强关联多电子体系的优化模型与算法

在电子结构计算领域,Kohn-Sham方程是最为广泛使用的数学模型之一.然而,由于现有的交换关联能近似仍存在缺陷,Kohn-Sham方程无法较好地描述强关联多电子体系.近年来,有学者从密度泛函理论的强相关极限出发,提出了严格关联电子能量的优化模型.该模型有望弥补Kohn-Sham方程的缺陷,从而拓宽密度泛函理论的应用面.由于在该模型中存在维数灾难,近年来,它的一些低维转化模型陆续被提出.在本文中,我们将介绍严格关联电子能量的优化模型、它的研究重点以及现有的一些低维转化模型.我们也将介绍这些转化模型的数值求解方法,并探讨未来的研究方向.     

电子结构模型与计算的若干数学问题 献给林群教授80华诞

又快又好地计算大规模的电子结构是极具挑战性的课题,而电子结构模型及其数学基础与数学性质在理解、分析与设计第一原理电子结构计算方法中发挥着重要作用.本文介绍作者所在小组在电子结构模型的数学基础和电子结构计算的方法与理论的研究中关注的若干数学问题.     

类Hartree-Fock方程的数值方法

本文的主要目的是介绍近年来大基组下的类Hartree-Fock方程数值求解的一些进展.类Hartree-Fock方程出现在Hartree-Fock理论和含杂化泛函的Kohn-Sham密度泛函理论中,是电子结构理论中一类重要的方程.该方程在复杂的化学和材料体系的电子结构计算中有广泛地应用.由于计算代价的原因,类Hartree-Fock方程一般只被用在较小规模的量子体系（含几十到几百个电子）的计算.从数学角度上讲,类Hartree-Fock方程是一个非线性积分-微分方程组,其计算代价主要来自于积分算子的部分,也就是Fock交换算子.通过发展和结合自适应压缩交换算子方法（ACE）,投影的C-DⅡS方法（PC-DⅡS）方法,以及插值可分密度近似方法（ISDF）,我们大大降低了杂化泛函密度泛函理论的计算代价.以含1000个硅原子的体系为例,我们将平面波基组下的杂化泛函的计算代价降至接近不含Fock交换算子的半局域泛函计算的水平.同时,我们发现类Hartree-Fock方程的数学结构也为一类特征值问题的迭代求解提供了新的思路.     

全空间中一类非局部问题非平凡解的存在性

利用变分方法研究了R~N上一类带有临界非线性项的p-Kirchhoff型问题非平凡解的存在性.首先得到了该问题的能量泛函并证明了其具有山路引理的几何结构.其次给出了山路值c的一个上界并且证明了相应的(PS)_c序列是有界的.最终利用集中紧性原理及其它相关知识证明了能量泛函满足(PS)_c条件,从而表明了能量泛函存在非零的临界点,即证明了该问题至少存在一个非平凡解.     

构建复杂系统的解景观

很多交叉科学的实际问题在数学上都可以被归为求解具有多个变量的非线性函数或泛函的极小值问题,如何有效地寻找其能量景观的全局极小和如何找到不同极小之间的关系是计算数学领域两个长久以来尚未解决的重要科学问题.本文着重介绍近年来提出的“解景观”概念和方法.我们将回顾解景观的概念、构建解景观的鞍点动力学方法、以及解景观在液晶和准晶方面的应用.     

晶体微观结构的数值计算

晶体微观结构是晶体材料在特定物理条件下其多个能量极小平衔态在空间形成的某种微尺度的规则分布．几何非线性的连续介质力学理论可以用能量极小化原理来解释晶体微观结构的形成，并用Young测度来刻画平衡态各变体在空间的概率分布．定性的理解与定量地分析和计算晶体材料的微观结构对于发展和改进高级晶体功能材料，如形状记忆合金、铁电体、磁至伸缩材料等，有重要的意义．本文回顾了近年来晶体微观结构数值计算方面的最新进展．介绍了计算晶体微观结构的几种数值方法及有关的数值分析结果。     

含有对数非线性项的p-Laplace方程的多重解

主要通过变分方法研究了有界区域上含有变号权函数和对数非线性项的一类p-Laplace方程Dirichlet边值问题的多解性.通过分解能量泛函的Nehari流形,利用对数Sobolev不等式,极小化序列方法及相关知识证明了能量泛函至少存在两个非零极小元,从而证明了问题至少存在两个非平凡解.     

带摩擦的弹性接触问题广义变分不等原理的简化证明

在弹性摩擦接触问题中 ,从变分原理出发来研究接触问题 ,可以将摩擦力纳入问题的能量泛函 .为了得到摩擦约束弹性接触问题的能量泛函 ,日前大多是用拉格朗日乘子法 ,但拉格朗日方法用在变分不等问题中 ,要利用非线性泛函分析和凸分析来证明 ,证明复杂 .本文利用向量分析的工具及巧妙的变换 ,对带摩擦约束的弹性接触问题的广义变分不等原理进行了严格的证明 ,由于只用到向量分析 ,简化了证明 .     

