Finite-Dimensionality of Tempered Random Uniform Attractors

Finite-dimensional attractors play an important role in finite-dimensional reduction of PDEs in mathematical modelization and numerical simulations. For non-autonomous random dynamical systems, Cui and Langa (J Differ Equ, 263:1225–1268, 2017) developed a random uniform attractor as a minimal compact random set which provides a certain description of the forward dynamics of the underlying system by forward attraction in probability. In this paper, we study the conditions that ensure a random uniform attractor to have finite fractal dimension. Two main criteria are given, one by a smoothing property and the other by a squeezing property of the system, and neither of the two implies the other. The upper bound of the fractal dimension consists of two parts: the fractal dimension of the symbol space plus a number arising from the smoothing/squeezing property. As an illustrative application, the random uniform attractor of a stochastic reaction–diffusion equation with scalar additive noise is studied, for which the finite-dimensionality in $$L^2$$ L 2 is established by the squeezing approach and that in $$H_0^1$$ H 0 1 by the smoothing framework. In addition, a random absorbing set that absorbs itself after a deterministic period of time is also constructed.

Asymptotic analysis and upper semicontinuity with respect to delay term of attractors to binary mixtures of solids

We investigated the asymptotic dynamics of a nonlinear system modeling binary mixture of solids with delay term. Using the recent quasi-stability methods introduced by Chueshov and Lasiecka, we prove the existence, smoothness and finite dimensionality of a global attractor. We also prove the existence of exponential attractors. Moreover, we study the upper semicontinuity of global attractors with respect to small perturbations of the delay terms.

On the Finite Dimensionality of Closed Subspaces in Lp(M,dμ)∩Lq(M,dν)

Finding effective finite-dimensional criteria for closed subspaces in Lp, endowed with some additional functional constraints, is a well-known and interesting problem. In this work, we are interested in some sufficient constraints on closed functional subspaces, Sp⊂Lp, whose finite dimensionality is not fixed a priori and can not be checked directly. This is often the case in diverse applications, when a closed subspace Sp⊂Lp is constructed by means of some additional conditions and constraints on Lp with no direct exemplification of the functional structure of its elements. We consider a closed topological subspace, Sp(q), of the functional Banach space, Lp(M,dμ), and, moreover, one assumes that additionally, Sp(q)⊂Lq(M,dν) is subject to a probability measure ν on M. Then, we show that closed subspaces of Lp(M,dμ)∩Lq(M,dν) for q>max{1,p},p>0 are finite dimensional. The finite dimensionality result concerning the case when q>p>0 is open and needs more sophisticated techniques, mainly based on analysis of the complementary subspaces to Lp(M,dμ)∩Lq(M,dν).

On a Characterization of Finite-Dimensional Vector Spaces

We provide a characterization of the finite dimensionality of vector spaces in terms of the right-sided invertibility of linear operators on them.

On stochastic accelerated gradient with non-strongly convexity

<abstract><p>In this paper, we consider stochastic approximation algorithms for least-square and logistic regression with no strong-convexity assumption on the convex loss functions. We develop two algorithms with varied step-size motivated by the accelerated gradient algorithm which is initiated for convex stochastic programming. We analyse the developed algorithms that achieve a rate of $ O(1/n^{2}) $ where $ n $ is the number of samples, which is tighter than the best convergence rate $ O(1/n) $ achieved so far on non-strongly-convex stochastic approximation with constant-step-size, for classic supervised learning problems. Our analysis is based on a non-asymptotic analysis of the empirical risk (in expectation) with less assumptions that existing analysis results. It does not require the finite-dimensionality assumption and the Lipschitz condition. We carry out controlled experiments on synthetic and some standard machine learning data sets. Empirical results justify our theoretical analysis and show a faster convergence rate than existing other methods.</p></abstract>