PDE-Constrained Equilibrium Problems under Uncertainty: Existence, Optimality Conditions and Regularization

In dieser Arbeit werden PDE-beschränkte Gleichgewichtsprobleme unter Unsicherheiten analysiert. Im Detail diskutieren wir eine Klasse von risikoneutralen verallgemeinerten Nash-Gleichgewichtsproblemen sowie eine Klasse von risikoaversen Nash Gleichgewichtsproblemen. Sowohl für die risikoneutralen PDE-beschränkten Optimierungsprobleme mit punktweisen Zustandsschranken als auch für die risikoneutralen verallgemeinerten Nash Gleichgewichtsprobleme wird die Existenz von Lösungen beziehungsweise Nash Gleichgewichten bewiesen und Optimalitätsbedingungen hergeleitet. Die Betrachtung von Ungleichheitsbedingungen an den stochastischen Zustand führt in beiden Fällen zu Komplikationen bei der Herleitung der Lagrange-Multiplikatoren. Nur durch höhere Regularität des stochastischen Zustandes können wir auf die bestehende Optimalitätstheorie für konvexe Optimierungsprobleme zurückgreifen. Die niedrige Regularität des Lagrange-Multiplikators stellt auch für die numerische Lösbarkeit dieser Probleme ein große Herausforderung dar. Wir legen den Grundstein für eine erfolgreiche numerische Behandlung risikoneutraler Nash Gleichgewichtsproblem mittels Moreau-Yosida Regularisierung, indem wir zeigen, dass dieser Regularisierungsansatz konsistent ist. Die Moreau-Yosida Regularisierung liefert eine Folge von parameterabhängigen Nash Gleichgewichtsproblemen und der Grenzübergang im Glättungsparameter zeigt, dass die stationären Punkte des regularisierten Problems gegen ein verallgemeinertes Nash Gleichgewicht des ursprünglich Problems schwach konvergieren. Die Theorie legt also nahe, dass auf der Moreau-Yosida Regularisierung eine numerische Methode aufgebaut werden kann. Darauf aufbauend werden Algorithmen vorgeschlagen, die aufzeigen, wie risikoneutrale PDE-beschränkte Optimierungsprobleme mit punktweisen Zustandsschranken und risikoneutrale PDE-beschränkte verallgemeinerte Nash Gleichgewichtsprobleme gelöst werden können. Für die Modellierung der Risikopräferenz in der Klasse von risikoaversen Nash Gleichgewichtsprobleme verwenden wir kohärente Risikomaße. Da kohärente Risikomaße im Allgemeinen nicht glatt sind, ist das resultierende PDE-beschränkte Nash Gleichgewichtsproblem ebenfalls nicht glatt. Daher glätten wir die kohärenten Risikomaße mit Hilfe einer Epi-Regularisierungstechnik. Sowohl für das ursprüngliche Nash Gleichgewichtsproblem als auch für die geglätteten parameterabhängigen Nash Gleichgewichtsprobleme wird die Existenz von Nash Gleichgewichten gezeigt, sowie Optimalitätsbedingungen hergeleitet. Wir liefern wertvolle Resultate dafür, dass dieser Glättungsansatz sich für die Entwicklung eines numerischen Verfahren eignet, indem wir beweisen können, dass sowohl eine Folge von stationären Punkten als auch eine Folge von Nash Gleichgewichten des epi-regularisierten Problems eine schwach konvergente Teilfolge hat, deren Grenzwert ein Nash Gleichgewicht des ursprünglichen Problems ist.In this paper, we analyze PDE-constrained equilibrium problems under uncertainty. In detail, we discuss a class of risk-neutral generalized Nash equilibrium problems and a class of risk-averse Nash equilibrium problems. For both, the risk-neutral PDE-constrained optimization problems with pointwise state constraints and the risk-neutral generalized Nash equilibrium problems, the existence of solutions and Nash equilibria, respectively, is proved and optimality conditions are derived. The consideration of inequality conditions on the stochastic state leads in both cases to complications in the derivation of the Lagrange multipliers. Only by higher regularity of the stochastic state we can resort to the existing optimality theory for convex optimization problems. The low regularity of the Lagrange multiplier also poses a major challenge for the numerical solvability of these problems. We lay the foundation for a successful numerical treatment of risk-neutral Nash equilibrium problems using Moreau-Yosida regularization by showing that this regularization approach is consistent. The Moreau-Yosida regularization yields a sequence of parameter-dependent Nash equilibrium problems and the boundary transition in the smoothing parameter shows that the stationary points of the regularized problem converge weakly against a generalized Nash equilibrium of the original problem. Thus, the theory suggests that a numerical method can be built on the Moreau-Yosida regularization. Based on this, algorithms are proposed to show how to solve risk-neutral PDE-constrained optimization problems with pointwise state bounds and risk-neutral PDE-constrained generalized Nash equilibrium problems. I n order to model risk preference in the class of risk-averse Nash equilibrium problems, we use coherent risk measures. Since coherent risk measures are generally not smooth, the resulting PDE-constrained Nash equilibrium problem is also not smooth. Therefore, we smooth the coherent risk measures using an epi-regularization technique. For both the original Nash equilibrium problem and the smoothed parameter-dependent Nash equilibrium problems, we show the existence of Nash equilibria, and derive optimality conditions. We provide valuable results for making this smoothing approach suitable for the development of a numerical method by proving that both, a sequence of stationary points and a sequence of Nash equilibria of the epi-regularized problem, have a weakly convergent subsequence whose limit is a Nash equilibrium of the original problem

