations, and with the choice of the functional spaces in which the problem at hand is well-posed. Our method of proof rests on compactness arguments in an essential way. The second part is devoted to the study of the blowup behavior of solutions to semilinear stochastic partial differential equations with Dirichlet boundary conditions driven by a class of differential operators including (not necessarily symmetric) Lévy processes and diffusion processes, and perturbed by a mixture of Brownian and fractional Brownian motions. Our aim is to understand the influence of the stochastic part and that of the differential operator on the blowup behavior of the solutions. In particular we derive explicit expressions for an upper and a lower bound of the blowup time of the solution and provide a sufficient condition for the existence of global positive solutions. Furthermore, we give estimates of the probability of finite time blowup and for the tail probabilities of an upper bound for the blowup time of the solutions
Royer, Manuela. "Équations différentielles stochastiques rétrogrades et martingales non linéaires". Rennes 1, 2003. http://www.theses.fr/2003REN1A018.
Texto completoHofmanová, Martina. "Degenerate parabolic stochastic partial differential equations". Phd thesis, École normale supérieure de Cachan - ENS Cachan, 2013. http://tel.archives-ouvertes.fr/tel-00916580.
Texto completoMadec, Pierre-Yves. "Equations différentielles stochastiques rétrogrades ergodiques et applications aux EDP". Thesis, Rennes 1, 2015. http://www.theses.fr/2015REN1S027/document.
Texto completoThis thesis deals with the study of ergodic BSDE and their applications to the study of the large time behaviour of solutions to semilinear parabolic PDE. In a first time, we establish some existence and uniqueness results to an ergodic BSDE with Neumann boundary conditions in an unbounded convex set in a weakly dissipative environment. Then we study their link with PDE with Neumann boundary condition and we give an application to an ergodic stochastic control problem. The second part consists of two sections. In the first one, we study the large time bahaviour of mild solutions to semilinear parabolic PDE in infinite dimension by a probabilistic method. This probabilistic method relies on a Basic coupling estimate result which gives us an exponential rate of convergence of the solution toward its asymptote. Let us mention that that this asymptote is fully determined by the solution of the ergodic semilinear PDE associated to the parabolic semilinear PDE. Then, we adapt this method to the sudy of the large time behaviour of viscosity solutions of semilinear parabolic PDE with Neumann boundary condition in a convex and bounded set in finite dimension. By regularization and penalization procedures, we obtain similar results as those obtained in the mild context, especially with an exponential rate of convergence for the solution toward its asymptote
Poncet, Romain. "Méthodes numériques pour la simulation d'équations aux dérivées partielles stochastiques non-linéaires en condensation de Bose-Einstein". Thesis, Université Paris-Saclay (ComUE), 2017. http://www.theses.fr/2017SACLX069/document.
Texto completoThis thesis is devoted to the numerical study of two stochastic models arising in Bose-Einstein condensation physics. They constitute two generalisations of the Gross-Pitaevskii Equation. This deterministic partial differential equation model the wave function dynamics of a Bose-Einstein condensate trapped in an external confining potential. The first chapter contains a simple presentation of the Bose-Einstein condensation phenomenon and of the experimental methods used to construct such systems.The first model considered enables to model the fluctuations of the confining potential intensity, and takes the form of a stochastic partial differential equation. In practice, these fluctuations lead to heating of the condensate and possibly to its collapse. In the second chapter we propose to build a numerical scheme to solve this model. It is based on a spectral space discretisation and a Crank-Nicolson discretisation in space. We show that the proposed scheme converges strongly at order at least one in probability. We also present numerical simulations to illustrate this result. The third chapter is devoted to the numerical and theoretical study of the dynamics of a stationary solution (for the deterministic equation) of vortex type. We study the influence of random disturbances of the confining potential on the solution. We show that the disturbed solution conserves the symmetries of the stationary solution for times up to at least the square of the inverse of the fluctuations intensity. These results are illustrated with numerical simulations based on a Monte-Carlo method suited to rare events estimation.The second model can be used to model the effects of the temperature on the dynamics of a Bose-Einstein condensate. In the case of finite temperature, the Bose-Einstein condensation is not complete and the condensate interacts with the non-condensed particles. These interactions are interesting to understand the dynamics of the phase transition and analyse the phenomena of symmetry breaking associated, like the spontaneous nucleation of vortices We have studied in the fourth and the fifth chapters some questions linked to the long time simulation of this model solutions. The fourth chapter is devoted to the construction of an unbiased sampling method of measures known up to a multiplicative constant. The distinctive feature of this Markov-Chain Monte-Carlo algorithm is that it enables to perform an unbiased non-reversible sampling based on an overdamped Langevin equation. It constitutes a generalization of the Metropolis-Adjusted Langevin Algorithm (MALA). The fifth chapter is devoted to the numerical study of metastable dynamics linked to the nucleation of vortices in rotating Bose-Einstein condensates. A numerical integrator and a suited Monte-Carlo methods for the simulation of metastable dynamics are proposed. This Monte-Carlo method is based on the Adaptive Multilevel Splitting (AMS) algorithm
Ondreját, Martin. "Equations d'évolution stochastiques dans les espaces de Banach : unicités abstraites, propriété de Markov forte, équations hyperboliques". Nancy 1, 2003. http://docnum.univ-lorraine.fr/public/SCD_T_2003_0046_ONDREJAT.pdf.
