We consider linear and semilinear stochastic partial differential equations that in some sense can be viewed as being at the "endpoints" of the classical variational theory by Krylov and Rozovskii [25]. In terms of regularity of the coeffcients, the minimal assumption is boundedness and measurability, and a unique L2- valued solution is then readily available. We investigate its further properties, such as higher order integrability, boundedness, and continuity. The other class of equations considered here are the ones whose leading operators do not satisfy the strong coercivity condition, but only a degenerate version of it, and therefore are not covered by the classical theory. We derive solvability in Wmp spaces and also discuss their numerical approximation through finite different schemes.

CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior / Esta tese trata do controle nulo local de um problema de fronteira-livre para a equaÃÃo do calor semilinear 1D com controles distribuÃdos (apoiado localmente no espaÃo) ou controles de fronteira (atuando em x = 0). provamos que, se o tempo final T à fixado e o estado inicial à suficientemente pequeno, existe controles que dirigem o estado exatamente para descansar no tempo t = T. AlÃm disso, analisamos a controlabilidade nulo de um sistema nÃo-linear 1D que modela a interaÃÃo de um fluido e sua fronteira. O fluido à governado pela equaÃÃo de Burgers viscosa e os controles distribuÃdos. Por Ãltimo, vamos lidar com o sistema de Navier-Stokes e Boussinesq 3D, definido em um cubo. Neste contexto, provamos um resultado sobre a sua controlabilidade aproximada global por meio de controles de fronteira que atuam em alguma parte da faces do cubo. / This Thesis deals with the local null control of a free-boundary problem for the 1D semilinear heat equation with distributed controls (locally supported in space) or boundary controls (acting at x = 0). we prove that, if the final time T is fixed and the initial state is sufficiently small, there exists controls that drive the state exactly to rest at time t = T. Furthermore, we analyze the null controllability of a 1D nonlinear system which models the interaction of a fluid and its boundary. The fluid is governed by the viscous Burgers equation and the distributed controls. Lastly, we deal with the 3D Navier-Stokes and Boussinesq system, posed in a cube. In this context, we prove a result concerning its global approximate controllability by means of boundary controls which act in some part of cube faces.

Linear degeneracy of a PDE is a concept that is related to a number of interesting geometric constructions. We first take a quadratic line complex, which is a three parameter family of lines in projective space P3 specified by a single quadratic relation in the Plucker coordinates. This complex supplies us with a conformal structure in P3. With this conformal structure, we associate a three-dimensional second order quasilinear wave equation. We show that any PDE arising in this way is linearly degenerate, furthermore, any linearly degenerate PDE can be obtained by this construction. We classify Segre types of quadratic complexes for which the structure is conformally flat, as well as Segre types for which the corresponding PDE is integrable. These results were published in [1]. We then introduce the notion of characteristic integrals, discuss characteristic integrals in 3D and show that, for certain classes of second-order linearly degenerate dispersionless integrable PDEs, the corresponding characteristic integrals are parameterised by points on the Veronese variety. These results were published in [2].

