Regularized Principal Component Analysis for Spatial Data

In many atmospheric and earth sciences, it is of interest to identify dominant spatial patterns of variation based on data observed at p locations and n time points with the possibility that p > n. While principal component analysis (PCA) is commonly applied to find the dominant patterns, the eigenimages produced from PCA may exhibit patterns that are too noisy to be physically meaningful when p is large relative to n. To obtain more precise estimates of eigenimages, we propose a regularization approach incorporating smoothness and sparseness of eigenimages, while accounting for their orthogonality. Our method allows data taken at irregularly spaced or sparse locations. In addition, the resulting optimization problem can be solved using the alternating direction method of multipliers, which is easy to implement, and applicable to a large spatial dataset. Furthermore, the estimated eigenfunctions provide a natural basis for representing the underlying spatial process in a spatial random-effects model, from which spatial covariance function estimation and spatial prediction can be efficiently performed using a regularized fixed-rank kriging method. Finally, the effectiveness of the proposed method is demonstrated by several numerical examples.     

由Campanato型函数和与薛定谔算子相关的Riesz变换生成的Toeplitz算子的有界性

本文研究了由Campanato型函数及与Schrödinger算子相关的Riesz变换生成的Toeplitz算子的有界性.利用Sharp极大函数估计得到了Toeplitz算子Θ^b在Lebesgue空间的有界性,拓广了已有交换子的结果.     

The transport dynamics in complex systems governing by anomalous diffusion modelled with Riesz fractional partial differential equations

In this paper, numerical solutions of fractional Fokker–Planck equations with Riesz space fractional derivatives have been developed. Here, the fractional Fokker–Planck equations have been considered in a finite domain. In order to deal with the Riesz fractional derivative operator, shifted Grünwald approximation and fractional centred difference approaches have been used. The explicit finite difference method and Crank–Nicolson implicit method have been applied to obtain the numerical solutions of fractional diffusion equation and fractional Fokker–Planck equations, respectively. Numerical results are presented to demonstrate the accuracy and effectiveness of the proposed numerical solution techniques. Copyright © 2016 John Wiley & Sons, Ltd.     

On the canonical equations of Kirchhoff-Love theory of shells

The paper outlines a procedure to derive the canonical system of equations of the classical theory of thin shells using Reissner’s variational principle and partial variational principles. The Hamiltonian form of the Reissner functional is obtained using Lagrange multipliers to include the kinematical conditions that follow from the Kirchhoff-Love hypotheses. It is shown that the canonical system of equations can be represented in three different forms: one conventional form (five equilibrium equations) and two forms that are equivalent to it. This can be proved by reducing them to the same system of three equations. For problems with separable active and passive variables, partial variational principles are formulated __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 10, pp. 99–107, October 2007.     

Bessel sequences of exponentials on fractal measures

Jorgensen and Pedersen have proven that a certain fractal measure ν has no infinite set of complex exponentials which form an orthonormal set in L²(ν). We prove that any fractal measure μ obtained from an affine iterated function system possesses a sequence of complex exponentials which forms a Riesz basic sequence, or more generally a Bessel sequence, in L²(μ) such that the frequencies have positive Beurling dimension.     

Free Boundary Value Problems for Abstract Elliptic Equations and Applications

The free boundary value problems for elliptic differential-operator equations are studied. Several conditions for the uniform maximal regularity with respect to boundary parameters and the Fredholmness in abstract L _p -spaces are given. In application, the nonlocal free boundary problems for finite or infinite systems of elliptic and anisotropic type equations are studied.     

Preconditioning trace coupled 3d‐1d systems using fractional Laplacian

Multiscale or multiphysics problems often involve coupling of partial differential equations posed on domains of different dimensionality. In this work, we consider a simplified model problem of a 3d‐1d coupling and the main objective is to construct algorithms that may utilize standard multilevel algorithms for the 3d domain, which has the dominating computational complexity. Preconditioning for a system of two elliptic problems posed, respectively, in a three‐dimensional domain and an embedded one dimensional curve and coupled by the trace constraint is discussed. Investigating numerically the properties of the well‐defined discrete trace operator, it is found that negative fractional Sobolev norms are suitable preconditioners for the Schur complement of the system. The norms are employed to construct a robust block diagonal preconditioner for the coupled problem.