A model for the evolution of laminates in finite-strain elastoplasticity

We study the time evolution in elastoplasticity within the rate-independent framework of generalized standard materials. Our particular interest is the formation and the evolution of microstructure. Providing models where existence proofs are possible is a challenging task since the presence of microstructure comes along with a lack of convexity and, hence, compactness arguments cannot be applied to prove the existence of solutions. In order to overcome this problem, we will incorporate information on the microstructure into the internal variable, which is still compatible with generalized standard materials. More precisely, we shall allow for such microstructure that is given by simple or sequential laminates. We will consider a model for the evolution of these laminates and we will prove a theorem on the existence of solutions to any finite sequence of time-incremental minimization problems. In order to illustrate the mechanical consequences of the theory developed some numerical results, especially dealing with the rotation of laminates, are presented

Formulation, analysis and solution algorithms for a model of gradient plasticity within a discontinuous Galerkin framework

Includes bibliographical references (p. [221]-239).An investigation of a model of gradient plasticity in which the classical von Mises yield function is augmented by a term involving the Laplacian of the equivalent plastic strain is presented. The theory is developed within the framework of non-smooth convex analysis by exploiting the equivalence between the primal and dual expressions of the plastic deformation evolution relations. The nonlocal plastic evolution relations for the case of gradient plasticity are approximated using a discontinuous Galerkin finite element formulation. Both the small- and finite-strain theories are investigated. Considerable attention is focused on developing a firm mathematical foundation for the model of gradient plasticity restricted to the infinitesimal-strain regime. The key contributions arising from the analysis of the classical plasticity problem and the model of gradient plasticity include demonstrating the consistency of the variational formulation, and analyses of both the continuous-in-time and fully-discrete approximations; the error estimates obtained correspond to those for the conventional Galerkin approximations of the classical problem. The focus of the analysis is on those properties of the problem that would ensure existence of a unique solution for both hardening and softening problems. It is well known that classical finite element method simulations of softening problems are pathologically dependent on the discretisation

Some Stationary and Evolution Problems Governed by Various Notions of Monotone Operators