Texto completoThis work consists of four chapters on some aspects of stochastic semilinear evolution equations (SPDE) in Banach spaces. The first chapter deals with different notions of uniqueness and existence (such as pathwise uniqueness, uniqueness in law, strong and weak existence) and the relations between them. We present an alternative construction of the stochastic integral in Banach spaces and we prove Burkholder's inequality, Fubini's theorem, the Chojnowska-Michalik theorem and Girsanov's theorem. We prove distribution preserving theorems for Bochner integrals, stochastic integrals and measurable selectors as well. The second chapter regards the Brownian representations of local cylindrical martingales in Banach spaces and the martingale problem in infinite dimensions. We use these results for illustrating the role of the notion "well-posedness" and for showing that weak existence and uniqueness in law for the equation in question imply the strong Markov property of the solutions. The third and the fourth chapter treats second order hyperbolic SPDE's driven by a spatially homogeneous Wiener process. We present sufficient conditions on the coefficients for the equation to have global strong and weak solutions, and we prove that the solutions propagate at finite speed
Martel, Sofiane. "Theoretical and numerical analysis of invariant measures of viscous stochastic scalar conservation laws". Thesis, Paris Est, 2019. http://www.theses.fr/2019PESC1040.
Texto completoThis devoted to the theoretical and numerical analysis of a certain class of stochastic partial differential equations (SPDEs), namely scalar conservation laws with viscosity and with a stochastic forcing which is an additive white noise in time. A particular case of interest is the stochastic Burgers equation, which is motivated by turbulence theory. We focus on the long time behaviour of the solutions of these equations through a study of the invariant measures. The theoretical part of the thesis constitutes the second chapter. In this chapter, we prove the existence and uniqueness of a solution in a strong sense. To this end, estimates on Sobolev norms up to the second order are established. In the second part of Chapter~2, we show that the solution of the SPDE admits a unique invariant measure. In the third chapter, we aim to approximate numerically this invariant measure. For this purpose, we introduce a numerical scheme whose spatial discretisation is of the finite volume type and whose temporal discretisation is a split-step backward Euler method. It is shown that this kind of scheme preserves some fundamental properties of the SPDE such as energy dissipation and L^1-contraction. Those properties ensure the existence and uniqueness of an invariant measure for the numerical scheme. Thanks to a few regularity estimates, we show that this discrete invariant measure converges, as the space and time steps tend to zero, towards the unique invariant measure for the SPDE in the sense of the second order Wasserstein distance. Finally, numerical experiments are performed on the Burgers equation in order to illustrate this convergence as well as some small-scale properties related to turbulence
El, Asri Brahim. "Switching optimal et équations différentielles stochastiques rétrogrades réfléchies". Le Mans, 2010. http://cyberdoc.univ-lemans.fr/theses/2010/2010LEMA1003.pdf.
Texto completoWe study optimal switching and Lр-solution for doubly reflected backward stochastic differential equations. In the first part, we show existence and uniqueness of a solution for a system of m variational partial differential inequalities with inter-connected obstacles. This system is the deterministic version of the Verification Theorem of the Markovian optimal m-states switching problem. The switching cost functions are arbitrary. In the second part we study the problem of the deterministic version of the Verification Theorem for the optimal m-states switching in infinite horizon under Markovian framework with arbitrary switching cost functions. The problem is formulated as an extended impulse control problem and solved by means of probabilistic tools such as the Snell envelop of processes and reflected backward stochastic differential equations. A viscosity solutions approach is employed to carry out a fine analysis on the associated system of m variational inequalities with inter-connected obstacles. We show that the vector of value functions of the optimal problem is the unique viscosity solution to the system. Finally in the third part, we deal the problem of existence and uniqueness of a solution for à backward stochastic differential equation (BSDE for short) with two strictly separated continuous reflecting barriers in the case when the terminal value, the generator and the obstacle process are Lр-integrable with р Є (1, 2). The main idea is to use the concept of local solution to construct the global one. As applications, we obtain new results in zerosum Dynkin games and in double obstacle variational inequalities theories
Hibon, Hélène. "Équations différentielles stochastiques rétrogrades quadratiques et réfléchies". Thesis, Rennes 1, 2018. http://www.theses.fr/2018REN1S007/document.
