Non-symmetric Relaxation Research Articles

A Bregman stochastic method for nonconvex nonsmooth problem beyond global Lipschitz gradient continuity

ABSTRACT In this paper, we consider solving a broad class of large-scale nonconvex and nonsmooth minimization problems by a Bregman proximal stochastic gradient (BPSG) algorithm. The objective function of the minimization problem is the composition of a differentiable and a nondifferentiable function, and the differentiable part does not admit a global Lipschitz continuous gradient. Under some suitable conditions, the subsequential convergence of the proposed algorithm is established. And under expectation conditions with the Kurdyka-Łojasiewicz (KL) property, we also prove that the proposed method converges globally. We also apply the BPSG algorithm to solve sparse nonnegative matrix factorization (NMF), symmetric NMF via non-symmetric relaxation, and matrix completion problems under different kernel generating distances, and numerically compare it with other algorithms. The results demonstrate the robustness and effectiveness of the proposed algorithm.

Characterization of the dissipative structure for the symmetric hyperbolic system with non-symmetric relaxation

This paper is concerned with the dissipative structure for the linear symmetric hyperbolic system with non-symmetric relaxation. If the relaxation matrix of the system has symmetric property, Shizuta and Kawashima in 1985 introduced the suitable stability condition called Classical Stability Condition in this paper, and Umeda, Kawashima and Shizuta in 1984 analyzed the dissipative structure of the standard type. On the other hand, Ueda, Duan and Kawashima in 2012 and 2018 focused on the system with non-symmetric relaxation, and got the partial result which is the extension of known results. Furthermore, they argued the new dissipative structure called the regularity-loss type. In this situation, our purpose of this paper is to extend the stability theory introduced by Shizuta and Kawashima in 1985 and Umeda, Kawashima and Shizuta in 1984 for our general system.

An $$\mathrm{S}$$ & $$\mathrm{K}$$ mixed condition for symmetric hyperbolic systems with non-symmetric relaxations

We establish asymptotic stability results and provide explicit decay rate for linear symmetric hyperbolic systems in terms of the dissipative structure. In particular, we are concerned with the model systems which have a non-symmetric relaxation term and therefore their decay estimates and energy estimates are of regularity-loss type. We present a kind of S & K mixed criterion which is applicable also to weakly dissipative models including the Timoshenko–Cattaneo system, for which any already known criterion did not work.

Global well-posedness and polynomial decay for a nonlinear Timoshenko–Cattaneo system under minimal Sobolev regularity

We consider the nonlinear stability of the Timoshenko–Cattaneo system in the one-dimensional whole space. The Timoshenko system consists of two coupled wave equations with non-symmetric relaxation, and describes vibrations of the beam with shear deformation and rotational inertia effect. Generally, if the relaxation is not symmetric, the dissipation is produced through the complicated interaction of the components of the system, and their decay estimates and the energy estimates are of regularity-loss type. In this paper, we introduce the mathematical method to control such a weak dissipativity by investigating the Timoshenko system with Cattaneo’s law, which is the first order approximation of Fourier’s law with its time-delay effect. Racke & Said-Houari (2012), showed the global existence and the decay estimate of solutions by assuming high regularity H8∩L1 on the small initial data to control their weak dissipativity. In contrast, we prove the global existence in H2 by energy methods without any negative weights. Our regularity assumption is the same as that needed to show the local existence. That is, we do not need to assume the extra higher regularity on the initial data. Besides, the optimal decay estimate in H2∩L1 is shown by using the time decay inequality of Lp-Lq-Lr type.