The purpose of this work is to explore some notions of monotonicity for operators between Banach spaces and the applications to the study of boundary value problems (BVPs) and initial boundary value problems (IBVPs) for partial differential equations (PDEs), with the possibility in the end to examine new problems and provide some solutions. Variational approach will be used to reformulate these problems into stationary equations (in the case of BVPs) and evolution equations (in the case of IBVPs), where the underlined operators constructed as realizations of those problems in appropriate function spaces. This is known as weak formulation, which allows us to find weak solutions of the problems in a larger functions space rather than classical solutions that are sufficiently smooth. The theory of monotone and pseudomonotone operators will be applied to find existence theorems for stationary equations and evolution equations. In addition, the existence theorem for evolution equations with locally monotone operator will also be presented as a generalisation of the one with monotone operators. Another type of monotonicity so-called strict p-quasimonotonicity, which is defined in term of Young measures. This type of weaker, integrated version of monotonicity is directly applied in the study of elliptic and parabolic system of PDEs, the difficulty arises from dealing with this monotonicity is overcome by the theory of Young measures. The application of these monotonicity in the study of variational inequality will also be discussed. In particular, there is a new setting for strict p-quasimonotonicity in a particular type of elliptic variational inequalities, the proof of the new existence theorem will also be presented. Some open problems on the application of strict p-quasimonotonicity in the study of parabolic variational inequalities will also be discussed. Finally, we mention the theory of monotone and pseudomonotone operators in the study of second order evolution equations. A new setting of the local monotonicity in the second order evolution equations will be presented as well as the new existence theorem

Long term dynamics of the subgradient method for Lipschitz path differentiable functions

We consider the long-term dynamics of the vanishing stepsize subgradient method in the case when the objective function is neither smooth nor convex. We assume that this function is locally Lipschitz and path differentiable, i.e., admits a chain rule. Our study departs from other works in the sense that we focus on the behavior of the oscillations, and to do this we use closed measures. We recover known convergence results, establish new ones, and show a local principle of oscillation compensation for the velocities. Roughly speaking, the time average of gradients around one limit point vanishes. This allows us to further analyze the structure of oscillations, and establish their perpendicularity to the general drift

The gap between a variational problem and its occupation measure relaxation

Recent works have proposed linear programming relaxations of variational optimization problems subject to nonlinear PDE constraints based on the occupation measure formalism. The main appeal of these methods is the fact that they rely on convex optimization, typically semidefinite programming. In this work we close an open question related to this approach. We prove that the classical and relaxed minima coincide when the dimension of the codomain of the unknown function equals one, both for calculus of variations and for optimal control problems, thereby complementing analogous results that existed for the case when the dimension of the domain equals one. In order to do so, we prove a generalization of the Hardt-Pitts decomposition of normal currents applicable in our setting. We also show by means of a counterexample that, if both the dimensions of the domain and of the codomain are greater than one, there may be a positive gap. The example we construct to show the latter serves also to show that sometimes relaxed occupation measures may represent a more conceptually-satisfactory "solution" than their classical counterparts, so that -- even though they may not be equivalent -- algorithms rendering accessible the minimum in the larger space of relaxed occupation measures remain extremely valuable. Finally, we show that in the presence of integral constraints, a positive gap may occur at any dimension of the domain and of the codomain.Comment: 46 pages, 10 figure

Statistical mechanics of two-dimensional and geophysical flows

International audienceThe theoretical study of the self-organization of two-dimensional and geophysical turbulent flows is addressed based on statistical mechanics methods. This review is a self-contained presentation of classical and recent works on this subject; from the statistical mechanics basis of the theory up to applications to Jupiter's troposphere and ocean vortices and jets. Emphasize has been placed on examples with available analytical treatment in order to favor better understanding of the physics and dynamics. The equilibrium microcanonical measure is built from the Liouville theorem. On this theoretical basis, we predict the output of the long time evolution of complex turbulent flows as statistical equilibria. This is applied to make quantitative models of two-dimensional turbulence, the Great Red Spot and other Jovian vortices, ocean jets like the Gulf-Stream, and ocean vortices. We also present recent results for non-equilibrium situations, for the studies of either the relaxation towards equilibrium or non-equilibrium steady states

H\"older and Sobolev regularity of optimal transportation potentials with rough measures

