arXiv subjects

Maria Infusino

Publications and source records attributed to Maria Infusino.

13 recordsLinked to original sources

arXiv2023

Infinite-dimensional moment-SOS hierarchy for nonlinear partial differential equations

We formulate a class of nonlinear {evolution} partial differential equations (PDEs) as linear optimization problems on moments of positive measures supported on infinite-dimensional vector spaces. Using sums of squares (SOS) representations of polynomials in these spaces, we can prove convergence of a hierarchy of finite-dimensional semidefinite relaxations solving approximately these infinite-dimensional optimization problems. As an illustration, we report on numerical experiments for solving the heat equation subject to a nonlinear perturbation.

math.OC↗

arXiv2023

The realizability problem as a special case of the infinite-dimensional truncated moment problem

The realizability problem is a well-known problem in the analysis of complex systems, which can be modeled as an infinite-dimensional moment problem. More precisely, as a truncated $K-$moment problem where $K$ is the space of all possible configurations of the components of the considered system. The power of this reformulation has been already exploited in \cite{KuLeSp11}, where necessary and sufficient conditions of Haviland type have been obtained for several instances of the realizability problem. In this article we exploit this same reformulation to apply to the realizability problem the recent advances obtained in \cite{CGIK2022} for the truncated moment problem for linear functionals on general unital commutative algebras. This provides alternative proofs and sometimes extensions of several results in \cite{KuLeSp11}, allowing to finally embed them in the unified framework for the infinite-dimensional truncated moment problem presented in \cite{CGIK2022}.

math.PR↗

arXiv2023

Moment problem for algebras generated by a nuclear space

We establish a criterion for the existence of a representing Radon measure for linear functionals defined on a unital commutative real algebra $A$, which we assume to be generated by a vector space $V$ endowed with a Hilbertian seminorm $q$. Such a general criterion provides representing measures with support contained in the space of characters of $A$ whose restrictions to $V$ are $q-$continuous. This allows us in turn to prove existence results for the case when $V$ is endowed with a nuclear topology. In particular, we apply our findings to the symmetric tensor algebra of a nuclear space.

math.FA↗

arXiv2022

An intrinsic characterization of moment functionals in the compact case

We consider the class of all linear functionals $L$ on a unital commutative real algebra $A$ that can be represented as an integral w.r.t. to a Radon measure with compact support in the character space of $A$. Exploiting a recent generalization of the classical Nussbaum theorem, we establish a new characterization of this class of moment functionals solely in terms of a growth condition intrinsic to the given linear functional. To the best of our knowledge, our result is the first to exactly identify the compact support of the representing Radon measure. We also describe the compact support in terms of the largest Archimedean quadratic module on which $L$ is non-negative and in terms of the smallest submultiplicative seminorm w.r.t. which $L$ is continuous. Moreover, we derive a formula for computing the measure of each singleton in the compact support, which in turn gives a necessary and sufficient condition for the support to be a finite set. Finally, some aspects related to our growth condition for topological algebras are also investigated.

math.FA↗

arXiv2020

The Truncated Moment Problem for Unital Commutative R-Algebras

We investigate when a linear functional $L$ defined on a linear subspace $B$ of a unital commutative real algebra $A$ admits an integral representation w.r.t. a positive Radon measure supported on a closed subset $K$ of the character space of $A$. We provide a criterion for the existence of such a representation for $L$ when $A$ is equipped with a submultiplicative seminorm. We then build on this result to prove our main theorem for $A$ not necessarily equipped with a topology. This allows us to extend well-known classical results on truncated moment problems.

math.FA↗

arXiv2019

Projective limits techniques for the infinite dimensional moment problem

We deal with the following general version of the classical moment problem: when can a linear functional on a unital commutative real algebra $A$ be represented as an integral with respect to a Radon measure on the character space $X(A)$ of $A$ equipped with the Borel $\sigma-$algebra generated by the weak topology? We approach this problem by constructing $X(A)$ as a projective limit of the character spaces of all finitely generated unital subalgebras of $A$. Using some fundamental results for measures on projective limits of measurable spaces, we determine a criterion for the existence of an integral representation of a linear functional on $A$ with respect to a measure on the cylinder $\sigma-$algebra on $X(A)$ (resp. a Radon measure on the Borel $\sigma-$algebra on $X(A)$) provided that for any finitely generated unital subalgebra of $A$ the corresponding moment problem is solvable. We also investigate how to localize the support of representing measures for linear functionals on $A$. These results allow us to establish infinite dimensional analogues of the classical Riesz-Haviland and Nussbaum theorems as well as a representation theorem for linear functionals non-negative on a "partially Archimedean" quadratic module of $A$. Our results in particular apply to the case when $A$ is the algebra of polynomials in infinitely many variables or the symmetric tensor algebra of a real infinite dimensional vector space, providing a unified setting which enables comparisons between some recent results for these instances of the moment problem.

math.FA↗

arXiv2018

The full moment problem on subsets of probabilities and point configurations

The aim of this paper is to study the full $K-$moment problem for measures supported on some particular non-linear subsets $K$ of an infinite dimensional vector space. We focus on the case of random measures, that is $K$ is a subset of all non-negative Radon measures on $\mathbb{R}^d$. We consider as $K$ the space of sub-probabilities, probabilities and point configurations on $\mathbb{R}^d$. For each of these spaces we provide at least one representation as a generalized basic closed semi-algebraic set to apply the main result in [J. Funct. Anal., 267 (2014) no.5: 1382--1418]. We demonstrate that this main result can be significantly improved by further considerations based on the particular chosen representation of $K$. In the case when $K$ is a space of point configurations, the correlation functions (also known as factorial moment functions) are easier to handle than the ordinary moment functions. Hence, we additionally express the main results in terms of correlation functions.

