Title: Ensemble Kalman filter on high and infinite dimensional spaces Author: Mgr. Ivan Kasanický Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Daniel Hlubinka, Ph.D., Department of Probability and Mathematical Statistics Consultant: prof. RNDr. Jan Mandel, CSc., Department of Mathematical and Statistical Sciences, University of Colorado Denver Abstract: The ensemble Kalman filter (EnKF) is a recursive filter, which is used in a data assimilation to produce sequential estimates of states of a hidden dynamical system. The evolution of the system is usually governed by a set of di↵erential equations, so one concrete state of the system is, in fact, an element of an infinite dimensional space. In the presented thesis we show that the EnKF is well defined on a infinite dimensional separable Hilbert space if a data noise is a weak random variable with a covariance bounded from below. We also show that this condition is su cient for the 3DVAR and the Bayesian filtering to be well posed. Additionally, we extend the already known fact that the EnKF converges to the Kalman filter in a finite dimension, and prove that a similar statement holds even in a infinite dimension. The EnKF su↵ers from a low rank approximation of a state covariance, so a covariance localization is required in...

Yes / Random sets are used to get a continuous partition of the cardinality of the union of many overlapping sets. The formalism uses Möbius transforms and adapts Shapley's methodology in cooperative game theory, into the context of set theory. These ideas are subsequently generalized into the context of finite-dimensional Hilbert spaces. Using random projectors into the subspaces spanned by states from a total set, we construct an infinite number of continuous resolutions of the identity, that involve Hermitian positive semi-definite operators. The simplest one is the diagonal continuous resolution of the identity, and it is used to expand an arbitrary vector in terms of a continuum of components. It is also used to define the function on the 'probabilistic quadrant' , which is analogous to the Wigner function for the harmonic oscillator, on the phase-space plane. Systems with finite-dimensional Hilbert space (which are naturally described with discrete variables) are described here with continuous probabilistic variables. / The full-text of this article will be released for public view at the end of the publisher embargo on 15 Oct 2020. / Research Development Fund Publication Prize Award winner, October 2019.

Este trabalho é dedicado ao estudo de propriedades teóricas dos operadores integrais positivos em \'L POT. 2\' (X; u), quando X é um espaço topológico localmente compacto ou primeiro enumerável e u é uma medida estritamente positiva. Damos ênfase à análise de propriedades espectrais relacionadas com extensões do Teorema de Mercer e ao estudo dos espaços de Hilbert de reprodução relacionados. Como aplicação, estudamos o decaimento dos autovalores destes operadores, em um contexto especial. Finalizamos o trabalho com a análise de propriedades de suavidade das funções do espaço de Hilbert de reprodução, quando X é um subconjunto do espaço euclidiano usual e u é a medida de Lebesgue usual de X / In this work we study theoretical properties of positive integral operators on \'L POT. 2\'(X; u), in the case when X is a topological space, either locally compact or first countable, and u is a strictly positive measure. The analysis is directed to spectral properties of the operator which are related to some extensions of Mercer\'s Theorem and to the study of the reproducing kernel Hilbert spaces involved. As applications, we deduce decay rates for the eigenvalues of the operators in a special but relevant case. We also consider smoothness properties for functions in the reproducing kernel Hilbert spaces when X is a subset of the Euclidean space and u is the Lebesgue measure of the space