Texto completoIn this thesis, we are interested in studying variously Backward Stochastic Differential Equations. A large proportion of the results are obtained under the assumption that the driver is of quadratic growth in its last variable. A first link between one-dimensional quadratic BSDEs and game theory leads us to develop results with convex drivers. Optimal control theory requires as for it to deal with the multidimensional case, in which global existence and uniqueness are obtained only for diagonaly quadratic drivers. Major achievements in reflected BSDEs (whose solution is constrained to remain in a domain) are reached for Lipschitz drivers. We develop a result of chaos propagation in this setting, with a constraint on the law of the solution rather than on its path. We finaly build bridge between quadratic BSDEs and reflected BSDEs thanks to mean field quadratic BSDEs. We give several new results on solvability of a quadratic BSDE whose driver depends also on the mean of both variables
Mtiraoui, Ahmed. "I. Etude des EDDSRs surlinéaires II. Contrôle des EDSPRs couplées". Thesis, Toulon, 2016. http://www.theses.fr/2016TOUL0010/document.
Texto completoIn this Phd thesis, we considers two parts. The first one establish the existence and the uniquness of the solutions of multidimensional backward doubly stochastic differential equations (BDSDEs in short) and the stochastic partial differential equations (SPDEs in short) in the superlinear growth generators. In the second part, we study the stochastic controls problems driven by a coupled Forward-Backward stochastic differentialequations (FBSDEs in short).• BDSDEs and SPDEs with a superlinear growth generators :We deal with multidimensional BDSDE with a superlinear growth generator and a square integrable terminal datum. We introduce new local conditions on the generator then we show that they ensure the existence and uniqueness as well as the stability of solutions. Our work go beyond the previous results on the subject. Although we are focused on multidimensional case, the uniqueness result we establish is new in one dimensional too. As application, we establish the existence and uniqueness of probabilistic solutions tosome semilinear SPDEs with superlinear growth generator. By probabilistic solution, we mean a solution which is representable throughout a BDSDEs.• Controlled coupled FBSDEs :We establish the existence of an optimal control for a system driven by a coupled FBDSE. The cost functional is defined as the initial value of the backward component of the solution. We construct a sequence of approximating controlled systems, for which we show the existence of a sequence of feedback optimal controls. By passing to the limit, we get the existence of a feedback optimal control. The convexity condition is used to ensure that the optimal control is strict. In this part, we study two cases of diffusions : degenerate and non-degenerate
Kopec, Marie. "Quelques contributions à l'analyse numérique d'équations stochastiques". Electronic Thesis or Diss., Rennes, École normale supérieure, 2014. http://www.theses.fr/2014ENSR0002.
Texto completoThis work presents some results about behavior in long time and in finite time of numerical methods for stochastic equations.In a first part, we are considered with overdamped Langevin Stochastic Differential Equations (SDE) and Langevin SDE. We show a weak backward error analysis result for its numerical approximations defined by implicit methods. In particular, we prove that the generator associated with the numerical solution coincides with the solution of a modified Kolmogorov equation up to high order terms with respect to the stepsize. This implies that every measure of the numerical scheme is close to a modified invariant measure obtained by asymptotic expansion. Moreover, we prove that, up to negligible terms, the dynamic associated with the implicit scheme considered is exponentially mixing.In a second part, we study the long-time behavior of fully discretized semilinear SPDEs with additive space-time white noise, which admit a unique invariant probability measure μ. We focus on the discretization in time thanks to a scheme of Euler type, and on a Finite Element discretization in space and we show that the average of regular enough test functions with respect to the (possibly non unique) invariant laws of the approximations are close to the corresponding quantity for μ.More precisely, we analyze the rate of the convergence with respect to the different discretization parameters. Finally, we are concerned with semilinear SPDEs with additive space-time white noise, which the nonlinear term is a polynomial function. We analyze the rate of the weak convergence for discretization in time with an implicit splitting method
Rosello, Angelo. "Limites d'échelles pour des modèles cinétiques stochastiques". Thesis, Rennes, École normale supérieure, 2020. http://www.theses.fr/2020ENSR0021.