The class of hyperbolic conservation laws model the phenomena of non-linear wave propagation, including the presence and propagation of discontinuities and expansion waves. Such nonlinear systems can generate discontinuities in the so-lution even for smooth initial conditions. Presence of discontinuities results in break down of a solution in the classical sense and to show existence, weak for-mulation of a problem is required. Moreover, closed form solutions are di cult to obtain and in some cases such solutions are even unavailable. Thus, numerical algorithms play an important role in solving such systems. There are several dis-cretization techniques to solve hyperbolic systems numerically and Finite Volume Method (FVM) is one of such important frameworks. Numerical algorithms based on FVM are broadly classi ed into two categories, central discretization methods and upwind discretization methods. Various upwind and central discretization methods developed so far di er widely in terms of robustness, accuracy and ef-ciency and an ideal scheme with all these characteristics is yet to emerge. In this thesis, novel upwind and central schemes are formulated for various hyper-bolic systems, with the aim of maintaining right balance between accuracy and robustness. This thesis is divided into two parts. First part consists of the formulation of upwind methods to simulate genuine weakly hyperbolic (GWH) systems. Such systems do not possess full set of linearly independent (LI) eigenvectors and some of the examples include pressureless gas dynamics system, modi ed Burgers' sys-tem and further modi ed Burgers' system. The main challenge while formulating an upwind solver for GWH systems, using the concept of Flux Di erence Splitting (FDS), is to recover full set of LI eigenvectors, which is done through addition of generalized eigenvectors using the theory of Jordan Canonical Forms. Once the defective set of LI eigenvectors are completed, a novel (FDS-J) solver is for-mulated in such a manner that it is independent of generalized eigenvectors, as they are not unique. FDS-J solver is capable of capturing various shocks such as -shocks, 0-shocks and 00-shocks accurately. In this thesis, the FDS-J schemes are proposed for those GWH systems each of which have one particular repeated eigenvalue with arithmetic multiplicity (AM) greater than one. Moreover, each ux Jacobian matrix corresponding to such systems is similar to a unique Jordan matrix. After the successful treatment of genuine weakly hyperbolic systems, this strategy is further applied to those weakly hyperbolic subsystems which result on employ-ing various convection-pressure splittings to the Euler ux function. For example, Toro-Vazquez (TV) splitting and Zha-Bilgen (ZB) type splitting approaches to split the Euler ux function yield genuine weakly hyperbolic convective parts and strict hyperbolic pressure parts. Moreover, the ux Jacobian of each convective part is similar to a Jordan matrix with at least two lower order Jordan blocks. Based on the lines of FDS-J scheme, we develop two numerical schemes for Eu-ler equations using TV splitting and ZB type splitting. Both the new ZBS-FDS and TVS-FDS schemes are tested on various 1-D shock tube problems and out of two, contact capturing ZBS-FDS scheme is extended to 2-dimensional Euler system where it is tested successfully on various test cases including many shock instability problems. Second part of the thesis is associated with the development of simple, robust and accurate central solvers for systems of hyperbolic conservation laws. The idea of splitting schemes together with the notion of FDS is not easily extendable to systems such as shallow water equations. Thus, a novel central solver Convection Isolated Discontinuity Recognizing Algorithm (CIDRA) is formulated for shallow water equations. As the name suggests, the convective ux is isolated from the total ux in such a way that other ux, in present case other ux represents celerity part, must possess non-zero eigenvalue contribution. FVM framework is applied to each part separately and ux equivalence principle is used to x the coe cient of numerical di usion. CIDRA for SWE is computed on various 1-D and 2-D benchmark problems and extended to Euler systems e ortlessly. As a further improvement, a scalar di usion based algorithm CIDRA-1 is designed for v Euler systems. The scalar di usion coe cient depends on that particular part of the Rankine-Hugoniot (R-H) condition which involves total energy of the system as a direct contribution. This algorithm is applied to a variety of shock tube test cases including a class of low density ow problems and also to various 2-D test problems successfully. vi

We develop expansion methods as a new computational approach towards high-dimensional partial differential equations (PDEs), particularly of such type as arising in the valuation of financial derivatives. The proposed methods are extended from [41] and use principal component analysis (PCA) of the underlying process in combination with a Taylor expansion of the value function into solutions to low-dimensional PDEs. They enable calculation of highly accurate approximate solutions with computational complexity polynomial in the number of dimensions for PDEs with a low number of dominant principal components. For the case of PDEs with constant coefficients, we show existence of expansion solutions and prove theoretical error bounds. We give a precise characterisation of when our methods can be applied and construct specific examples of a first and second order version. We provide numerical results showing that the empirically observed convergence speeds are in agreement with the theoretical predictions. For the case of PDEs with varying coefficients, we give a heuristic motivation using the Parametrix approach and empirically test the methods' accuracy for a range of variable parameter stock models. We demonstrate the applicability of our expansion methods to real-world securities pricing problems by considering path-dependent and early-exercise options in the LIBOR market model. Using the example of Bermudan swaptions and Ratchet floors, which are considered difficult benchmark problems, we give a careful analysis of the numerical accuracy and computational complexity. We are able to demonstrate that for problems with medium to high dimensionality, around 60-100, and moderate time horizons, the presented PDE methods deliver results comparable in accuracy to benchmark state-of-the-art Monte Carlo methods in similar or (significantly) faster run time.

No description available.

Dans certains phénomènes physiques modélisés par des EDP, les coefficients intervenant dans les équations ne sont pas des fonctions déterministes fixées, et dépendent de paramètres qui peuvent varier. Ceci se produit par exemple dans le cadre de la modélisation des écoulements en milieu poreux lorsqu’on décrit le champ de perméabilité par un processus stochastique pour tenir compte de l’incertitude sur ce champs. Dans d’autres cadres, il peut s’agir de paramètres déterministes que l’on cherche à ajuster, par exemple pour optimiser un certain critère sur la solution. La solution u dépend donc non seulement de la variable x d’espace/temps mais aussi d’un vecteur y = (yj) de paramètres potentiellement nombreux, voire en nombre infinis. L’approximation numérique en y de l’application (x,y)-> u(x, y) est donc impossible par les méthodes classiques de type éléments finis, et il faut envisager des approches adaptées aux grandes dimensions. Cette thèse est consacrée à l’étude théorique et l’approximation numérique des EDP paramétriques en grandes dimensions. Pour une large classe d’EDP avec une certaine dépendance anisotrope en les paramètres yj, on étudie de la régularité en y de l’application u et on propose des méthodes d’approximation numérique dont les performances ne subissent pas les détériorations classiquement observées en grande dimension. On cherche en particulier à évaluer la complexité de la classe des solutions {u(y)}, par exemple au sens des épaisseurs de Kolmogorov, afin de comprendre les limites inhérentes des méthodes numériques. On analyse en pratique les propriétés de convergences de diverses méthodes d’approximation avec des polynômes creux. / For certain physical phenomenon that are modelled by PDE, the coefficients intervening in the equations are not fixed deterministic functions, but depend on parameters that may vary.

