Infinite Dimensional Domains Research Articles

A unifying modeling abstraction for infinite-dimensional optimization

Infinite-dimensional optimization (InfiniteOpt) problems involve modeling components (variables, objectives, and constraints) that are functions defined over infinite-dimensional domains. Examples include continuous-time dynamic optimization (time is an infinite domain and components are functions of time), PDE optimization problems (space and time are infinite domains and components are functions of space-time), as well as stochastic and semi-infinite optimization (random space is an infinite domain and components are a function of such random space). InfiniteOpt problems also arise from combinations of these problem classes (e.g., stochastic PDE optimization). Given the infinite-dimensional nature of objectives and constraints, one often needs to define appropriate quantities (measures) to properly pose the problem. Moreover, InfiniteOpt problems often need to be transformed into a finite dimensional representation so that they can be solved numerically. In this work, we present a unifying abstraction that facilitates the modeling, analysis, and solution of InfiniteOpt problems. The proposed abstraction enables a general treatment of infinite-dimensional domains and provides a measure-centric paradigm to handle associated variables, objectives, and constraints. This abstraction allows us to transfer techniques across disciplines and with this identify new, interesting, and useful modeling paradigms (e.g., event constraints and risk measures defined over time domains). Our abstraction serves as the backbone of an intuitive Julia-based modeling package that we call InfiniteOpt.jl. We demonstrate the developments using diverse case studies arising in engineering.

Generation of Conditional Densities in Nonlinear Filtering for Infinite-Dimensional Systems

Consider a signal process that takes values in an infinite-dimensional space and assume that this process is observed through a nonlinear diffusion process. In the nonlinear filtering theory literature, recursive expressions for such problems in the form of conditional expectations and, consequently, measure-valued stochastic partial differential equations on the underlying infinite-dimensional space are obtained. In this work, state estimation problems for such stochastic systems are considered, and nonlinear filtering equations in the form of conditional densities are obtained. This is achieved by first deriving the filtering equations in the form of conditional expectations for a Banach space-valued state process whose dynamics depends on a finite-dimensional Brownian motion. Next, a recursion for the conditional density, which we assume to exist with respect to a $\sigma$ -finite measure, is obtained. For such an analysis, we obtain the adjoint operator for the generator of the state process, which requires to use the theory of integration by parts in the infinite-dimensional domains. Finally, we discuss some potential application of the derived filtering equations in a portfolio optimization problem for portfolios, which consist of bonds whose price curve is interpreted as a stochastic partial differential equation taking values in a Banach space. Applications in mean field games are also demonstrated.

On the size of special families of linear operators

We continue the study, started in [12], of the search for algebraic structures one can find within the sets of injective linear functions. We shall focus on the cases when the operators are considered both on finite dimensional and infinite dimensional domains. We also study the set of continuous surjective linear operators.

Maximal Sobolev regularity in Neumann problems for gradient systems in infinite dimensional domains

Nous considerons une equation de Kolmogorov elliptique $\lambda u-Ku=f$ dans un sous-ensemble convexe $\mathcal{C}$ d’un espace de Hilbert separable $X$. L’operateur de Kolmogorov $K$ est une realisation de $u\mapsto\frac{1}{2}\operatorname{Tr} [D^{2}u(x)]+\langle Ax-DU(x),Du(x)\rangle$, ou $A$ est un operateur auto-adjoint dans $X$ et $U:X\mapsto\mathbb{R}\cup\{+\infty\}$ est une fonction convexe. Nous prouvons que pour $\lambda>0$ et $f\in L^{2}(\mathcal{C},\nu)$ la solution faible $u$ appartient a l’espace de Sobolev $W^{2,2}(\mathcal{C},\nu)$, ou $\nu$ est la mesure log-concave associee au systeme. Nous prouvons aussi des estimations maximales sur le gradient de $u$ qui permettent de montrer que $u$ satisfait des conditions au bord de Neumann au sens des traces a la frontiere de $\mathcal{C}$. Les resultats generaux sont appliques aux equations de reaction–diffusion de Kolmogorov et a l’equation de Cahn–Hilliard stochastique dans des ensembles convexes d’espaces de Hilbert appropries.

