Mixed-Norm Amalgam Spaces and Their Predual

In this paper, we introduce mixed-norm amalgam spaces (Lp→,Ls→)α(Rn) and prove the boundedness of maximal function. Then, the dilation argument obtains the necessary and sufficient conditions of fractional integral operators’ boundedness. Furthermore, the strong estimates of linear commutators [b,Iγ] generated by b∈BMO(Rn) and Iγ on mixed-norm amalgam spaces (Lp→,Ls→)α(Rn) are established as well. In order to obtain the necessary conditions of fractional integral commutators’ boundedness, we introduce mixed-norm Wiener amalgam spaces (Lp→,Ls→)(Rn). We obtain the necessary and sufficient conditions of fractional integral commutators’ boundedness by the duality theory. The necessary conditions of fractional integral commutators’ boundedness are a new result even for the classical amalgam spaces. By the equivalent norm and the operators Str(p)(f)(x), we study the duality theory of mixed-norm amalgam spaces, which makes our proof easier. In particular, note that predual of the primal space is not obtained and the predual of the equivalent space does not mean the predual of the primal space.

Local saddles of relaxed averaged alternating reflections algorithms on phase retrieval

Phase retrieval can be expressed as a non-convex constrained optimization problem to identify one phase minimizer one a torus. Many iterative transform techniques have been proposed to identify the minimizer, e.g., relaxed averaged alternating reflections(RAAR) algorithms. In this paper, we present one optimization viewpoint on the RAAR algorithm. RAAR algorithm is one alternating direction method of multipliers(ADMM) with one penalty parameter. Pairing with multipliers (dual vectors), phase vectors on the primal space are lifted to higher-dimensional vectors, the RAAR algorithm is one continuation algorithm, which searches for local saddles in the primal-dual space. The dual iteration approximates one gradient ascent flow, which drives the corresponding local minimizers in a positive-definite Hessian region. Altering penalty parameters, the RAAR eliminates the stagnation of these corresponding local minimizers in the primal space and thus screens out many stationary points corresponding to non-local minimizers.

Alternating Primal-Dual Algorithm for Minimizing Multiple-Summed Separable Problems with Application to Image Restoration

In order to discover the difference among dual strategies, we propose an alternating primal-dual algorithm (APDA) that can be considered as a general version for minimizing problem which is multiple-summed separable convex but not necessarily smooth. First, the original multiple-summed problem is transformed into two subproblems. Second, one subproblem is solved in the primal space and the other is solved in the dual space. Finally, the alternating direction method is executed between the primal and the dual part. Furthermore, the classical alternating direction method of multipliers (ADMM) is extended to solve the primal subproblem which is also multiple summed, therefore, the extended ADMM can be seen as a parallel method for the original problem. Thanks to the flexibility of APDA, different dual strategies for image restoration are analyzed. Numerical experiments show that the proposed method performs better than some existing algorithms in terms of both speed and accuracy.

Training robust support vector regression machines for more general noise

Symmetric loss functions are widely used in regression algorithms to focus on estimating the means. Huber loss, a symmetric smooth loss function, has been proved that it can be optimized with high efficiency and certain robustness. However, mean estimators may be poor when the noise distribution is asymmetric (even outliers caused heavy-tailed distribution noise) and estimators beyond the means are necessary. Under the circumstances, quantile regression is a natural choice which estimates quantiles instead of means through asymmetric loss functions. In this paper, an asymmetric Huber loss function is proposed to implement different penalty for overestimation and underestimation so as to deal with more general noise. Moreover, a smooth truncated version of the proposed loss is introduced to enhance stronger robustness to outliers. Concave-convex procedure is developed in the primal space with the proof of convergence to handle the non-convexity of the involved truncated objective. Experiments are carried out on both artificial and benchmark datasets and robustness of the proposed methods are verified.

Spectral primal spaces

Let [Formula: see text] be a mapping. Consider [Formula: see text] Then, according to Echi, [Formula: see text] is an Alexandroff topology. A topological space [Formula: see text] is called a primal space if its topology coincides with an [Formula: see text] for some mapping [Formula: see text]. We denote by [Formula: see text] the set of all fixed points of [Formula: see text], and [Formula: see text] the set of all periodic points of [Formula: see text]. The topology [Formula: see text] induces a preorder [Formula: see text] defined on [Formula: see text] by: [Formula: see text] if and only if [Formula: see text], for some integer [Formula: see text]. The main purpose of this paper is to provide necessary and sufficient algebraic conditions on the function [Formula: see text] in order to get [Formula: see text] (respectively, the one-point compactification of [Formula: see text]) a spectral topology. More precisely, we show the following results. (1) [Formula: see text] is spectral if and only if [Formula: see text] is a finite set and every chain in the ordered set [Formula: see text] is finite. (2) The one-point(Alexandroff) compactification of [Formula: see text] is a spectral topology if and only if [Formula: see text] and every nonempty chain of [Formula: see text] has a least element. (3) The poset [Formula: see text] is spectral if and only if every chain is finite. As an application the main theorem [12, Theorem 3. 5] of Echi–Naimi may be derived immediately from the general setting of the above results.

Isogeometric BDDC deluxe preconditioners for linear elasticity

Balancing Domain Decomposition by Constraints (BDDC) preconditioners have been shown to provide rapidly convergent preconditioned conjugate gradient methods for solving many of the very ill-conditioned systems of algebraic equations which often arise in finite element approximations of a large variety of problems in continuum mechanics. These algorithms have also been developed successfully for problems arising in isogeometric analysis. In particular, the BDDC deluxe version has proven very successful for problems approximated by Non-Uniform Rational B-Splines (NURBS), even those of high order and regularity. One main purpose of this paper is to extend the theory, previously fully developed only for scalar elliptic problems in the plane, to problems of linear elasticity in three dimensions. Numerical experiments supporting the theory are also reported. Some of these experiments highlight the fact that the development of the theory can help to decrease substantially the dimension of the primal space of the BDDC algorithm, which provides the necessary global component of these preconditioners, while maintaining scalability and good convergence rates.