Wir modellieren einen Finanzmarkt mit unendlich vielen Wertpapieren als stochastischen Prozeß X in stetiger Zeit mit Werten in einem separablen Hilbertraum H. In diesem Rahmen zeigen wir die Äquivalenz von Vollständigkeit des Marktes und der Eindeutigkeit des äquivalenten Martingalmaßes unter der Bedingung, daß X stetige Pfade besitzt. Weiter zeigen wir, daß (unter gewissen technischen Bedingungen) für X die Abwesenheit von asymptotischer Arbitrage der ersten/zweiten Art (im Sinne von Kabanov/Kramkov) äquivalent zur Absolutstetigkeit des Referenzmaßes zu einem eindeutigen, lokal äquivalenten Martingalmaß ist. Hat X stetige Pfade, so ist die Abwesenheit von allgemeiner asymptotischer Arbitrage äquivalent zur Existenz eines äquivalenten lokalen Martingalmaßes. Außerdem geben wir ein Kriterium für die Existenz einer optionalen Zerlegung von X an. Dies wenden wir auf das Problem der Risikominimierung bei vorgegebener Investitionsobergrenze (effizientes Hedgen (Föllmer/Leukert)) an, um dieses im unendlichdimensionalen Kontext zu behandeln. Außerdem stellen wir eine unendlichdimensionale Erweiterung des Heath-Jarrow-Morton-Modells vor und nutzen den Potentialansatz nach Rodgers, um zwei weitere Zinsstrukturmodelle zu konstruieren. Als Beitrag zur allgemeinen stochastischen Analysis in Hilberträumen beweisen wir eine pfadweise Version der Itoformel für stochastische Prozesse mit stetigen Pfaden in einem separablen Hilbertraum. Daraus läßt sich eine pfadweise Version des Satzes über die Vertauschbarkeit von stochastischem und Lebesgue-Integral ableiten. Außerdem zeigen wir eine Version der Clark-Formel für eine Brownsche Bewegung mit Werten in einem Hilbertraum. / We model a financial market with infinitely many assets as a stochastic process X with values in a separable Hilbert space H. In this setting we show the equivalence of market completeness and the uniqueness of the equivalent martingale measure, if X has continuous paths. Another result for our model is, that under some technical conditions, the absence of asymptotic arbitrage of the first/second kind (in the sense of Kabanov/Kramkov) is equivalent to the absolute continuity of the reference measure to a unique, locally equivalent, martingale measure. If X has continuous paths, the absence of general asymptotic arbitrage is equivalent to the existence of an equivalent local martingale measure. Furthermore, we give a sufficient condition for the existence of the optional decomposition of X. We apply this result to the problem of risk minimization with given upper limit for investion (efficient hedging (Föllmer/Leukert)). This allows us to solve this optimization problem in our infinite dimensional context. Another result is an infinite dimensional extension of the Heath-Jarrow-Morton term structure model. Two further term structure models are constructed, using the Markov potential approach developed by Rodgers. As a contribution to the theory of stochastic analysis in Hilbert spaces, we proof a pathwise version of the Ito formula for stochastic processes with continuous paths in a separable Hilbert space. This leads to a pathwise version of the interchangability theorem for stochastic and Lebesgue integrals. We also show a version of the Clark formula for Hilbert space valued Brownian motion.

In this thesis we analyse the high-order in time discontinuous Galerkin nite element method (DGFEM) for second-order in time linear abstract wave equations. Our abstract approximation analysis is a generalisation of the approach introduced by Claes Johnson (in Comput. Methods Appl. Mech. Engrg., 107:117-129, 1993), writing the second order problem as a system of fi rst order problems. We consider abstract spatial (time independent) operators, highorder in time basis functions when discretising in time; we also prove approximation results in case of linear constraints, e.g. non-homogeneous boundary data. We take the two steps approximation approach i.e. using high-order in time DGFEM; the discretisation approach in time introduced by D Schötzau (PhD thesis, Swiss Federal institute of technology, Zürich, 1999) to fi rst obtain the semidiscrete scheme and then conformal spatial discretisation to obtain the fully-discrete formulation. We have shown solvability, unconditional stability and conditional a priori error estimates within our abstract framework for the fully discretized problem. The skew-symmetric spatial forms arising in our abstract framework for the semi- and fully-discrete schemes do not full ll the underlying assumptions in D. Schötzau's work. But the semi-discrete and fully discrete forms satisfy an Inf-sup condition, essential for our proofs; in this sense our approach is also a generalisation of D. Schötzau's work. All estimates are given in a norm in space and time which is weaker than the Hilbert norm belonging to our abstract function spaces, a typical complication in evolution problems. To the best of the author's knowledge, with the approximation approach we used, these stability and a priori error estimates with their abstract structure have not been shown before for the abstract variational formulation used in this thesis. Finally we apply our abstract framework to the acoustic and an elasto-dynamic linear equations with non-homogeneous Dirichlet boundary data.