math.FA↗

arXiv2016

Infinite dimensional moment problem: open questions and applications

Infinite dimensional moment problems have a long history in diverse applied areas dealing with the analysis of complex systems but progress is hindered by the lack of a general understanding of the mathematical structure behind them. Therefore, such problems have recently got great attention in real algebraic geometry also because of their deep connection to the finite dimensional case. In particular, our most recent collaboration with Murray Marshall and Mehdi Ghasemi about the infinite dimensional moment problem on symmetric algebras of locally convex spaces revealed intriguing questions and relations between real algebraic geometry, functional and harmonic analysis. Motivated by this promising interaction, the principal goal of this paper is to identify the main current challenges in the theory of the infinite dimensional moment problem and to highlight their impact in applied areas. The last advances achieved in this emerging field and briefly reviewed throughout this paper led us to several open questions which we outline here.

math.FA↗

arXiv2016

On the determinacy of the moment problem for symmetric algebras of a locally convex space

This note aims to show a uniqueness property for the solution (whenever exists) to the moment problem for the symmetric algebra $S(V)$ of a locally convex space $(V, \tau)$. Let $\mu$ be a measure representing a linear functional $L: S(V)\to\mathbb{R}$. We deduce a sufficient determinacy condition on $L$ provided that the support of $\mu$ is contained in the union of the topological duals of $V$ w.r.t. to countably many of the seminorms in the family inducing $\tau$. We compare this result with some already known in literature for such a general form of the moment problem and further discuss how some prior knowledge on the support of the representing measure influences its determinacy.

math.FA↗

arXiv2015

Translation invariant realizability problem on the $d-$dimensional lattice: an explicit construction

We consider a particular instance of the truncated realizability problem on the $d-$dimensional lattice. Namely, given two functions $\rho_1({\bf i})$ and $\rho_2({\bf i},{\bf j})$ non-negative and symmetric on $\mathbb{Z}^d$, we ask whether they are the first two correlation functions of a translation invariant point process. We provide an explicit construction of such a realizing process for any $d\geq 2$ when the radial distribution has a specific form. We also derive from this construction a lower bound for the maximal realizable density and compare it with the already known lower bounds.

math.PR↗

arXiv2014

Uniform distribution of sequences of points and partitions

The interest for uniformly distributed (u.d.) sequences of points, in particular for sequences with small discrepancy, arises from various applications. For instance, low-discrepancy sequences, which are sequences with a discrepancy of order $((\log N)^d)/N$ ($d$ is the dimension of the space where the sequence lies), are a fundamental tool for getting faster rate of convergence in approximation involving Quasi-Monte Carlo methods. The objectives of this work can be summarized as follows (1)The research of explicit techniques for introducing new classes of u.d. sequences of points and of partitions on $[0,1]$ and also on fractal sets (2) A quantitative analysis of the distribution behaviour of a class of generalized Kakutani's sequences on $[0,1]$ through the study of their discrepancy. Concerning (1), we propose an algorithm to construct u.d. sequences of partitions and of points on fractals generated by an Iterated Function System (IFS) of similarities having the same ratio and satisfying a natural separation condition of their components called Open Set Condition (OSC). We also provide an estimate for the elementary discrepancy of these sequences. We generalize these results to a wider class of fractals by using a recent generalization of Kakutani's splitting procedure on $[0,1]$, namely the technique of $\rho-$refinements. First, we focus on (2) and get precise bounds for the discrepancy of a large class of generalized Kakutani's sequences, exploiting a correspondence between the tree representation associated to successive $\rho-$refinements and the tree generated by Khodak's coding algorithm. Then we adapt the $\rho-$refinements method to the new class of fractals and prove bounds for the elementary discrepancy of the sequences of partitions constructed with such a procedure.

math.PR↗

arXiv2014

Quasi-analyticity and determinacy of the full moment problem from finite to infinite dimensions

This paper is aimed to show the essential role played by the theory of quasi-analytic functions in the study of the determinacy of the moment problem on finite and infinite-dimensional spaces. In particular, the quasi-analytic criterion of self-adjointness of operators and their commutativity are crucial to establish whether or not a measure is uniquely determined by its moments. Our main goal is to point out that this is a common feature of the determinacy question in both the finite and the infinite-dimensional moment problem, by reviewing some of the most known determinacy results from this perspective. We also collect some properties of independent interest concerning the characterization of quasi-analytic classes associated to log-convex sequences.

math.FA↗

arXiv2013

The full infinite dimensional moment problem on semi-algebraic sets of generalized functions

We consider a generic basic semi-algebraic subset $\mathcal{S}$ of the space of generalized functions, that is a set given by (not necessarily countably many) polynomial constraints. We derive necessary and sufficient conditions for an infinite sequence of generalized functions to be realizable on $\mathcal{S}$, namely to be the moment sequence of a finite measure concentrated on $\mathcal{S}$. Our approach combines the classical results about the moment problem on nuclear spaces with the techniques recently developed to treat the moment problem on basic semi-algebraic sets of $\mathbb{R}^d$. In this way, we determine realizability conditions that can be more easily verified than the well-known Haviland type conditions. Our result completely characterizes the support of the realizing measure in terms of its moments. As concrete examples of semi-algebraic sets of generalized functions, we consider the set of all Radon measures and the set of all the measures having bounded Radon-Nikodym density w.r.t. the Lebesgue measure.

math.PR↗