Infinite Setting Research Articles

Convergence Rate Analysis of Inertial Krasnoselskii–Mann Type Iteration with Applications

It is well known that the Krasnoselskii–Mann iteration of nonexpansive operators find applications in many areas of mathematics and known to be weakly convergent in the infinite dimensional setting. In this present paper, we provide a nonasymptotic convergence rate result for a Krasnoselskii–Mann iteration with inertial extrapolation step in real Hilbert spaces. We give some applications of our results to the Douglas–Rachford splitting method and the alternating projection method by John von Neumann. Our result serves as supplement to many existing results on convergence rate of Krasnoselskii–Mann iteration in the literature.

Stability results for 2D Navier–Stokes equations with unbounded delay

Some results related to 2D Navier–Stokes equations when the external force contains hereditary characteristics involving unbounded delays are analyzed. First, the existence and uniqueness of solutions is proved by Galerkin approximations and the energy method. The existence of stationary solution is then established by means of the Lax–Milgram theorem and the Schauder fixed point theorem. The local stability analysis of stationary solutions is studied by several different methods: the classical Lyapunov function method, the Razumikhin–Lyapunov technique and by constructing appropriate Lyapunov functionals. Finally, we also verify the polynomial stability of the stationary solution in a particular case of unbounded variable delay. Exponential stability in this infinite delay setting remains as an open problem.

Clique Trees of Infinite Locally Finite Chordal Graphs

We investigate clique trees of infinite locally finite chordal graphs. Our main contribution is a bijection between the set of clique trees and the product of local finite families of finite trees. Even more, the edges of a clique tree are in bijection with the edges of the corresponding collection of finite trees. This allows us to enumerate the clique trees of a chordal graph and extend various classic characterisations of clique trees to the infinite setting.

Slice regular semigroups

In this paper we introduce the notion of slice regular right linear semigroup in a quaternionic Banach space. It is an operatorial function which is slice regular (a noncommutative counterpart of analyticity) and which satisfies a noncommutative semigroup law characterizing the exponential function in an infinite dimensional noncommutative setting. We prove that a right linear operator semigroup in a quaternionic Banach space is slice regular if and only if its generator is spherical sectorial. This result provides a connection between the slice regularity and the noncommutative semigroups theory, and characterizes those semigroups which can be represented by a noncommutative Cauchy integral formula. All our results are generalized to Banach two-sided modules having as a set of scalar any real associative *-algebra, Clifford R_n algebras included.

A note on Anderson's theorem in the infinite-dimensional setting

Anderson's theorem states that if the numerical range W(A) of an n-by-n matrix A is contained in the unit disk D‾ and intersects with the unit circle at more than n points, then W(A)=D‾. An analogue of this result for compact A in an infinite dimensional setting was established by Gau and Wu. We consider here the case of A being the sum of a normal and compact operator.

Real Algebraic Geometry With a View Toward Moment Problems and Optimization

Continuing the tradition initiated inMFO workshop held in 2014, the aim of this workshop was to foster the interaction between real algebraic geometry, operator theory, optimization, and algorithms for systems control. A particular emphasis was given to moment problems through an interesting dialogue between researchers working on these problems in finite and infinite dimensional settings, from which emerged new challenges and interdisciplinary applications.

Initial value problem for the free-boundary magnetohydrodynamics with zero magnetic boundary condition

We show local existence and uniqueness of plasma(fluid)-vaccum free boundary problem of magnetohydrodynamic flow in three-dimensional space with infinite depth setting when magnetic field is zero on the free boundary. We use Sobolev-Slobodetskii space which was used in usual free boundary problem in [3,5,6,7,8,9]. We also show that this solution can be extended as long as we want for sufficiently small initial data. Using the result of this paper we will get a unique solution of (kinematic inviscid) - (magnetic non diffusive) free boundary magnetohydrodynamics problem via (kinematic viscosity) - (magnetic diffusivity) limit in [10].

A Subspace Method for Large-Scale Eigenvalue Optimization

We consider the minimization or maximization of the $J$th largest eigenvalue of an analytic and Hermitian matrix-valued function, and build on Mengi, Yildirim, and Kilic [SIAM J. Matrix Anal. Appl., 35, pp. 699--724, 2014]. This work addresses the setting when the matrix-valued function involved is very large. We describe subspace procedures that convert the original problem into a small-scale one by means of orthogonal projections and restrictions to certain subspaces, and that gradually expand these subspaces based on the optimal solutions of small-scale problems. Global convergence and superlinear rate-of-convergence results with respect to the dimensions of the subspaces are presented in the infinite dimensional setting, where the matrix-valued function is replaced by a compact operator depending on parameters. In practice, it suffices to solve eigenvalue optimization problems involving matrices with sizes on the scale of tens, instead of the original problem involving matrices with sizes on the scale...

