Authors:Stephan Dahlke, Cornelia Schneider Pages: 235 - 291 Abstract: Analysis and Applications, Volume 17, Issue 02, Page 235-291, March 2019. This paper is concerned with the regularity of solutions to linear and nonlinear evolution equations on nonsmooth domains. In particular, we study the smoothness in the specific scale [math] of Besov spaces. The regularity in these spaces determines the approximation order that can be achieved by adaptive and other nonlinear approximation schemes. We show that for all cases under consideration the Besov regularity is high enough to justify the use of adaptive algorithms. Citation: Analysis and Applications PubDate: 2019-03-11T10:30:27Z DOI: 10.1142/S0219530518500306
Authors:Cheng Wang, Ting Hu Pages: 293 - 322 Abstract: Analysis and Applications, Volume 17, Issue 02, Page 293-322, March 2019. Minimum error entropy (MEE) criterion is an important optimization method in information theoretic learning (ITL) and has been widely used and studied in various practical scenarios. In this paper, we shall introduce the online MEE algorithm for dealing with big datasets, associated with reproducing kernel Hilbert spaces (RKHS) and unbounded sampling processes. Explicit convergence rate will be given under the conditions of regularity of the regression function and polynomially decaying step sizes. Besides its low complexity, we will also show that the learning ability of online MEE is superior to the previous work in the literature. Our main techniques depend on integral operators on RKHS and probability inequalities for random variables with values in a Hilbert space. Citation: Analysis and Applications PubDate: 2019-03-11T10:30:21Z DOI: 10.1142/S0219530518500148
Authors:Pedro Caro, Tapio Helin, Matti Lassas Pages: 1 - 55 Abstract: Analysis and Applications, Ahead of Print. In this paper, we consider an inverse problem for the [math]-dimensional random Schrödinger equation [math]. We study the scattering of plane waves in the presence of a potential [math] which is assumed to be a Gaussian random function such that its covariance is described by a pseudodifferential operator. Our main result is as follows: given the backscattered far field, obtained from a single realization of the random potential [math], we uniquely determine the principal symbol of the covariance operator of [math]. Especially, for [math] this result is obtained for the full nonlinear inverse backscattering problem. Finally, we present a physical scaling regime where the method is of practical importance. Citation: Analysis and Applications PubDate: 2019-01-25T11:18:49Z DOI: 10.1142/S0219530519500015
Authors:Shenglan Yuan, Jianyu Hu, Xianming Liu, Jinqiao Duan Pages: 1 - 35 Abstract: Analysis and Applications, Ahead of Print. This work is concerned with the dynamics of a class of slow–fast stochastic dynamical systems driven by non-Gaussian stable Lévy noise with a scale parameter. Slow manifolds with exponentially tracking property are constructed, and then we eliminate the fast variables to reduce the dimensions of these stochastic dynamical systems. It is shown that as the scale parameter tends to zero, the slow manifolds converge to critical manifolds in distribution, which helps to investigate long time dynamics. The approximations of slow manifolds with error estimate in distribution are also established. Furthermore, we corroborate these results by three examples from biological sciences. Citation: Analysis and Applications PubDate: 2019-01-25T11:18:48Z DOI: 10.1142/S0219530519500027
Authors:Lei Shi Pages: 1 - 29 Abstract: Analysis and Applications, Ahead of Print. We investigate the distributed learning with coefficient-based regularization scheme under the framework of kernel regression methods. Compared with the classical kernel ridge regression (KRR), the algorithm under consideration does not require the kernel function to be positive semi-definite and hence provides a simple paradigm for designing indefinite kernel methods. The distributed learning approach partitions a massive data set into several disjoint data subsets, and then produces a global estimator by taking an average of the local estimator on each data subset. Easy exercisable partitions and performing algorithm on each subset in parallel lead to a substantial reduction in computation time versus the standard approach of performing the original algorithm on the entire samples. We establish the first mini-max optimal rates of convergence for distributed coefficient-based regularization scheme with indefinite kernels. We thus demonstrate that compared with distributed KRR, the concerned algorithm is more flexible and effective in regression problem for large-scale data sets. Citation: Analysis and Applications PubDate: 2019-01-09T02:47:56Z DOI: 10.1142/S021953051850032X