Texto completoThis thesis aims at providing an understanding of certain scaling limits for kinetic models perturbed with some random noise, where the limiting object remains of stochastic nature, governed by a stochastic partial differential equation. In the first chapter, the transition from a mesoscopic to a macroscopic description is studied through a kinetic system of equations – corresponding to the behavior of a “spray” of particles embedded in an ambient fluid perturbed by a mixing Markov process. Under a suitable scaling, relying on the perturbed test function method, we establish the convergence of the density of particles to a hydrodynamic limit which can be expressed as the solution of a stochastic conservation equation driven by a Wiener process.Next, we focus on stochastic kinetic equations derived from biological models of collective motion. This study is split into two different works, devoted to distinct models. In chapter 2, we first examine the mean-field limits of a few different particle systems which correspond to random perturbations of the classical Cucker-Smale model. Then, in chapter 3, we establish the existence of martingale solutions for some more advanced model, which allows local interactions between individuals
Guerrier, Claire. "Multi-scale modeling and asymptotic analysis for neuronal synapses and networks". Thesis, Paris 6, 2015. http://www.theses.fr/2015PA066518/document.
Texto completoIn the present PhD thesis, we study neuronal structures at different scales, from synapses to neural networks. Our goal is to develop mathematical models and their analysis, in order to determine how the properties of synapses at the molecular level shape their activity and propagate to the network level. This change of scale can be formulated and analyzed using several tools such as partial differential equations, stochastic processes and numerical simulations. In the first part, we compute the mean time for a Brownian particle to arrive at a narrow opening defined as the small cylinder joining two tangent spheres. The method relies on Möbius conformal transformation applied to the Laplace equation. We also estimate, when the particle starts inside a boundary layer near the hole, the splitting probability to reach the hole before leaving the boundary layer, which is also expressed using a mixed boundary-value Laplace equation. Using these results, we develop model equations and their corresponding stochastic simulations to study vesicular release at neuronal synapses, taking into account their specific geometry. We then investigate the role of several parameters such as channel positioning, the number of entering ions, or the organization of the active zone. In the second part, we build a model for the pre-synaptic terminal, formulated in an initial stage as a reaction-diffusion problem in a confined microdomain, where Brownian particles have to bind to small target sites. We coarse-grain this model into two reduced ones. The first model couples a system of mass action equations to a set of Markov equations, which allows to obtain analytical results. We develop in a second phase a stochastic model based on Poissonian rate equations, which is derived from the mean first passage time theory and the previous analysis. This model allows fast stochastic simulations, that give the same results than the corresponding naïve and endless Brownian simulations. In the final part, we present a neural network model of bursting oscillations in the context of the respiratory rhythm. We build a mass action model for the synaptic dynamic of a single neuron and show how the synaptic activity between individual neurons leads to the emergence of oscillations at the network level. We benchmark the model against several experimental studies, and confirm that respiratory rhythm in resting mice is controlled by recurrent excitation arising from the spontaneous activity of the neurons within the network
Bréhier, Charles-Edouard. "Analyse numérique d'EDP Stochastiques hautement oscillantes". Phd thesis, École normale supérieure de Cachan - ENS Cachan, 2012. http://tel.archives-ouvertes.fr/tel-00763340.
Texto completoDe, Moor Sylvain. "Limites diffusives pour des équations cinétiques stochastiques". Electronic Thesis or Diss., Rennes, École normale supérieure, 2014. http://www.theses.fr/2014ENSR0001.
Texto completoThis thesis presents several results about stochastic partial differential equations. The main subject is the study of diffusive limits of kinetic models perturbed with a random term. We also present a result about the regularity of a class of stochastic partial differential equations and a result of existence and uniqueness of invariant measures for a stochastic Fokker-Planck equation.First, we give three results of approximation-diffusion in a stochastic context. The first one deals with the case of a kinetic equation with a linear operator of relaxation whose velocity equilibrium has a power tail distribution at ininity. The equation is perturbed with a Markovian process. This gives rise to a stochastic fluid fractional limit. The two remaining results consider the case of the radiative transfer equation which is a non-linear kinetic equation. The equation is perturbed successively with a cylindrical Wiener process and with a Markovian process. In both cases, we are led to a stochastic Rosseland fluid limit.Then, we introduce a result of regularity for a class of quasilinear stochastic partial differential equations of parabolic type whose random term is driven by a cylindrical Wiener process.Finally, we study a Fokker-Planck equation with a noisy force governed by a cylindrical Wiener process. We prove existence and uniqueness of solutions to the problem and then existence and uniqueness of invariant measures to the equation
Caillerie, Nils. "Équations cinétiques stochastiques et déterministes dans le contexte des mathématiques appliquées à la biologie". Thesis, Lyon, 2017. http://www.theses.fr/2017LYSE1117/document.
