Time Discrete Approximation of Weak Solutions to Stochastic Equations of Geophysical Fluid Dynamics and Applications

As a first step towards the numerical analysis of the stochastic primitive equations of the atmosphere and the oceans,the time discretization of these equations by an implicit Euler scheme is studied.From the deterministic point of view,the 3D primitive equations are studied in their full form on a general domain and with physically realistic boundary conditions.From the probabilistic viewpoint,this paper deals with a wide class of nonlinear,state dependent,white noise forcings which may be interpreted in either the It6 or the Stratonovich sense.The proof of convergence of the Euler scheme,which is carried out within an abstract framework,covers the equations for the oceans,the atmosphere,the coupled oceanic-atmospheric system as well as other related geophysical equations.The authors obtain the existence of solutions which are weak in both the PDE and probabilistic sense,a result which is new by itself to the best of our knowledge.     

On the approximation of Laplacian eigenvalues in graph disaggregation

Graph disaggregation is a technique used to address the high cost of computation for power law graphs on parallel processors. The few high-degree vertices are broken into multiple small-degree vertices, in order to allow for more efficient computation in parallel. In particular, we consider computations involving the graph Laplacian, which has significant applications, including diffusion mapping and graph partitioning, among others. We prove results regarding the spectral approximation of the Laplacian of the original graph by the Laplacian of the disaggregated graph. In addition, we construct an alternate disaggregation operator whose eigenvalues interlace those of the original Laplacian. Using this alternate operator, we construct a uniform preconditioner for the original graph Laplacian.     

Approximations of spectra of Schrödinger operators with complex potentials on ℝd

We study spectral approximations of Schrödinger operators T = ?Δ+Q with complex potentials on Ω = ?^d, or exterior domains Ω??^d, by domain truncation. Our weak assumptions cover wide classes of potentials Q for which T has discrete spectrum, of approximating domains Ω_n, and of boundary conditions on ?Ω_n such as mixed Dirichlet/Robin type. In particular, Re Q need not be bounded from below and Q may be singular. We prove generalized norm resolvent convergence and spectral exactness, i.e. approximation of all eigenvalues of T by those of the truncated operators T_n without spectral pollution. Moreover, we estimate the eigenvalue convergence rate and prove convergence of pseudospectra. Numerical computations for several examples, such as complex harmonic and cubic oscillators for d = 1,2,3, illustrate our results.     

基于改进的GM(1,1)-Markov链组合模型广东省单位GDP能耗预测

本文以灰色系统理论的GM(1,1)模型和随机过程理论的Markov链模型为基础构建了一个动态GM(1,1)-Markov链组合预测模型。该模型同时利用了GM(1,1)模型对序列趋势因素良好的拟合能力和Markov链模型对残差序列信息的提取能力。为进一步提高该模型的预测精度,用泰勒(Taylor)近似方法和新信息优先的思想对该模型进行了改进。最后,以1991-2014年广东省单位GDP能耗数据实证了该模型的预测效果。     

Theoretical investigation of the weak interactions of rare gas atoms with silver clusters by resonance Raman spectroscopy modeling

The interactions of rare gas atoms (Rg = Ar, Kr, and Xe) with small neutral and cationic silver clusters have been investigated by density functional methods and the effect of these weak interactions on the resonance Raman spectra of the complexes has been evaluated. The resonance Raman technique that depends on the properties of ground and excited state, seems deeply sensitive to the weak rare gas–metal cluster interactions, and the use of inert gases has been proven to be an excellent approach to recognize the ability of this technique to detect extremely weak interactions. In this work, for , and complexes the IR, normal and resonance Raman spectra have been calculated and the effect of rare gas–cluster stretching vibration ( ) on the pattern and the relative intensities of different spectra have been investigated. The resonance Raman spectra for the weakly interacted complexes (with the interaction energies less than ?2.0 kcal/mol) exhibit the vibration with the detectable intensity that its intensity increases by going from Ag₆–Ar to Ag₆–Xe complex. Moreover, the resonance Raman spectra (based on the excited state gradient approximation) for high intensity nearly degenerate excited states, proved the effect of accumulation of the excited state charge density on the relative intensity of vibration.     

About the compatibility between ansatzes and constraints for a local formulation of orbital‐free density functional theory

Functional properties that are exact for the Hohenberg–Kohn functional may turn into mutually exclusive constraints at a given level of ansatz. This is exemplarily shown for the local density approximation. Nevertheless, it is possible to reach exactly the Kohn–Sham data from an orbital‐free density functional framework based on simple one‐point functionals by starting from the Levy–Perdew–Sahni formulation. The energy value is obtained from the density‐potential pair, and therefore does not refer to the functional dependence of the potential expression. Consequently, the potential expression can be obtained from any suitable model and is not required to follow proper scaling behavior.     

Remarks on the Born approximation and the Factorization Method

In this paper, we compare the far-field operators for the full nonlinear inverse scattering problem with the Born approximation as its linearization. The Factorization Method shows that both operators share the same behavior with respect to illposedness of the inverse problem. The results are derived for acoustic and electromagnetic scattering problems and the corresponding problem in electrical impedance tomography.