Neste trabalho analisamos o papel das funções layout de um núcleo positivo definido K sobre um espaço topológico de Hausdor E com relação a duas propriedades específicas: a universalidade de K e a ortogonalidade no espaço de Hilbert de reprodução de K a partir de suportes disjuntos. As funções layout sempre existem mas podem não ser únicas. De uma maneira geral, a função layout e uma aplicação que transfere, convenientemente, informações do espaço E para um espaço com produto interno de dimensão alta, onde métodos lineares podem ser usados. Tanto a universalidade quanto a ortogonalidade pressupõem a continuidade do núcleo. O primeiro conceito exige que para cada compacto não vazio X de E, o conjunto de \"seções\" {K(., y) : y \'PERTENCE\' X} seja total no espaço de todas as funções contínuas com domínio X, munido da topologia da convergência uniforme. Um dos resultados principais do trabalho caracteriza a universalidade de um núcleo K através de uma propriedade de universalidade semelhante da função layout. A ortogonalidade a partir de suportes disjuntos almeja então a ortogonalidade de quaisquer duas funções do espaço de Hilbert de reprodução de K quando seus suportes não se intersectam / We analyze the role of feature maps of a positive denite kernel K acting on a Hausdorff topological space E in two specific properties: the universality of K and the orthogonality in the reproducing kernel Hilbert space of K from disjoint supports. Feature maps always exist but may not be unique. A feature map may be interpreted as a kernel based procedure that maps the data from the original input space E into a potentially higher dimensional \"feature space\" in which linear methods may then be used. Both properties, universality and orthogonality from disjoint supports, make sense under continuity of the kernel. Universality of K is equivalent to the fundamentality of {K(. ; y) : y \'IT BELONGS\' X} in the space of all continuous functions on X, with the topology of uniform convergence, for all nonempty compact subsets X of E. One of the main results in this work is a characterization of the universality of K from a similar concept for the feature map. Orthogonality from disjoint supports seeks the orthogonality of any two functions in the reproducing kernel Hilbert space of K when the functions have disjoint supports

Este trabalho é dedicado ao estudo de propriedades teóricas dos operadores integrais positivos em \'L POT. 2\' (X; u), quando X é um espaço topológico localmente compacto ou primeiro enumerável e u é uma medida estritamente positiva. Damos ênfase à análise de propriedades espectrais relacionadas com extensões do Teorema de Mercer e ao estudo dos espaços de Hilbert de reprodução relacionados. Como aplicação, estudamos o decaimento dos autovalores destes operadores, em um contexto especial. Finalizamos o trabalho com a análise de propriedades de suavidade das funções do espaço de Hilbert de reprodução, quando X é um subconjunto do espaço euclidiano usual e u é a medida de Lebesgue usual de X / In this work we study theoretical properties of positive integral operators on \'L POT. 2\'(X; u), in the case when X is a topological space, either locally compact or first countable, and u is a strictly positive measure. The analysis is directed to spectral properties of the operator which are related to some extensions of Mercer\'s Theorem and to the study of the reproducing kernel Hilbert spaces involved. As applications, we deduce decay rates for the eigenvalues of the operators in a special but relevant case. We also consider smoothness properties for functions in the reproducing kernel Hilbert spaces when X is a subset of the Euclidean space and u is the Lebesgue measure of the space