Mixing properties for toral extensions of slowly mixing dynamical systems with finite and infinite measure

We prove results on mixing and mixing rates for toral extensions of nonuniformly expanding maps with subexponential decay of correlations. Both the finite and infinite measure settings are considered. Under a Dolgo-pyat-type condition on nonexistence of approximate eigenfunctions, we prove that existing results for (possibly non-Markovian) nonuniformly expanding maps hold also for their toral extensions.

Strong convergence of the forward–backward splitting method with multiple parameters in Hilbert spaces

Many problems arising from machine learning, signal & image recovery, and compressed sensing can be casted into a monotone inclusion problem for finding a zero of the sum of two monotone operators. The forward–backward splitting algorithm is one of the most powerful and successful methods for solving such a problem. However, this algorithm has only weak convergence in the infinite dimensional settings. In this paper, we propose a new modification of the FBA so that it possesses a norm convergent property. Moreover, we establish two strong convergence theorems of the proposed algorithms under more general conditions.

Properties of Chance Constraints in Infinite Dimensions with an Application to PDE Constrained Optimization

Chance constraints represent a popular tool for finding decisions that enforce the satisfaction of random inequality systems in terms of probability. They are widely used in optimization problems subject to uncertain parameters as they arise in many engineering applications. Most structural results of chance constraints (e.g., closedness, convexity, Lipschitz continuity, differentiability etc.) have been formulated in finite dimensions. The aim of this paper is to generalize some of these well-known semi-continuity and convexity properties as well as a stability result to an infinite dimensional setting. The abstract results are applied to a simple PDE constrained control problem subject to (uniform) state chance constraints.

Time preferences and the pricing of complementary durables and consumables

There is strong empirical evidence that consumers discount at significantly higher rates than firms. Yet, most research abstracts from the effect of discount rates on marketing decisions such as pricing. We study the effects of a consumers' discount rate that is higher than a firm's discount rate on prices, profits and consumer surplus of complementary products in four competitive settings and an infinite time setting: the firm is a monopolist or competes in the durable market and either ties the consumable to the durable or sells untied products. Our analysis yields five main results: First, a higher time preference of consumers than the firm never increases the optimal durable price and never decreases the optimal consumable price. Second, the optimal consumable price of tied goods is always higher than the optimal consumable price of untied goods, whereas the optimal durable price is always higher when goods are untied. Third, a higher time preference of consumers than the firm never increases profit, always decreases consumer surplus and, as a result, always decreases welfare. Fourth, the ability of the firm to commit to future prices and of consumers to commit to future purchases benefits both consumers and the firm. Fifth, if the firm competes in the durable market, then tying increases consumer surplus when consumers commit to purchasing the consumable. We discuss the implications of our results for firms' pricing strategies.

Global Infinite Energy Solutions for the 2D Gravity Water Waves System

We prove global existence and a modified scattering property for the solutions of the 2D gravity water waves system in the infinite depth setting for a class of initial data, which is only required to be small above the level . No assumption is made below this level. Therefore, the nonlinear solution can have infinite energy. As a direct consequence, the momentum condition assumed on the physical velocity in all previous small energy results by Ionescu‐Pusateri, Alazard‐Delort, and Ifrim‐Tataru is removed.© 2017 Wiley Periodicals, Inc.

Infinite dimensional Jordan algebras and symmetric cones

A celebrated result of Koecher and Vinberg asserts the one-one correspondence between the finite dimensional formally real Jordan algebras and Euclidean symmetric cones. We extend this result to the infinite dimensional setting.

Solutions to inexact resolvent inclusion problems with applications to nonlinear analysis and optimization

Many problems in nonlinear analysis and optimization, among them variational inequalities and minimization of convex functions, can be reduced to finding zeros (namely, roots) of set-valued operators. Hence numerous algorithms have been devised in order to achieve this task. A lot of these algorithms are inexact in the sense that they allow perturbations to appear during the iterative process, and hence they enable one to better deal with noise and computational errors, as well as superiorization. For many years a certain fundamental question has remained open regarding many of these known inexact algorithmic schemes in various finite and infinite dimensional settings, namely whether there exist sequences satisfying these inexact schemes when errors appear. We provide a positive answer to this question. Our results also show that various theorems discussing the convergence of these inexact schemes have a genuine merit beyond the exact case. As a by-product we solve the standard and the strongly implicit inexact resolvent inclusion problems, introduce a promising class of functions (fully Legendre functions), establish continuous dependence (stability) properties of the solution of the inexact resolvent inclusion problem and continuity properties of the protoresolvent, and generalize the notion of strong monotonicity.

