scholarly journals Connecting a direct and a Galerkin approach to slow manifolds in infinite dimensions

2021 ◽  
Vol 8 (21) ◽  
pp. 252-266
Author(s):  
Maximilian Engel ◽  
Felix Hummel ◽  
Christian Kuehn
Keyword(s):  
Invariant Manifolds ◽  
Evolution Equations ◽  
Galerkin Approximation ◽  
Infinite Dimensions ◽  
Galerkin Approach ◽  
Infinite Dimensional ◽  
Slow Manifolds ◽  
Finite Dimensional ◽  
Abstract Evolution Equation ◽  
Scales Of Banach Spaces

In this paper, we study slow manifolds for infinite-dimensional evolution equations. We compare two approaches: an abstract evolution equation framework and a finite-dimensional spectral Galerkin approximation. We prove that the slow manifolds constructed within each approach are asymptotically close under suitable conditions. The proof is based upon Lyapunov-Perron methods and a comparison of the local graphs for the slow manifolds in scales of Banach spaces. In summary, our main result allows us to change between different characterizations of slow invariant manifolds, depending upon the technical challenges posed by particular fast-slow systems.

The Bargmann Constraint and Integrable System for a Second-Order Spectral Problem

2015 ◽  
Vol 70 (11) ◽  
pp. 913-917
Author(s):  
Wei Liu ◽  
Yafeng Liu ◽  
Shujuan Yuan
Keyword(s):  
Coordinate System ◽  
Spectral Problem ◽  
Integrable System ◽  
Evolution Equations ◽  
Lagrange Equations ◽  
Lax Pairs ◽  
Infinite Dimensional ◽  
Hamilton System ◽  
Finite Dimensional ◽  
Dimensional Soliton

AbstractIn this article, the Bargmann system related to the spectral problem (∂2+q∂+∂q+r)φ=λφ+λφx is discussed. By the Euler–Lagrange equations and the Legendre transformations, a suitable Jacobi–Ostrogradsky coordinate system is obtained. So the Lax pairs of the aforementioned spectral problem are nonlinearised. A new kind of finite-dimensional Hamilton system is generated. Moreover, the involutive solutions of the evolution equations for the infinite-dimensional soliton system are derived.

scholarly journals Infinite-dimensional Ellentuck spaces and Ramsey-classification theorems

2016 ◽  
Vol 16 (01) ◽  
pp. 1650003 ◽  
Author(s):  
Natasha Dobrinen
Keyword(s):  
Partial Order ◽  
Equivalence Relations ◽  
Infinite Dimensions ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Iterative Construction

We extend the hierarchy of finite-dimensional Ellentuck spaces to infinite dimensions. Using uniform barriers [Formula: see text] on [Formula: see text] as the prototype structures, we construct a class of continuum many topological Ramsey spaces [Formula: see text] which are Ellentuck-like in nature, and form a linearly ordered hierarchy under projections. We prove new Ramsey-classification theorems for equivalence relations on fronts, and hence also on barriers, on the spaces [Formula: see text], extending the Pudlák–Rödl theorem for barriers on the Ellentuck space. The inspiration for these spaces comes from continuing the iterative construction of the forcings [Formula: see text] to the countable transfinite. The [Formula: see text]-closed partial order [Formula: see text] is forcing equivalent to [Formula: see text], which forces a non-p-point ultrafilter [Formula: see text]. This work forms the basis for further work classifying the Rudin–Keisler and Tukey structures for the hierarchy of the generic ultrafilters [Formula: see text].

scholarly journals On the Relation between States and Maps in Infinite Dimensions

2007 ◽  
Vol 14 (04) ◽  
pp. 355-370 ◽  
Author(s):  
Janusz Grabowski ◽  
Marek Kuś ◽  
Giuseppe Marmo
Keyword(s):  
Hilbert Spaces ◽  
Trace Class ◽  
Quantum Systems ◽  
Bounded Operators ◽  
Infinite Dimensions ◽  
Infinite Dimensional ◽  
Continuous Map ◽  
Finite Dimensional ◽  
Trace Class Operators ◽  
Product Realization

Relations between states and maps, which are known for quantum systems in finite-dimensional Hilbert spaces, are formulated rigorously in geometrical terms with no use of coordinate (matrix) interpretation. In a tensor product realization they are represented simply by a permutation of factors. This leads to natural generalizations for infinite-dimensional Hilbert spaces and a simple proof of a generalized Choi Theorem. The natural framework is based on spaces of Hilbert-Schmidt operators [Formula: see text] and the corresponding tensor products [Formula: see text] of Hilbert spaces. It is proved that the corresponding isomorphisms cannot be naturally extended to compact (or bounded) operators, nor reduced to the trace-class operators. On the other hand, it is proven that there is a natural continuous map [Formula: see text] from trace-class operators on [Formula: see text] (with the nuclear norm) into compact operators mapping the space of all bounded operators on [Formula: see text] into trace class operators on [Formula: see text] (with the operator-norm). Also in the infinite-dimensional context, the Schmidt measure of entanglement and multipartite generalizations of state-maps relations are considered in the paper.