Authors:Yangyang Zhang, Dachun Yang, Wen Yuan, Songbai Wang Pages: 1 - 68 Abstract: Analysis and Applications, Ahead of Print. In this paper, the authors first introduce a class of Orlicz-slice spaces which generalize the slice spaces recently studied by Auscher et al. Based on these Orlicz-slice spaces, the authors then introduce a new kind of Hardy-type spaces, the Orlicz-slice Hardy spaces, via the radial maximal functions. This new scale of Orlicz-slice Hardy spaces contains the variant of the Orlicz–Hardy space of Bonami and Feuto as well as the Hardy-amalgam space of de Paul Ablé and Feuto as special cases. Their characterizations via the atom, the molecule, various maximal functions, the Poisson integral and the Littlewood–Paley functions are also obtained. As an application of these characterizations, the authors establish their finite atomic characterizations, which further induce a description of their dual spaces and a criterion on the boundedness of sublinear operators from these Orlicz-slice Hardy spaces into a quasi-Banach space. Then, applying this criterion, the authors obtain the boundedness of [math]-type Calderón–Zygmund operators on these Orlicz-slice Hardy spaces. All these results are new even for slice Hardy spaces and, moreover, for Hardy-amalgam spaces, the Littlewood–Paley function characterizations, the dual spaces and the boundedness of [math]-type Calderón–Zygmund operators on these Hardy-type spaces are also new. Citation: Analysis and Applications PubDate: 2018-12-19T01:48:17Z DOI: 10.1142/S0219530518500318
Authors:G. L. Myleiko, S. Pereverzyev, S. G. Solodky Pages: 1 - 23 Abstract: Analysis and Applications, Ahead of Print. In the supervised learning, the Nyström type subsampling is considered as a tool for reducing the computational complexity of regularized kernel methods in the big data setting. Up to now, the theoretical analysis of this approach has been done almost exclusively in the context of the regression learning and in the case where the smoothness of the target functions is restricted to the Hölder type source conditions. Such conditions do not cover the case of target functions with high and low smoothness, which are also of practical interest. Moreover, in the case of the Hölder source conditions, there is no need to consider a regularization with high enough qualification because order-optimal learning rates are achieved by the simple Tikhonov regularization known also as the kernel ridge regression. At the same time, this learning method does not improve its performance for any smoothness higher than Hölder ones. Therefore, in this paper, our goal is to extend previous analysis of the Nyström type subsampling to the case of the general source conditions, and to the regularization schemes with high enough qualification. We also show that under rather natural assumption, our results can be easily reformulated in the ranking learning setting. Citation: Analysis and Applications PubDate: 2018-11-07T01:58:10Z DOI: 10.1142/S021953051850029X
Authors:Lingwei Ma, Zhenqiu Zhang, Qi Xiong Pages: 1 - 28 Abstract: Analysis and Applications, Ahead of Print. Pointwise estimates of weak solution pairs to a stationary Stokes system with small [math] semi-norm coefficients are established in Reifenberg flat domains by using the restricted sharp maximal function. These pointwise estimates provide a unified treatment of the Calderón–Zygmund estimates for the solution pair to Stokes systems in [math] and [math] spaces. Citation: Analysis and Applications PubDate: 2018-11-07T01:58:09Z DOI: 10.1142/S0219530518500288
Authors:J. A. Carrillo, M. G. Delgadino, F. S. Patacchini Pages: 1 - 31 Abstract: Analysis and Applications, Ahead of Print. We analyze free energy functionals for macroscopic models of multi-agent systems interacting via pairwise attractive forces and localized repulsion. The repulsion at the level of the continuous description is modeled by pressure-related terms in the functional making it energetically favorable to spread, while the attraction is modeled through nonlocal forces. We give conditions on general entropies and interaction potentials for which neither ground states nor local minimizers exist. We show that these results are sharp for homogeneous functionals with entropies leading to degenerate diffusions while they are not sharp for fast diffusions. The particular relevant case of linear diffusion is totally clarified giving a sharp condition on the interaction potential under which the corresponding free energy functional has ground states or not. Citation: Analysis and Applications PubDate: 2018-09-19T02:42:13Z DOI: 10.1142/S0219530518500276
Authors:Lili Fan, Guiqiong Gong, Shaojun Tang Pages: 1 - 24 Abstract: Analysis and Applications, Ahead of Print. This paper is concerned with the Cauchy problem of heat-conductive ideal gas without viscosity, where the far field states are prescribed. When the corresponding Riemann problem for the compressible Euler system has the solution consisting of a contact discontinuity and rarefaction waves, we show that if the strengths of the wave patterns and the initial perturbation are suitably small, the unique global-in-time solution exists and asymptotically tends to the corresponding composition of a viscous contact wave with rarefaction waves, which extended the results by Huang–Li–Matsumura [Asymptotic stability of combination of viscous contact wave with rarefaction waves for one-dimensional compressible Navier–Stokes system, Arch. Ration. Mech. Anal. 197 (2010) 89–116.], where they treated the viscous and heat-conductive ideal gas. Citation: Analysis and Applications PubDate: 2018-07-27T03:08:50Z DOI: 10.1142/S0219530518500239