New structural conditions on decay property with regularity-loss for symmetric hyperbolic systems with non-symmetric relaxation

This paper is concerned with the weak dissipative structure for linear symmetric hyperbolic systems with relaxation. The authors of this paper had already analyzed the new dissipative structure called the regularity-loss type in [Y. Ueda, R. Duan and S. Kawashima, Decay structure for symmetric hyperbolic systems with non-symmetric relaxation and its application, Arch. Ration. Mech. Anal. 205 (2012) 239–266]. Compared with the dissipative structure of the standard type in [T. Umeda, S. Kawashima and Y. Shizuta, On the devay of solutions to the linearized equations of electro-magneto-fluid dynamics, Japan J. Appl. Math. 1 (1984) 435–457; Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J. 14 (1985) 249–275], the regularity-loss type possesses a weaker structure in the high-frequency region in the Fourier space. Furthermore, there are some physical models which have more complicated structure, which we discussed in [Y. Ueda, R. Duan and S. Kawashima, Decay structure of two hyperbolic relaxation models with regularity loss, Kyoto J. Math. 57(2) (2017) 235–292]. Under this situation, we introduce new concepts and extend our previous results developed in [Y. Ueda, R. Duan and S. Kawashima, Decay structure for symmetric hyperbolic systems with non-symmetric relaxation and its application, Arch. Ration. Mech. Anal. 205 (2012) 239–266] to cover those complicated models.

Decay structure of two hyperbolic relaxation models with regularity loss

The paper aims at investigating two types of decay structure for linear symmetric hyperbolic systems with non-symmetric relaxation. Precisely, the system is of the type $(p,q)$ if the real part of all eigenvalues admits an upper bound $-c|\xi|^{2p}/(1+|\xi|^2)^{q}$, where $c$ is a generic positive constant and $\xi$ is the frequency variable, and the system enjoys the regularity-loss property if $p<q$. It is well known that the standard type $(1,1)$ can be assured by the classical Kawashima-Shizuta condition. A new structural condition was introduced in \cite{UDK} to analyze the regularity-loss type $(1,2)$ system with non-symmetric relaxation. In the paper, we construct two more complex models of the regularity-loss type corresponding to $p=m-3$, $q=m-2$ and $p=(3m-10)/2$, $q=2(m-3)$, respectively, where $m$ denotes phase dimensions. The proof is based on the delicate Fourier energy method as well as the suitable linear combination of series of energy inequalities. Due to arbitrary higher dimensions, it is not obvious to capture the energy dissipation rate with respect to the degenerate components. Thus, for each model, the analysis always starts from the case of low phase dimensions in order to understand the basic dissipative structure in the general case, and in the mean time, we also give the explicit construction of the compensating symmetric matrix $K$ and skew-symmetric matrix $S$.

Decay Structure for Symmetric Hyperbolic Systems with Non-Symmetric Relaxation and its Application

This paper is concerned with the decay structure for linear symmetric hyperbolic systems with relaxation. When the relaxation matrix is symmetric, the dissipative structure of the systems is completely characterized by the Kawashima-Shizuta stability condition formulated in \cite{UKS84,SK85}, and we obtain the asymptotic stability result together with the explicit time-decay rate under that stability condition. However, some physical models which satisfy the stability condition have non-symmetric relaxation term (cf.~the Timoshenko system and the Euler-Maxwell system). Moreover, it had been already known that the dissipative structure of such systems is weaker than the standard type and is of the regularity-loss type (cf.~\cite{D,IHK08,IK08,USK,UK}). Therefore our purpose of this paper is to formulate a new structural condition which include the Kawashima-Shizuta condition, and to analyze the weak dissipative structure for general systems with non-symmetric relaxation.

Decay Structure for Symmetric Hyperbolic Systems with Non-Symmetric Relaxation

AbstractThis article is concerned with the decay structure for linear symmetric hyperbolic systems with relaxation. When the relaxation matrix is symmetric, the dissipative structure of the systems is completely characterized by the Kawashima-Shizuta stability condition formulated in Umeda et al. (1984); Shizuta et al. (1985). However, some physical models which satisfy the stability condition have non-symmetric relaxation term (cf. the Timoshenko system and the Euler-Maxwell system). Therefore our purpose of this article is to formulate a new structural condition and to analyze the weak dissipative structure for general systems with non-symmetric relaxation.