Texto completoIn this thesis, we study some biology inspired mathematical models. More precisely, we focus on kinetic partial differential equations. The fields of application of such equations are numerous but we focus here on propagation phenomena for invasive species, the Escherichia coli bacterium and the cane toad Rhinella marina, for example. The first part of this this does not establish any mathematical result. We build several models for the dispersion of the cane toad in Australia. We confront those very models to multiple statistical data (birth rate, survival rate, dispersal behaviors) to test their validity. Those models are based on velocity-jump processes and kinetic equations. In the second part, we study propagation phenomena on simpler kinetic models. We illustrate several methods to mathematically establish propagation speed in this models. This part leads us to establish convergence results of kinetic equations to Hamilton-Jacobi equations by the perturbed test function method. We also show how to use the Hamilton-Jacobi framework to establish spreading results et finally, we build travelling wave solutions for reaction-transport model. In the last part, we establish a stochastic diffusion limit result for a kinetic equation with a random term. To do so, we adapt the perturbed test function method on the formulation of a stochastic PDE in term of infinitesimal generators. The thesis also contains an annex which presents the data on toads’ trajectories used in the first part."
Yang, Jie. "Solving Partial Differential Equations by Taylor Meshless Method". Thesis, Université de Lorraine, 2018. http://www.theses.fr/2018LORR0032/document.
Texto completoBased on Taylor Meshless Method (TMM), the aim of this thesis is to develop a simple, robust, efficient and accurate numerical method which is capable of solving large scale engineering problems and to provide a new idea for the follow-up study on meshless methods. To this end, the influence of the key factors in TMM has been studied by solving three-dimensional and non-linear Partial Differential Equations (PDEs). The main idea of TMM is to use high order polynomials as shape functions which are approximated solutions of the PDE and the discretization concerns only the boundary. To solve the unknown coefficients, boundary conditions are accounted by collocation procedures associated with least-square method. TMM that needs only boundary collocation without integration process, is a true meshless method. The main contributions of this thesis are as following: 1) Based on TMM, a general and efficient algorithm has been developed for solving three-dimensional PDEs; 2) Three coupling techniques in piecewise resolutions have been discussed and tested in cases of large-scale problems, including least-square collocation method and two coupling methods based on Lagrange multipliers; 3) A general numerical method for solving non-linear PDEs has been proposed by combining Newton Method, TMM and Automatic Differentiation technique; 4) To apply TMM for solving problems with singularities, the singular solutions satisfying the control equation are introduced as complementary shape functions, which provides a theoretical basis for solving singular problems
Autret, Solenn. "Retournement temporel des ondes acoustiques en milieux aléatoires". Toulouse 3, 2006. http://www.theses.fr/2006TOU30252.
Texto completoThis thesis is dedicated to the study of the distribution of the acoustic waves in one dimensional random media. The studied acoustic waves are those incoming from a time reversal experiment (TR) organized by Mathias Fink and his team in the Laboratory " Waves and Acoustics. " We bring to light why the profile of the waves obtained after TR does not depend on realizations of the random media unlike the waves not returned temporarily which have for profile a Gaussian process such as its power spectral density is solution of a system of equations of determinist transport, the coefficients of which depend on macroscopic variations of the media. Furthermore, the profile of the wave ater TR can spell as the married of the incidental signal with the power spectral density. The robustness of TR is then tested when errors of recording are made and when the random media changes between both phases of TR
Morancey, Morgan. "Contrôle d'équations de Schrödinger et d'équations paraboliques dégénérées singulières". Phd thesis, Ecole Polytechnique X, 2013. http://tel.archives-ouvertes.fr/tel-00910985.
Texto completoBellingeri, Carlo. "Formules d'Itô pour l'équation de la chaleur stochastique à travers les théories des chemins rugueux et des structures de regularité". Thesis, Sorbonne université, 2019. http://www.theses.fr/2019SORUS028.
Texto completoIn this thesis we develop a general theory to prove the existence of several Itô formulae on the one dimensional stochastic heat equation driven by additive space-time white noise. That is denoting by u the solution of this SPDE for any smooth enough function f we define some new notions of stochastic integrals defined upon u, which cannot be defined classically, in order to deduce new identities involving f(u) and some non trivial corrections. These new relations are obtained by using the theory of regularity structures and the theory of rough paths. In the first chapter we obtain a differential and an integral identity involving the reconstruction of some modelled distributions. Then we discuss a general change of variable formula over any Hölder continuous path in the context of rough paths, relating it to the notion of quasi-shuffle algebras and the family of so called quasi-geometric rough paths. Finally we apply the general results on quasi-geometric rough paths to the time evolution of u. Using the Gaussian behaviour of the process u, most of the terms involved in these equations are also identified with some classical constructions of stochastic calculus
Huré, Come. "Numerical methods and deep learning for stochastic control problems and partial differential equations". Thesis, Sorbonne Paris Cité, 2019. http://www.theses.fr/2019USPCC052.