Authors:Claudianor O. Alves, Vincenzo Ambrosio, César E. Torres Ledesma Pages: 1 - 27 Abstract: Analysis and Applications, Ahead of Print. In this paper, we study the existence of heteroclinic solution for a class of nonlocal problems of the type (−Δ)αu + a(ðœ–x)V′(u) = 0,x ∈ ℝ,limx→−∞u(x) = −1andlimx→+∞u(x) = 1, where [math], [math] are continuous functions verifying some technical conditions. For example [math] can be asymptotically periodic and potential [math] can be the Ginzburg–Landau potential, that is, [math]. Citation: Analysis and Applications PubDate: 2018-07-27T03:08:50Z DOI: 10.1142/S0219530518500252
Authors:Gerlind Plonka, Kilian Stampfer, Ingeborg Keller Pages: 1 - 32 Abstract: Analysis and Applications, Ahead of Print. We employ the generalized Prony method in [T. Peter and G. Plonka, A generalized Prony method for reconstruction of sparse sums of eigenfunctions of linear operators, Inverse Problems 29 (2013) 025001] to derive new reconstruction schemes for a variety of sparse signal models using only a small number of signal measurements. By introducing generalized shift operators, we study the recovery of sparse trigonometric and hyperbolic functions as well as sparse expansions into Gaussians chirps and modulated Gaussian windows. Furthermore, we show how to reconstruct sparse polynomial expansions and sparse non-stationary signals with structured phase functions. Citation: Analysis and Applications PubDate: 2018-07-27T03:08:49Z DOI: 10.1142/S0219530518500240
Authors:Robert J. Martin, Ionel-Dumitrel Ghiba, Patrizio Neff Pages: 1 - 13 Abstract: Analysis and Applications, Ahead of Print. Adapting a method introduced by Ball, Muite, Schryvers and Tirry, we construct a polyconvex isotropic energy function [math] which is equal to the classical Hencky strain energy WH(F) = μ∥devnlogU∥2 + κ 2 [tr(logU)]2 = μ∥logU∥2 + Λ 2 [tr(logU)]2 in a neighborhood of the identity matrix [math]; here, [math] denotes the set of [math]-matrices with positive determinant, [math] denotes the deformation gradient, [math] is the corresponding stretch tensor, [math] is the principal matrix logarithm of [math], tr is the trace operator, [math] is the Frobenius matrix norm and [math] is the deviatoric part of [math]. The extension can also be chosen to be coercive, in which case Ball’s classical theorems for the existence of energy minimizers under appropriate boundary conditions are immediately applicable. We also generalize the approach to energy functions [math] in the so-called Valanis–Landel form WVL(F) =∑i=1nw(λ i) with [math], where [math] denote the singular values of [math]. Citation: Analysis and Applications PubDate: 2018-07-13T01:56:57Z DOI: 10.1142/S0219530518500173
Authors:Jianbin Yang Pages: 1 - 25 Abstract: Analysis and Applications, Ahead of Print. Shift-invariant spaces play an important role in approximation theory, wavelet analysis, finite elements, etc. In this paper, we consider the stability and reconstruction algorithm of random sampling in multiply generated shift-invariant spaces [math]. Under some decay conditions of the generator [math], we approximate [math] with finite-dimensional subspaces and prove that with overwhelming probability, the stability of sampling set conditions holds uniformly for all functions in certain compact subsets of [math] when the sampling size is sufficiently large. Moreover, we show that this stability problem is connected with properties of the random matrix generated by [math]. In the end, we give a reconstruction algorithm for the random sampling of functions in [math]. Citation: Analysis and Applications PubDate: 2018-07-13T01:56:53Z DOI: 10.1142/S0219530518500185
Authors:Philippe G. Ciarlet, Maria Malin, Cristinel Mardare Pages: 1 - 30 Abstract: Analysis and Applications, Ahead of Print. A nonlinear Korn inequality on a surface is any estimate of the distance, up to a proper isometry of [math], between two surfaces measured by some appropriate norms (the “left-hand side” of the inequality) in terms of the distances between their three fundamental forms measured by some appropriate norms (the “right-hand side” of the inequality). The first objective of this paper is to provide several extensions of a nonlinear Korn inequality on a surface obtained in 2006 by the first and third authors and Gratie, then measured by means of [math]-norms on the left-hand side and [math]-norms on the right-hand side. First, we extend this inequality to [math]-norms on the left-hand side and [math]-norms on the right-hand side for any [math] and [math] that satisfy [math]; second, we show how the third fundamental forms can be disposed in the right-hand side; and third, we show that there is no need to introduce proper isometries of [math] in the left-hand side if the surfaces satisfy appropriate boundary conditions. The second objective is to provide nonlinear Korn inequalities on a surface where the left-hand sides are now measured by means of [math]-norms while the right-hand sides are measured by means of [math]-norms, for any [math]. These nonlinear Korn inequalities on a surface themselves rely on various nonlinear Korn inequalities in a domain in [math], recently obtained by the first and third authors in 2015 and by the first author and Sorin Mardare in 2016. Citation: Analysis and Applications PubDate: 2018-03-29T09:55:40Z DOI: 10.1142/S0219530518500136
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