Trajectory controllability of a semilinear parabolic system

This paper introduces the trajectory controllability for a semilinear parabolic system in infinite dimensional space setting. Under suitable conditions the trajectory controllability of semilinear system has been established. Numerical experiments are conducted using backward Euler discontinuous finite volume methods with interpolated coefficients to approximate the semilinear system and to illustrate our theoretical results.

Demonstration of Multisetting One-Way Einstein-Podolsky-Rosen Steering in Two-Qubit Systems.

Einstein-Podolsky-Rosen (EPR) steering describes the ability of one party to remotely affect another's state through local measurements. One of the most distinguishable properties of EPR steering is its asymmetric aspect. Steering can work in one direction but fail in the opposite direction. This type of one-way steering, which is different from the symmetry concepts of entanglement and Bell nonlocality, has garnered much interest. However, an experimental demonstration of genuine one-way EPR steering in the simplest scenario, i.e., one that employs two-qubit systems, is still lacking. In this Letter, we experimentally demonstrate one-way EPR steering with multimeasurement settings for a class of two-qubit states, which are still one-way steerable even with infinite settings. The steerability is quantified by the steering radius, which represents a necessary and sufficient steering criterion. The demonstrated one-way steering in the simplest bipartite quantum system is of fundamental interest and may provide potential applications in one-way quantum information tasks.

Towards a Perron–Frobenius theory for eventually positive operators

This article is a contribution to the spectral theory of so-called eventually positive operators, i.e. operators T which may not be positive but whose powers Tn become positive for large enough n. While the spectral theory of such operators is well understood in finite dimensions, the infinite dimensional case has received much less attention in the literature. We show that several sensible notions of “eventual positivity” can be defined in the infinite dimensional setting, and in contrast to the finite dimensional case those notions do not in general coincide. We then prove a variety of typical Perron–Frobenius type results: we show that the spectral radius of an eventually positive operator is contained in the spectrum; we give sufficient conditions for the spectral radius to be an eigenvalue admitting a positive eigenvector; and we show that the peripheral spectrum of an eventually positive operator is a cyclic set under quite general assumptions. All our results are formulated for operators on Banach lattices, and many of them do not impose any compactness assumptions on the operator.

Generalized Krasnoselskii–Mann-type iterations for nonexpansive mappings in Hilbert spaces

The Krasnoselskii---Mann iteration plays an important role in the approximation of fixed points of nonexpansive operators; it is known to be weakly convergent in the infinite dimensional setting. In this present paper, we provide a new inexact Krasnoselskii---Mann iteration and prove weak convergence under certain accuracy criteria on the error resulting from the inexactness. We also show strong convergence for a modified inexact Krasnoselskii---Mann iteration under suitable assumptions. The convergence results generalize existing ones from the literature. Applications are given to the Douglas---Rachford splitting method, the Fermat---Weber location problem as well as the alternating projection method by John von Neumann.

Distributed Parameter Control Strategies for the Wave Equation with Fractional Order Derivative

In this paper, the control strategies are studied for the fractional order uncertain wave equation subject to persistent external disturbances in Hilbert spaces. The twisting and super-twisting fractional order sliding mode controllers (SMCs) are designed for the infinite dimensional setting and they are applied for addressing the asymptotic state tracking of the fractional order perturbed wave equation. Furthermore, by introducing the adaptive control law to the twisting controller, the bound of the external disturbances which is unknown is dealt with, and for the design of the super-twisting SMC, a fractional order sliding mode manifold is utilized which results in a continuous input control and a chattering free signal. Both of the controllers are associated with the fractional order parameter, which influences the convergence rate of the proposed control algorithms. In addition, the relative theorem involved in the paper for the proof of the stability is proved. Then, the control algorithms are extended to globally asymptotically stabilize the fractional order uncertain wave equation by choosing the appropriate Lyapunov functional. Finally, numerical simulations are presented to verify the viability and efficiency of the proposed fractional order controllers.