Texto completoThe present thesis deals with numerical schemes to solve Markov Decision Problems (MDPs), partial differential equations (PDEs), quasi-variational inequalities (QVIs), backward stochastic differential equations (BSDEs) and reflected backward stochastic differential equations (RBSDEs). The thesis is divided into three parts.The first part focuses on methods based on quantization, local regression and global regression to solve MDPs. Firstly, we present a new algorithm, named Qknn, and study its consistency. A time-continuous control problem of market-making is then presented, which is theoretically solved by reducing the problem to a MDP, and whose optimal control is accurately approximated by Qknn. Then, a method based on Markovian embedding is presented to reduce McKean-Vlasov control prob- lem with partial information to standard MDP. This method is applied to three different McKean- Vlasov control problems with partial information. The method and high accuracy of Qknn is validated by comparing the performance of the latter with some finite difference-based algorithms and some global regression-based algorithm such as regress-now and regress-later.In the second part of the thesis, we propose new algorithms to solve MDPs in high-dimension. Neural networks, combined with gradient-descent methods, have been empirically proved to be the best at learning complex functions in high-dimension, thus, leading us to base our new algorithms on them. We derived the theoretical rates of convergence of the proposed new algorithms, and tested them on several relevant applications.In the third part of the thesis, we propose a numerical scheme for PDEs, QVIs, BSDEs, and RBSDEs. We analyze the performance of our new algorithms, and compare them to other ones available in the literature (including the recent one proposed in [EHJ17]) on several tests, which illustrates the efficiency of our methods to estimate complex solutions in high-dimension.Keywords: Deep learning, neural networks, Stochastic control, Markov Decision Process, non- linear PDEs, QVIs, optimal stopping problem BSDEs, RBSDEs, McKean-Vlasov control, perfor- mance iteration, value iteration, hybrid iteration, global regression, local regression, regress-later, quantization, limit order book, pure-jump controlled process, algorithmic-trading, market-making, high-dimension
Bachouch, Achref. "Numerical Computations for Backward Doubly Stochastic Differential Equations and Nonlinear Stochastic PDEs". Thesis, Le Mans, 2014. http://www.theses.fr/2014LEMA1034/document.
Texto completoThe purpose of this thesis is to study a numerical method for backward doubly stochastic differential equations (BDSDEs in short). In the last two decades, several methods were proposed to approximate solutions of standard backward stochastic differential equations. In this thesis, we propose an extension of one of these methods to the doubly stochastic framework. Our numerical method allows us to tackle a large class of nonlinear stochastic partial differential equations (SPDEs in short), thanks to their probabilistic interpretation. In the last part, we study a new particle method in the context of shielding studies
Rakotonasy, Solonjaka Hiarintsoa. "Modèle fractionnaire pour la sous-diffusion : version stochastique et edp". Phd thesis, Université d'Avignon, 2012. http://tel.archives-ouvertes.fr/tel-00839892.
Texto completoStoltz, Gabriel. "Quelques méthodes mathématiques pour la simulation moléculaire et multiéchelle". Phd thesis, Ecole des Ponts ParisTech, 2007. http://tel.archives-ouvertes.fr/tel-00166728.
Texto completoBréhier, Charles-Edouard. "Numerical analysis of highly oscillatory Stochastic PDEs". Phd thesis, École normale supérieure de Cachan - ENS Cachan, 2012. http://tel.archives-ouvertes.fr/tel-00824693.
Texto completoCharrier, Julia. "Analyse numérique d’équations aux dérivées aléatoires, applications à l’hydrogéologie". Thesis, Cachan, Ecole normale supérieure, 2011. http://www.theses.fr/2011DENS0030/document.