Asymptotics of Gaussian integrals in infinite dimensions

2019 ◽  
Vol 22 (01) ◽  
pp. 1950004 ◽  
Author(s):  
Sergio Albeverio ◽  
Victoria Steblovskaya
Keyword(s):  
Hilbert Space ◽  
Phase Function ◽  
Original Form ◽  
Relevant Parameter ◽  
Laplace Method ◽  
Infinite Dimensions ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Dimensional Version ◽  
General Phase

We introduce an infinite-dimensional version of the classical Laplace method, in its original form, relative to a canonical Gaussian measure associated with a Hilbert space, and for a general phase function. Particular attention is given to the case of a phase function with finite-dimensional degeneracy. Explicit results on expansions in the form of power series in the relevant parameter, with estimates on remainders, are provided.

Infinite-Dimensional Polyhedrality

2004 ◽  
Vol 56 (3) ◽  
pp. 472-494 ◽  
Author(s):  
Vladimir P. Fonf ◽  
Libor Veselý
Keyword(s):  
Banach Space ◽  
Convex Subset ◽  
General Definition ◽  
Infinite Dimensions ◽  
Open Problems ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Isomorphic Classification

AbstractThis paper deals with generalizations of the notion of a polytope to infinite dimensions. The most general definition is the following: a bounded closed convex subset of a Banach space is called a polytope if each of its finite-dimensional affine sections is a (standard) polytope.We study the relationships between eight known definitions of infinite-dimensional polyhedrality. We provide a complete isometric classification of them, which gives solutions to several open problems. An almost complete isomorphic classification is given as well (only one implication remains open).

scholarly journals Isometries of infinite dimensional Hilbert geometries

2018 ◽  
Vol 10 (04) ◽  
pp. 941-959
Author(s):  
Bas Lemmens ◽  
Mark Roelands ◽  
Marten Wortel
Keyword(s):  
Infinite Dimensions ◽  
Isometry Groups ◽  
Infinite Dimensional ◽  
Strictly Convex ◽  
Analytic Methods ◽  
Finite Dimensional

In this paper we extend two classical results concerning the isometries of strictly convex Hilbert geometries, and the characterisation of the isometry groups of Hilbert geometries on finite dimensional simplices, to infinite dimensions. The proofs rely on a mix of geometric and functional analytic methods.

Beyond fractional derivatives: local approximation of other convolution integrals

2009 ◽  
Vol 466 (2114) ◽  
pp. 563-581 ◽  
Author(s):  
Satwinder Jit Singh ◽  
Anindya Chatterjee
Keyword(s):  
Numerical Solution ◽  
Fractional Order ◽  
Galerkin Approximation ◽  
Local Approximation ◽  
Dimensional System ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Convolution Integrals ◽  
Damped Systems ◽  
Special Case

Dynamic systems involving convolution integrals with decaying kernels, of which fractionally damped systems form a special case, are non-local in time and hence infinite dimensional. Straightforward numerical solution of such systems up to time  t needs computations owing to the repeated evaluation of integrals over intervals that grow like  t . Finite-dimensional and local approximations are thus desirable. We present here an approximation method which first rewrites the evolution equation as a coupled infinite-dimensional system with no convolution, and then uses Galerkin approximation with finite elements to obtain linear, finite-dimensional, constant coefficient approximations for the convolution. This paper is a broad generalization, based on a new insight, of our prior work with fractional order derivatives ( Singh & Chatterjee 2006 Nonlinear Dyn. 45 , 183–206). In particular, the decaying kernels we can address are now generalized to the Laplace transforms of known functions; of these, the power law kernel of fractional order differentiation is a special case. The approximation can be refined easily. The local nature of the approximation allows numerical solution up to time t with computations. Examples with several different kernels show excellent performance. A key feature of our approach is that the dynamic system in which the convolution integral appears is itself approximated using another system, as distinct from numerically approximating just the solution for the given initial values; this allows non-standard uses of the approximation, e.g. in stability analyses.

scholarly journals An FDA-Based Approach for Clustering Elicited Expert Knowledge

Stats ◽  
2021 ◽  
Vol 4 (1) ◽  
pp. 184-204
Author(s):  
Carlos Barrera-Causil ◽  
Juan Carlos Correa ◽  
Andrew Zamecnik ◽  
Francisco Torres-Avilés ◽  
Fernando Marmolejo-Ramos
Keyword(s):  
Functional Data ◽  
Expert Knowledge ◽  
Probability Distributions ◽  
Knowledge Elicitation ◽  
Parallel Analysis ◽  
Data Sets ◽  
Dimensional Problem ◽  
Prior Distributions ◽  
Infinite Dimensional ◽  
Finite Dimensional

Expert knowledge elicitation (EKE) aims at obtaining individual representations of experts’ beliefs and render them in the form of probability distributions or functions. In many cases the elicited distributions differ and the challenge in Bayesian inference is then to find ways to reconcile discrepant elicited prior distributions. This paper proposes the parallel analysis of clusters of prior distributions through a hierarchical method for clustering distributions and that can be readily extended to functional data. The proposed method consists of (i) transforming the infinite-dimensional problem into a finite-dimensional one, (ii) using the Hellinger distance to compute the distances between curves and thus (iii) obtaining a hierarchical clustering structure. In a simulation study the proposed method was compared to k-means and agglomerative nesting algorithms and the results showed that the proposed method outperformed those algorithms. Finally, the proposed method is illustrated through an EKE experiment and other functional data sets.

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