We consider a Kantorovich potential associated to an optimal transportation problem between measures that are not necessarily absolutely continuous with respect to the Lebesgue measure, but are comparable to the Lebesgue measure when restricted to balls with radius greater than some $\delta>0$. Our main results extend the classical regularity theory of optimal transportation to this framework. In particular, we establish both H\"older and Sobolev regularity results for Kantorovich potentials up to some critical length scale depending on $\delta$. Our assumptions are very natural in the context of the numerical computation of optimal maps, which often involves approximating by sums of Dirac masses some measures that are absolutely continuous with densities bounded away from zero and infinity on their supports

Some results on cohesive energies: approximation, lower semicontinuity and quasistatic evolution

In this thesis, cohesive fracture is investigated under three different perspectives. First we study the asymptotic behaviour of a variational model for damaged elasto-plastic materials in the case of antiplane shear. The energy functionals we consider depend on a small parameter, which forces damage concentration on regions of codimension one. We determine the Gamma-limit as the small parameter tends to zero and show that it contains an energy term involving the crack opening. The second problem we consider is the lower semicontinuity of some free discontinuity functionals with linear growth defined on the space of functions with bounded deformation. The volume term is convex and depends only on the Euclidean norm of the symmetrised gradient. We introduce a suitable class of cohesive surface terms, which make the functional lower semicontinuous with respect to L^1 convergence. Finally, we prove the existence of quasistatic evolutions for a cohesive fracture on a prescribed crack surface, in small-strain antiplane elasticity. The main feature of the model is that the density of the energy dissipated in the fracture process depends on the total variation of the amplitude of the jump. Thus, any change in the crack opening entails a loss of energy, until the crack is complete. In particular this implies a fatigue phenomenon, i.e., a complete fracture may be produced by oscillation of small jumps

Computing approximations and generalized solutions using moments and positive polynomials