Texto completoThis work presents some results about probabilistic and deterministic numerical methods for partial differential equations with stochastic coefficients, with applications to hydrogeology. We first consider the steady flow equation in porous media with a homogeneous lognormal permeability coefficient, including the case of a low regularity covariance function. We establish error estimates, both in strong and weak senses, of the error in the solution resulting from the truncature of the Karhunen-Loève expansion of the coefficient. Then we establish finite element error estimates, from which we deduce an extension of the existing error estimate for the stochastic collocation method along with an error estimate for a multilevel Monte-Carlo method. We finally consider the coupling of the previous flow equation with an advection-diffusion equation, in the case when the uncertainty is important and the correlation length is small. We propose the numerical analysis of a numerical method, which aims at computing the mean velocity of the expansion of a pollutant. The method consists in a Monte-Carlo method, combining a finite element method for the flow equation and an Euler scheme for the stochastic differential equation associated to the advection-diffusion equation, seen as a Fokker-Planck equation
Le, cavil Anthony. "Représentation probabiliste de type progressif d'EDP nonlinéaires nonconservatives et algorithmes particulaires". Electronic Thesis or Diss., Université Paris-Saclay (ComUE), 2016. http://www.theses.fr/2016SACLY023.
Texto completoThis thesis performs forward probabilistic representations of nonlinear and nonconservative Partial Differential Equations (PDEs), which allowto numerically estimate the corresponding solutions via an interacting particle system algorithm, mixing Monte-Carlo methods and non-parametric density estimates.In the literature, McKean typeNonlinear Stochastic Differential Equations (NLSDEs) constitute the microscopic modelof a class of PDEs which are conservative. The solution of a NLSDEis generally a couple (Y,u) where Y is a stochastic process solving a stochastic differential equation whose coefficients depend on u and at each time t, u(t,.) is the law density of the random variable Yt.The main idea of this thesis is to consider this time a non-conservative PDE which is the result of a conservative PDE perturbed by a term of the type Lambda(u, nabla u) u. In this case, the solution of the corresponding NLSDE is again a couple (Y,u), where again Y is a stochastic processbut where the link between the function u and Y is more complicated and once fixed the law of Y, u is determined by a fixed pointargument via an innovating Feynmann-Kac type formula
Darrigade, Léo. "Modélisation du dialogue hôte-microbiote au voisinage de l’épithélium de l'intestin distal". Thesis, université Paris-Saclay, 2020. http://www.theses.fr/2020UPASM008.
Texto completoHuman health and physiology relie both on human cells activity and on intestinal microbiota activity, as well as on their interactions. In Mammals, the intestinal epithelium is very densely folded, and the smallest fold is called the intestinal crypt. It is also the simplest unit of the host-microbiota crosstalk. The size of a crypt, which is made of approximately 700 cells, and the detailed knowledge that we have of its functionning are amenable to build a detailed mathematical model. We constructed an individual-based model of epithelial cells interacting with chemicals produced by the microbiota which diffuse in the crypt lumen. This model is formalised as a piecewise determinist Markov process. It accounts for: local interactions due to cell contact (among which are mechanical interactions); cell proliferation, differenciation and extrusion which are regulated spatially or by chemicals concentrations; chemicals diffusing and reacting with cells. We demonstrate the convergence of the stochastic individual-based model to a deterministic model under the assumption of large population. A second limit model is obtained formally when the size of cells goes to 0. We aim at obtaining deterministic limit model because they can be simulated much faster, and can be used to improve understanding of the stochastic model or can be coupled with a model of the whole colon. The three models are implemented, convergence of models is shown numerically, and simulations of the individual-based model are analysed to show that qualitative behaviour of the crypt is reproduced
Le, cavil Anthony. "Représentation probabiliste de type progressif d'EDP nonlinéaires nonconservatives et algorithmes particulaires". Thesis, Université Paris-Saclay (ComUE), 2016. http://www.theses.fr/2016SACLY023.
Texto completoThis thesis performs forward probabilistic representations of nonlinear and nonconservative Partial Differential Equations (PDEs), which allowto numerically estimate the corresponding solutions via an interacting particle system algorithm, mixing Monte-Carlo methods and non-parametric density estimates.In the literature, McKean typeNonlinear Stochastic Differential Equations (NLSDEs) constitute the microscopic modelof a class of PDEs which are conservative. The solution of a NLSDEis generally a couple $(Y,u)$ where $Y$ is a stochastic process solving a stochastic differential equation whose coefficients depend on $u$ and at each time $t$, $u(t,cdot)$ is the law density of the random variable $Y_t$.The main idea of this thesis is to consider this time a non-conservative PDE which is the result of a conservative PDE perturbed by a term of the type $Lambda(u, nabla u) u$. In this case, the solution of the corresponding NLSDE is again a couple $(Y,u)$, where again $Y$ is a stochastic processbut where the link between the function $u$ and $Y$ is more complicated and once fixed the law of $Y$, $u$ is determined by a fixed pointargument via an innovating Feynmann-Kac type formula
Scotti, Simone. "Applications of the error theory using Dirichlet forms". Phd thesis, Université Paris-Est, 2008. http://tel.archives-ouvertes.fr/tel-00349241.