Sobolev functions on infinite-dimensional domains

We study extensions of Sobolev and BV functions on infinite-dimensional domains. Along with some positive results we present a negative solution of the long-standing problem of existence of Sobolev extensions of functions in Gaussian Sobolev spaces from a convex domain to the whole space.

Classes of functions of bounded variation on infinite-dimensional domains

Classes of functions of bounded variation on infinite-dimensional domains

Starlike and convex rational mappings on infinite dimensional domains

We give starlike criteria for a class of rational mappings on the open unit ball of a complex Banach space. We also give a sufficient condition for these mappings to be convex when they are defined in Hilbert spaces. These criteria facilitate the construction of concrete examples of starlike and convex mappings on infinite dimensional domains (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Approximate Maximum a Posteriori with Gaussian Process Priors

A maximum a posteriori method has been developed for Gaussian priors over infinite-dimensional function spaces. In particular, variational equations based on a generalisation of the representer theorem and an equivalent optimisation problem are presented. This amounts to a generalisation of the ordinary Bayesian maximum a posteriori approach which is nontrivial as infinite-dimensional domains do not admit any probability densities. Instead of the gradient of the density, the logarithmic gradient of the probability distribution is used. Galerkin methods are proposed for the approximate solution of the variational equations. In summary, a framework and some foundations are provided which are required for the application of numerical approximation to an important class of machine learning problems.

Invertibility in Fréchet Algebras

Using tensor products and a generalised spectrum, we extend to infinite dimensional domains classical results on the ideals generated by a finite or countable set of elements in a Frechet algebra.

Nearly trivial homotopy classes between finite complexes

We construct examples of essential maps of finite complexes f:X→Y which are trivial of order ⩾n. This latter condition implies that for any space K with cone length ⩽n, the induced map f∗=0:[K,X]→[K,Y]. The main result establishes a connection between the skeleta of the infinite dimensional domains of essential phantom maps and the finite dimensional domains of maps which are trivial of order ⩾n. In particular, there are essential maps f:Σ2i(CPt/S2)→M(Z/ps,2l+3) which are trivial of order ⩾n.

Unitary Highest Weight Representations in Hilbert Spaces of Holomorphic Functions on Infinite Dimensional Domains

Automorphism groups of symmetric domains in Hilbert spaces form a natural class of infinite dimensional Lie algebras and corresponding Banach Lie groups. We give a classification of the algebraic category of unitary highest weight modules for such Lie algebras and show that infinite dimensional versions of the Lie algebras so(2,n) have no unitary highest weight representations and thus do not meet the physical requirement of having positive energy. Highest weight modules correspond to unitary representations of global Banach Lie groups realized in Hilbert spaces of vector valued holomorphic functions on the relevant domains in Hilbert spaces. The construction of such holomorphic representations of certain Banach Lie groups, followed by the application of the general framework of Harish-Chandra type groups in an appropriate Banach setting, leads to the integration of the Lie algebra representation to a group representation. The extension of this theory to infinite dimensional settings is explored.

Compact Sets of Holomorphic Mappings

Compactness of locally bounded sets of holomorphic functions with infinite dimensional domains is connected, using Heinrich's density condition, to the Schwartz and semi-Montel properties on the domain. The metrizability of bounded subsets for various spaces of holomorphic functions is investigated.

HOLOMORPHIC MAPS ON DOMAINS WITH EMPTY INTERIOR

Abstract We bring to light further infinite dimensional domains with empty interior on which holomorphic maps can be constructed.

On composition operators on Banach spaces of holomorphic functions

We discuss some spectral properties of composition operators in Banach spaces of holomorphic functions over infinite-dimensional domains. Our results generalize known properties of such operators in the finite-dimensional case.