On étudie les propriétés de rigidité symplectique des difféomorphismes hamiltoniens en dimension finie et en dimension infinie. En dimension finie, les outils principaux qu'on utilise sont les fonctions génératrices et les capacités symplectiques. En dimension infinie on regarde les flots des équations en dérivées partielles (EDPs) hamiltoniennes et, en particulier, les flots qui peuvent être approchés uniformément par des flots hamiltoniens de dimension finie.Dans la première partie de la thèse on étudie les sélecteurs d'action définies à partir des fonctions génératrices et on construit des invariants hamiltoniens pour les sous-ensembles de $R^{2m}times T^*T^k$. Cela nous permet de démontrer un théorème non-squeezing coisotrope pour les difféomorphismes hamiltoniens à support compact de $R^{2n}$. On montre à continuation que cette propriété apparaisse dans certains cas non compacts. Finalement, on explique comment ce résultat donne aussi l'information sur le problème de rigidité symplectique en dimension intermédiaire. Encore en dimension finie, on démontre qu'on peut utiliser le théorème du chameau symplectique pour produire des sous-ensembles invariants compacts dans des surfaces d'energie.Dans la deuxième partie on étudie les propriétés de rigidité symplectique des flots des EDPs hamiltoniennes. On se place dans le contexte introduit par Kuksin et on étudie une classe particulière de EDPs semi-linéaires qui peuvent être approchées par flots hamiltoniens de dimension finie. D'abord on donne une nouvelle construction de capacité symplectique en dimension infinie à partir des capacités de Viterbo. Puis on démontre l'analogue de la rigidité intermédiaire pour certaines EDPs hamiltoniennes. Cette classe inclue l'équation d'ondes en dimension 1 avec une non-linéarité bornée, comme par exemple l'équation de Sine-Gordon. Dans la dernière partie de la thèse on s'intéresse à un analogue de la conjecture d'Arnold pour l'équation de Schrödinger périodique avec une non linéarité de convolution. / We study symplectic rigidity properties in both finite and infinite dimension. In finite dimension, the main tools that we use are generating functions and symplectic capacities. In infinite dimension we study flows of Hamiltonian partial differential equations (PDEs) and, in particular, flows which can be uniformly approximated by finite dimensional Hamiltonian diffeomorphisms.In the first part of this thesis we study the action selectors defined from generating functions and we build Hamiltonian invariants for subsets of $R^{2m}times T^*T^k$. This allows us to prove a coisotropic non-squeezing theorem for compactly supported Hamiltonian diffeomorphisms of $R^{2n}$. We then extend this result to some non-compact settings. Finally we explain how this result can give information about the middle dimensional symplectic rigidity problem. Still in finite dimensions, we show that it is possible to use the symplectic camel theorem to create energy surfaces with compact invariant subsets.In the second part of the thesis we study symplectic rigidity properties of flows of Hamiltonian PDEs. We work in the context introduced by Kuksin and study a particular class of semi-linear Hamiltonian PDEs that can be approximated by finite dimensional Hamiltonian diffeomorphisms. We first give a new construction of an infinite dimensional capacity using Viterbo's capacities. The main result of this part is the proof of the analogue of the middle dimensional rigidity for certain types of Hamiltonian PDEs. These include nonlinear string equations with bounded nonlinearity such as the Sine-Gordon equation. In the final part of this thesis we study an analogue of Arnold's conjecture for the periodic Schrödinger equations with a convolution nonlinearity.

No description available.

This thesis describes initial language extensions to the Modelica language to define a more general language called PDEModelica, with built-in support for modeling with partial differential equations (PDEs). Modelica® is a standardized modeling language for objectoriented, equation-based modeling. It also supports component-based modeling where existing components with modified parameters can be combined into new models. The aim of the language presented in this thesis is to maintain the advantages of Modelica and also add partial differential equation support. Partial differential equations can be defined using a coefficient-based approach, where a predefined PDE is modified by changing its coefficient values. Language operators to directly express PDEs in the language are also discussed. Furthermore, domain geometry description is handled and language extensions to describe geometries are presented. Boundary conditions, required for a complete PDE problem definition, are also handled. A prototype implementation is described as well. The prototype includes a translator written in the relational meta-language, RML, and interfaces to external software such as mesh generators and PDE solvers, which are needed to solve PDE problems. Finally, a few examples modeled with PDEModelica and solved using the prototype are presented. / <p>Report code: LiU-Tek-Lic-2002:63.</p>