Texto completoSabbagh, Wissal. "Some Contributions on Probabilistic Interpretation For Nonlinear Stochastic PDEs". Thesis, Le Mans, 2014. http://www.theses.fr/2014LEMA1019/document.
Texto completoThe objective of this thesis is to study the probabilistic representation (Feynman-Kac for- mula) of different classes ofStochastic Nonlinear PDEs (semilinear, fully nonlinear, reflected in a domain) by means of backward doubly stochastic differential equations (BDSDEs). This thesis contains four different parts. We deal in the first part with the second order BDS- DEs (2BDSDEs). We show the existence and uniqueness of solutions of 2BDSDEs using quasi sure stochastic control technics. The main motivation of this study is the probabilistic representation for solution of fully nonlinear SPDEs. First, under regularity assumptions on the coefficients, we give a Feynman-Kac formula for classical solution of fully nonlinear SPDEs and we generalize the work of Soner, Touzi and Zhang (2010-2012) for deterministic fully nonlinear PDE. Then, under weaker assumptions on the coefficients, we prove the probabilistic representation for stochastic viscosity solution of fully nonlinear SPDEs. In the second part, we study the Sobolev solution of obstacle problem for partial integro-differentialequations (PIDEs). Specifically, we show the Feynman-Kac formula for PIDEs via reflected backward stochastic differentialequations with jumps (BSDEs). Specifically, we establish the existence and uniqueness of the solution of the obstacle problem, which is regarded as a pair consisting of the solution and the measure of reflection. The approach is based on stochastic flow technics developed in Bally and Matoussi (2001) but the proofs are more technical. In the third part, we discuss the existence and uniqueness for RBDSDEs in a convex domain D without any regularity condition on the boundary. In addition, using the approach based on the technics of stochastic flow we provide the probabilistic interpretation of Sobolev solution of a class of reflected SPDEs in a convex domain via RBDSDEs. Finally, we are interested in the numerical solution of BDSDEs with random terminal time. The main motivation is to give a probabilistic representation of Sobolev solution of semilinear SPDEs with Dirichlet null condition. In this part, we study the strong approximation of this class of BDSDEs when the random terminal time is the first exit time of an SDE from a cylindrical domain. Thus, we give bounds for the discrete-time approximation error.. We conclude this part with numerical tests showing that this approach is effective
Godinho, David. "Contribution à l'étude des équations de Boltzmann, Kac et Keller-Segel à l'aide d'équations différentielles stochastiques non linéaires". Phd thesis, Université Paris-Est, 2013. http://tel.archives-ouvertes.fr/tel-00931392.
Texto completoBonnaillie-Noël, Virginie. "Analyse asymptotique, spectrale et numérique pour quelques problèmes elliptiques issus de la physique ou de la mécanique". Habilitation à diriger des recherches, Université Rennes 1, 2011. http://tel.archives-ouvertes.fr/tel-00650033.
Texto completoCambon, Sebastien. "Méthode d'éléments finis d'ordre élevé et d'équations intégrales pour la résolution de problème de furtivité radar d'objets à symétrie de révolution". Thesis, Toulouse, INSA, 2012. http://www.theses.fr/2012ISAT0047/document.
Texto completoIn this thesis, we are interested in modeling diffraction of electromagnetic waves by axisymmetric and highly heterogeneous objects. Our method consists in a coupling between partial differential equations and integral equations. This idea is mainly interesting for two reasons : heterogeneities are handled naturally in the formulation and integral equations give an analytical representation of solutions outside the object based on surface currents.These advantages allow us to limit the domain of simulation to the object itself. In addition,using Fourier series combined with the rotational invariance property of the object, the 3D problem is reduced to a countable set of 2D problems. The study of these problems is split into several parts. Each part has to deal with aspecific problem as for example the numerical integration of singular integrals which is difficult to achieve. As a first step, we study time-harmonic Maxwell’s equations in a bounded domain for which we develop a new high-order finite element method and present its efficiency and accuracy on many examples. Secondly, we consider the diffraction of plane waves by perfect electric conductors to analyse integral equations for these kind of object.The boundary finite element method applied is defined by extension of the previous one via tangential trace operator. Then, we solve the coupled problem using a well chosen formulation based on the previous studies for which our finite element method is naturally adapted by construction. In order to evaluate its efficiency, a comparison is performed between our program « AxiMax » and one based on a purely 3D model. To conclude, in the last two chapters, we present the numerical integration method and the multi-processing algorithm developed in AxiMax. In all cases, we put forward the fact that our finite element method provides accurate results depending on the quality of the simulation parameters