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This thesis poses and solves a distribution free learning problem with consensus that arises in the study of estimation and control strategies for distributed sensor networks. Each node i for i = 1, . . . , n of the sensor network collects independent and identically distributed local measurements {z i} := {z i j}j∈N := {(x i j , yi j )}j∈N ⊆ X × Y := Z that are generated by the probability measure ρ i on Z. Each node i for i = 1, . . . , n of the network constructs a sequence of estimates {f i k }k∈N from its local measurements {z i} and from information functionals whose values are exchanged with other nodes as specified by the communication graph G for the network. The optimal estimate of the distribution free learning problem with consensus is cast as a saddle point problem which characterizes the consensus-constrained optimal estimate. This thesis introduces a two stage learning dynamic wherein local estimation is carried out via local least square approximations based on wavelet constructions and information exchange is associated with the Lagrange multipliers of the saddle point problem. Rates of convergence for the two stage learning dynamic are derived based on certain recent probabilistic bounds derived for wavelet approximation of regressor functions. / Master of Science

We consider the set of all spacelike embeddings of the circle S1 into a spacetime R1 × S1 with a metric globally conformal to the Minkowski metric. We identify this set and the group of conformal isometries of this spacetime as quotients of semidirect products involving diffeomorphism groups and give a transitive action of the conformal group on the set of spacelike embeddings. We provide results showing that the group of conformal isometries is a topological group and that its action on the set of spacelike embeddings is continuous. Finally, we point out some directions for future research.

We consider infinite-dimensional diffusions where the interaction between the coordinates has a finite extent both in space and time. In particular, it is not supposed to be smooth or Markov. The initial state of the system is Gibbs, given by a strong summable interaction. If the strongness of this initial interaction is lower than a suitable level, and if the dynamical interaction is bounded from above in a right way, we prove that the law of the diffusion at any time t is a Gibbs measure with absolutely summable interaction. The main tool is a cluster expansion in space uniformly in time of the Girsanov factor coming from the dynamics and exponential ergodicity of the free dynamics to an equilibrium product measure.

Following the financial crisis of 2008, the need for more robust techniques to quantify the capital charge for risk management has become a pressing problem. Under Basel II/III, banks are allowed to calculate the capital charge using internally developed models subject to regulatory approval. An interesting problem for the regulator is to compare the resulting figures against the required capital under worst case scenarios. The existing literature on the latter problem, which is based on the marginal problem, assumes that no a-priori information is known about the dependencies of contributing risks. These problems are linear optimisation problems over a constrained set of probability measures, discretisation of which leads to large scale LPs. But this approach is very conservative and cannot be implemented robustly in practice, due to the scarcity of historical data. In our approach, we take a less conservative strategy by incorporating dependence information contained in the data in a form that still leads to LPs, an important feature of such problems due to their high dimensionality. Conceptually, our model is the discretisation of an infinite dimensional linear optimisation problem over a set of probability measures. For some specific cases we can prove strong duality, opening up the approach of discretising the dual instead of the primal. This approach is preferable, as it yields better numerical results. In this work we also apply our model to model-free path-dependent option pricing. Use of delayed column generation techniques allows us to solve problems several orders of magnitude larger than via the standard simplex algorithm. For high-dimensional LPs we also implement Nesterov's smoothing technique to solve the problems.

Tubular reactors cover a large class of processes in chemical and biochemical engineering. They are typically reactors in which the medium is not homogeneous (like fixed-bed reactors, packed-bed reactors, fluidized-bed reactors,...) and possibly involve diferent phases (liquid/solid/gas). The dynamics of nonisothermal axial dispersion or plug flow tubular reactors are described by semilinear partial differential equations (PDE's) derived from mass and energy balances. The main source of nonlinearities in such dynamics is concentrated in the kinetics terms of the model equations. Like tubular reactors many physical phenomena are modelled by partial differential equations (PDE's). Such systems are called distributed parameter systems. Control problems of these systems can be formulated in state-space form in a way analogous to those of lumped parameter systems (those described by ordinary differential equations) if one introduces a suitable infinite-dimensional state-space and suitable operators instead of the usual matrices. This thesis deals with the synthesis of optimal control laws with a view to regulate the temperature and the reactant concentration of a nonisothermal plug flow reactor model. Several tools of linear and semilinear infinite-dimensional system theory are extended and/or developed, and applied to this model. On the one hand, the concept of asymptotic stability is studied for a class of infinite-dimensional semilinear Banach state- space systems. Asymptotic stability criteria are established, which are based on the concept of strictly m-dissipative operator. This theory is applied to a nonisothermal plug flow reactor. On the other hand, the concept of optimal Linear-Quadratic (LQ) feedback is studied for class of infinite-dimensional linear systems. This theory is applied to a linearized plug flow reactor model in order to design an LQ optimal feedback controller. Then the resulting nonlinear closed-loop system performances are analyzed. Finally this control design strategy is extended to a large class of first-order hyperbolic PDE's systems.

We prove in this paper an existence result for infinite-dimensional stationary interactive Brownian diffusions. The interaction is supposed to be small in the norm ||.||∞ but otherwise is very general, being possibly non-regular and non-Markovian. Our method consists in using the characterization of such diffusions as space-time Gibbs fields so that we construct them by space-time cluster expansions in the small coupling parameter.

We establish in this paper the existence of weak solutions of infinite-dimensional shift invariant stochastic differential equations driven by a Brownian term. The drift function is very general, in the sense that it is supposed to be neither small or continuous, nor Markov. On the initial law we only assume that it admits a finite specific entropy. Our result strongly improves the previous ones obtained for free dynamics with a small perturbative drift. The originality of our method leads in the use of the specific entropy as a tightness tool and on a description of such stochastic differential equation as solution of a variational problem on the path space.

[pt] O objetivo deste trabalho é o estudo das condições para a estabilidade de sistemas lineares discretos de dimensão infinita invariantes no tempo, evoluindo em um espaço de Hilbert. Apresentaremos uma vasta coleção de resultados sobre estabilidade assintótica uniforme, incluindo uma condição espectral equivalente. Em relação à estabilidade assintótica fraca, analisaremos tanto a dificuldade de se estabelecer uma condição necessária e suficiente sobre o espectro do operador, como também sua relação com similaridade a contração. Por último, apresentaremos alguns resultados disponíveis sobre estabilidade assintótica forte para algumas classes específicas de operadores. / [en] The purpose of this work is to analyse stability conditions for infinity-dimensional linear discrete systems operating in a Hilbert space. Whe shall present a wide collections of results on uniform asymptotic stability, incluiding an equivalent spectral condition. Concerning the weak asymptotic stability, we shall analyse the dificulty associated to the problem of attempting to establish a necessary and sufficient condition involving the spectral of the system operator. The relation between weak asymptotic stability and similarity to a contraction will be analysed as well. Finally, we shall present some of the available results concerning strong asymptotic stability for particular classes of operators.

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