Stochastic model reduction: convergence and applications to climate equations

AbstractWe study stochastic model reduction for evolution equations in infinite-dimensional Hilbert spaces and show the convergence to the reduced equations via abstract results of Wong–Zakai type for stochastic equations driven by a scaled Ornstein–Uhlenbeck process. Both weak and strong convergence are investigated, depending on the presence of quadratic interactions between reduced variables and driving noise. Finally, we are able to apply our results to a class of equations used in climate modeling.

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Almost Automorphic Solutions to Nonautonomous Stochastic Functional Integrodifferential Equations

Abstract and Applied Analysis ◽

10.1155/2013/473969 ◽

2013 ◽

Vol 2013 ◽

pp. 1-13

Author(s):

Li Xi-liang ◽

Han Yu-liang

Keyword(s):

Hilbert Spaces ◽

Evolution Equations ◽

Mild Solutions ◽

Stochastic Equations ◽

Stochastic Evolution ◽

Stochastic Evolution Equations ◽

Almost Automorphic ◽

Banach Contraction ◽

Stability Results ◽

Stochastic Functional

This paper concerns the square-mean almost automorphic solutions to a class of abstract semilinear nonautonomous functional integrodifferential stochastic evolution equations in real separable Hilbert spaces. Using the so-called “Acquistapace-Terreni” conditions and Banach contraction principle, the existence, uniqueness, and asymptotical stability results of square-mean almost automorphic mild solutions to such stochastic equations are established. As an application, square-mean almost automorphic solution to a concrete nonautonomous integro-differential stochastic evolution equation is analyzed to illustrate our abstract results.

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Multiple-Set Split Feasibility Problems forκ-Strictly Pseudononspreading Mapping in Hilbert Spaces

Abstract and Applied Analysis ◽

10.1155/2013/342545 ◽

2013 ◽

Vol 2013 ◽

pp. 1-5 ◽

Cited By ~ 4

Author(s):

Jing Quan ◽

Shih-sen Chang ◽

Xiang Zhang

Keyword(s):

Iterative Method ◽

Strong Convergence ◽

Hilbert Spaces ◽

Weak And Strong Convergence ◽

Convergence Theorems ◽

Infinite Dimensional ◽

Split Feasibility Problems ◽

Split Feasibility ◽

Strictly Pseudononspreading Mapping

The purpose of this paper is to prove some weak and strong convergence theorems for solving the multiple-set split feasibility problems forκ-strictly pseudononspreading mapping in infinite-dimensional Hilbert spaces by using the proposed iterative method. The main results presented in this paper extend and improve the corresponding results of Xu et al. (2006), of Osilike et al. (2011), and of many other authors.

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ON DIFFERENTIAL-NON-AUTONOMOUS REPRESENTATION INTEGRATIVE ACTIVITY OF NEUROPOPULATION BILINEAR SECOND-ORDER MODEL WITH A DELAY

Izvestiya of Samara Scientific Center of the Russian Academy of Sciences ◽

10.37313/1990-5378-2021-23-2-115-126 ◽

2021 ◽

Vol 23 (2) ◽

pp. 115-126

Author(s):

A.V. Daneev ◽

◽

A.V. Lakeev ◽

V.A. Rusanov ◽

P.A. Plesnev ◽

...

Keyword(s):

Hilbert Space ◽

Hilbert Spaces ◽

Evolution Equations ◽

Basic Research ◽

Second Order ◽

Infinite Dimensional ◽

Machine Interface ◽

Additive Combination ◽

Basic Research Project ◽

Principle Of Maximum

For neuromorphic processes specified by the behavior of a local neuropopulation (for example, processes induced by a brain-machine interface platform of the Neuralink type), we study the solvability of the problem of the existence of a differential realization of these processes in the class of bilinear nonstationary ordinary differential equations of the second order (with delay) in separable Hilbert space. This formulation belongs to the type of inverse problems for an additive combination of nonstationary linear and bilinear operators of evolution equations in an infinite-dimensional Hilbert space. The metalanguage of the theory being developed is the constructions of tensor products of Hilbert spaces, lattice structures with orthocompletion, the functional apparatus of the nonlinear Rayleigh-Ritz operator, and the principle of maximum entropy. It is shown that the property of sublinearity of this operator, allows you to obtain conditions for the existence of such differential realizations; along the way, metric conditions for the continuity of the projectivization of this operator are substantiated with the calculation of the fundamental group of its compact image. This work was financially supported by the Russian Foundation for Basic Research (project no. 19-01-00301).

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