Le problème généralisé des moments (PGM) est un problème d'optimisation linéaire sur des espaces de mesures. Il permet de modéliser simplement un grand nombre d'applications. En toute généralité il est impossible à résoudre mais si ses données sont des polynômes et des ensembles semi-algébriques alors on peut définir une hiérarchie de relaxations semidéfinies (SDP) - la hiérarchie moments-sommes-de-carrés (moments-SOS) - qui permet en principe d'approcher la valeur optimale avec une précision arbitraire. Le travail contenu dans cette thèse adresse deux facettes concernants le PGM et la hiérarchie moments-SOS: Une première facette concerne l'évolution des relaxations SDP pour le PGM. Le degré des poids SOS dans la hiérarchie moments-SOS augmente avec l'ordre de relaxation. Lorsque le nombre de variables n'est pas modeste, on obtient rapidement des programmes SDP de taille trop grande pour les logiciels de programmation SDP actuels, sauf si l'on peut utiliser des symétries ou une parcimonie structurée souvent présente dans beaucoup d'applications de grande taille. On présente donc un nouveau certificat de positivité sur un compact semi-algébrique qui (i) exploite la parcimonie présente dans sa description, et (ii) dont les polynômes SOS ont un degré borné à l'avance. Grâce à ce nouveau certificat on peut définir une nouvelle hiérarchie de relaxations SDP pour le PGM qui exploite la parcimonie et évite l'explosion de la taille des matrices semidéfinies positives liée au degré des poids SOS dans la hiérarchie standard. Une deuxième facette concerne (i) la modélisation de nouvelles applications comme une instance particulière du PGM, et (ii) l'application de la méthodologie moments-SOS pour leur résolution. En particulier on propose des approximations déterministes de contraintes probabilistes, un problème difficile car le domaine des solutions admissibles associées est souvent non-convexe et même parfois non connecté. Dans notre approche moments-SOS le domaine admissible est remplacé par un ensemble plus petit qui est le sous-niveau d'un polynôme dont le vecteur des coefficients est une solution optimale d'un certain SDP. La qualité de l'approximation (interne) croît avec le degré du polynôme et la taille du SDP. On illustre cette approche dans le problème du calcul du flux de puissance optimal dans les réseaux d'énergie, une application stratégique où la prise en compte des contraintes probabilistes devient de plus en plus cruciale (e.g., pour modéliser l'incertitude liée á l'énergie éolienne et solaire). En outre on propose une extension des cette procedure qui est robuste à l'incertitude sur la distribution sous-jacente. Des garanties de convergence sont fournies. Une deuxième contribution concerne l'application de la méthodologie moments-SOS pour l'approximation de solutions généralisés en commande optimale. Elle permet de capturer le comportement limite d'une suite minimisante de commandes et de la suite de trajectoires associée. On peut traiter ainsi le cas de phénomènes simultanés de concentrations de la commande et de discontinuités de la trajectoire. Une troisième contribution concerne le calcul de solutions mesures pour les lois de conservation hyperboliques scalaires dont l'exemple typique est l'équation de Burgers. Cette classe d'EDP non linéaire peut avoir des solutions discontinues difficiles à approximer numériquement avec précision. Sous certaines hypothèses, la solution mesurepeut être identifiée avec la solution classique (faible) à la loi de conservation. Notre approche moment-SOS fournit alors une méthode alternative pour approcher des solutions qui contrairement aux méthodes existantes évite une discrétisation du domaine.The generalized moment problem (GMP) is a linear optimization problem over spaces of measures. It allows to model many challenging mathematical problems. While in general it is impossible to solve the GMP, in the case where all data are polynomial and semialgebraic sets, one can define a hierarchy of semidefinite relaxations - the moment-sums-of-squares (moment-SOS) hierachy - which in principle allows to approximate the optimal value of the GMP to arbitrary precision. The work presented in this thesis addresses two facets concerning the GMP and the moment-SOS hierarchy: One facet is concerned with the scalability of relaxations for the GMP. The degree of the SOS weights in the moment-SOS hierarchy grows when augmenting the relaxation order. When the number of variables is not small, this leads quickly to semidefinite programs (SDPs) that are out of range for state of the art SDP solvers, unless one can use symmetries or some structured sparsity which is typically present in large scale applications. We provide a new certificate of positivity which (i) is able to exploit the structured sparsity and (ii) only involves SOS polynomials of fixed degree. From this, one can define a new hierarchy of SDP relaxations for the GMP which can take into account sparsity and at the same time prevents from explosion of the size of SDP variables related to the increasing degree of the SOS weights in the standard hierarchy. The second facet focusses on (i) modelling challenging problems as a particular instance of the GMP and (ii) solving these problems by applying the moment-SOS hierarchy. In particular we propose deterministic approximations of chance constraints a difficult problem as the associated set of feasible solutions is typically non-convex and sometimes not even connected. In our approach we replace this set by a (smaller) sub-level-set of a polynomial whose vector of coefficients is a by-product of the moment-SOS hierarchy when modeling the problem as an instance of the GMP. The quality of this inner approximation improves when increasing the degree of the SDP relaxation and asymptotic convergence is guaranteed. The procedure is illustrated by approximating the feasible set of an instance of the chance-constrained AC Optimal Power Flow problem (a nonlinear problem in the management of energy networks) which nowadays becomes more and more important as we rely increasingly on uncertain energy sources such as wind and solar power. Furthermore, we propose an extension of this framework to the case where the underlying distribution itself is uncertain and provide guarantees of convergence. Another application of the moment-SOS methodology discussed in this thesis consider measure valued solutions to optimal control problems. We show how this procedure can capture the limit behavior of an optimizing sequence of control and its corresponding sequence of trajectories. In particular we address the case of concentrations of control and discontinuities of the trajectory may occur simultaneously. In a final contribution, we compute measure valued solutions to scalar hyperbolic conservation laws, such as Burgers equation. It is known that this class of nonlinear partial differential equations has potentially discontinuous solutions which are difficult to approximate numerically with accuracy. Under some conditions the measure valued solution can be identified with the classical (weak) solution to the conservation law. In this case our moment-SOS approach provides an alternative numerical scheme to compute solutions which in contrast to existing methods, does not rely on discretization of the domain