Neste trabalho analisamos o papel das funções layout de um núcleo positivo definido K sobre um espaço topológico de Hausdor E com relação a duas propriedades específicas: a universalidade de K e a ortogonalidade no espaço de Hilbert de reprodução de K a partir de suportes disjuntos. As funções layout sempre existem mas podem não ser únicas. De uma maneira geral, a função layout e uma aplicação que transfere, convenientemente, informações do espaço E para um espaço com produto interno de dimensão alta, onde métodos lineares podem ser usados. Tanto a universalidade quanto a ortogonalidade pressupõem a continuidade do núcleo. O primeiro conceito exige que para cada compacto não vazio X de E, o conjunto de \"seções\" {K(., y) : y \'PERTENCE\' X} seja total no espaço de todas as funções contínuas com domínio X, munido da topologia da convergência uniforme. Um dos resultados principais do trabalho caracteriza a universalidade de um núcleo K através de uma propriedade de universalidade semelhante da função layout. A ortogonalidade a partir de suportes disjuntos almeja então a ortogonalidade de quaisquer duas funções do espaço de Hilbert de reprodução de K quando seus suportes não se intersectam / We analyze the role of feature maps of a positive denite kernel K acting on a Hausdorff topological space E in two specific properties: the universality of K and the orthogonality in the reproducing kernel Hilbert space of K from disjoint supports. Feature maps always exist but may not be unique. A feature map may be interpreted as a kernel based procedure that maps the data from the original input space E into a potentially higher dimensional \"feature space\" in which linear methods may then be used. Both properties, universality and orthogonality from disjoint supports, make sense under continuity of the kernel. Universality of K is equivalent to the fundamentality of {K(. ; y) : y \'IT BELONGS\' X} in the space of all continuous functions on X, with the topology of uniform convergence, for all nonempty compact subsets X of E. One of the main results in this work is a characterization of the universality of K from a similar concept for the feature map. Orthogonality from disjoint supports seeks the orthogonality of any two functions in the reproducing kernel Hilbert space of K when the functions have disjoint supports

Let H be a separable Hilbert space over the complex field. The class S := {N|M : N is normal on H and M is an invariant subspace for Ng of subnormal operators. This notion was introduced by Halmos. The minimal normal extension Ň of a subnormal operator S was introduced by σ (S) and then Bram proved that Halmos. Halmos proved that σ(Ň) (S) is obtained by filling certain number of holes in the spectrum (Ň) of the minimal normal extension Ň of a subnormal operator S. Let σ (S) := σ (Ň) be the spectrum of the minimal normal extension Ň of S; which is called the normal spectrum of a subnormal operator S: This notion is due to Abrahamse and Douglas. We give several well-known characterization of subnormality. Let C* (S1) and C* (S2) be the C*- algebras generated by S1 and S2 respectively, where S1 and S2 are bounded operators on H: Next we give a characterization for subnormality which is purely C - algebraic. We also establish an intrinsic characterization of the normal spectrum for a subnormal operator, which enables us to answer the fol-lowing two questions. Let II be a *- representation from C* (S1) onto C* (S2) such that II(S1) = S2. If S1 is subnormal, then does it follow that S2 is subnormal? What is the relation between σ (S1) and σ (S2)? The first question was asked by Bram and second was asked by Abrahamse and Douglas. Answers to these questions were given by Bunce and Deddens.

Let H be a separable Hilbert space over the complex field. The class S := {N|M : N is normal on H and M is an invariant subspace for Ng of subnormal operators. This notion was introduced by Halmos. The minimal normal extension Ň of a subnormal operator S was introduced by σ (S) and then Bram proved that Halmos. Halmos proved that σ(Ň) (S) is obtained by filling certain number of holes in the spectrum (Ň) of the minimal normal extension Ň of a subnormal operator S. Let σ (S) := σ (Ň) be the spectrum of the minimal normal extension Ň of S; which is called the normal spectrum of a subnormal operator S: This notion is due to Abrahamse and Douglas. We give several well-known characterization of subnormality. Let C* (S1) and C* (S2) be the C*- algebras generated by S1 and S2 respectively, where S1 and S2 are bounded operators on H: Next we give a characterization for subnormality which is purely C - algebraic. We also establish an intrinsic characterization of the normal spectrum for a subnormal operator, which enables us to answer the fol-lowing two questions. Let II be a *- representation from C* (S1) onto C* (S2) such that II(S1) = S2. If S1 is subnormal, then does it follow that S2 is subnormal? What is the relation between σ (S1) and σ (S2)? The first question was asked by Bram and second was asked by Abrahamse and Douglas. Answers to these questions were given by Bunce and Deddens.