一般壳体组合结构在均布压力下的平衡与稳定

本文采用几何非线性理论建立一般壳体组合结构的能量泛函,再根据能量极值原理提出了这类组合结构在均布压力下的平衡和稳定的非线性有限元计算方法。计算结果与实验结果符合较好。     

关于含非齐次Dirichlet边值的Brzis-Nirenberg问题的研究

该文研究了含非齐次Dirichlet边值的Brezis-Nirenberg方程对应泛函的Nehari流形的结构.并结合Lusternik-Schnirelman畴数理论和极大极小原理,证明了含非齐次Dirichlet边值的Brezis-Nirenberg方程存在4个正解.     

h — P Finite Element Approximation for Pull-Potential Electronic Structure Calculations

The （continuous） finite element approximations of different orders for the computation of the solution to electronic structures were proposed in some papers and the performance of these approaches is becoming appreciable and is now well understood. In this publication, the author proposes to extend this discretization for full-potential electronic structure calculations by combining the refinement of the finite element mesh, where the solution is most singular with the increase of the degree of the polynomial approximations in the regions where the solution is mostly regular. This combination of increase of approximation properties, done in an a priori or a posteriori manner, is well-known to generally produce an optimal exponential type convergence rate with respect to the number of degrees of freedom even when the solution is singular. The analysis performed here sustains this property in the case of Hartree-Fock and Kohn-Sham problems.     

Convergent and Orthogonality Preserving Schemes for Approximating the Kohn-Sham Orbitals

To obtain convergent numerical approximations without using any orthogonalization operations is of great importance in electronic structure calculations. In this paper, we propose and analyze a class of iteration schemes for the discretized Kohn-Sham Density Functional Theory model, with which the iterative approximations are guaranteed to converge to the Kohn-Sham orbitals without any orthogonalization as long as the initial orbitals are orthogonal and the time step sizes are given properly. In addition, we present a feasible and efficient approach to get suitable time step sizes and report some numerical experiments to validate our theory.     

An Implicit Solver for the Time-Dependent Kohn-Sham Equation

The implicit numerical methods have the advantages on preserving the physical properties of the quantum system when solving the time-dependent Kohn-Sham equation. However, the efficiency issue prevents the practical applications of those implicit methods. In this paper, an implicit solver based on a class of Runge-Kutta methods and the finite element method is proposed for the time-dependent Kohn-Sham equation. The efficiency issue is partially resolved by three approaches, i.e., an $h$-adaptive mesh method is proposed to effectively restrain the size of the discretized problem, a complex-valued algebraic multigrid solver is developed for efficiently solving the derived linear system from the implicit discretization, as well as the OpenMP based parallelization of the algorithm. The numerical convergence, the ability on preserving the physical properties, and the efficiency of the proposed numerical method are demonstrated by a number of numerical experiments.     

Solving non-linear portfolio optimization problems with the primal-dual interior point method

Stochastic programming is recognized as a powerful tool to help decision making under uncertainty in financial planning. The deterministic equivalent formulations of these stochastic programs have huge dimensions even for moderate numbers of assets, time stages and scenarios per time stage. So far models treated by mathematical programming approaches have been limited to simple linear or quadratic models due to the inability of currently available solvers to solve NLP problems of typical sizes. However stochastic programming problems are highly structured. The key to the efficient solution of such problems is therefore the ability to exploit their structure. Interior point methods are well-suited to the solution of very large non-linear optimization problems. In this paper we exploit this feature and show how portfolio optimization problems with sizes measured in millions of constraints and decision variables, featuring constraints on semi-variance, skewness or non-linear utility functions in the objective, can be solved with the state-of-the-art solver.     

All-electron calculations with finite elements

The Kohn-Sham equations resemble a nonlinear eigenvalue problem for the determination of the electronic structure of an atomic system, where the electrons are exposed to an effective potential, accounting for the Coulomb and quantum mechanical interactions between the particles. The effectiveness of the potential requires an iterative solution procedure, until self-consistency is reached. This work illustrates the implementation of the self consistent field algorithm based on nested finite elements spaces and analyzes its properties in the case of all-electron calculations on atoms as large as the noble gas Xenon. All-electron calculations have maximal requirements onto the numerical basis, as it must be able to represent all the orthogonal electronic wavefunctions simultaneously together with the electrostatic potential, showing singularities at the positions of the atoms. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Properties and applications of forests of quadtrees for pictorial data representation

Region representation as a quadtree data structure is a rich field in computer science with many different approaches. Forests of quadtrees offer space savings over regular quadtrees by concentrating the vital information [4, 5, 6]. They scavenge unused and unneeded space (i.e., node containing no information). This paper investigates several properties of forests of quadtrees which can be used to design manipulation algorithms for forest-quadtree data structure. In addition, the paper discusses the space saving and shows how the basic operations that can be performed on a quadtree can also be done on the more space efficient representation (a forest of quadtrees).     

An efficient computational approach for multiframe blind deconvolution

Obtaining high resolution images of space objects from ground based telescopes involves using a combination of sophisticated hardware and computational post-processing techniques. An important, and often highly effective, computational post processing tool is multiframe blind deconvolution (MFBD). Mathematically, MFBD is modeled as a nonlinear inverse problem that can be solved using a flexible, variable projection optimization approach. In this paper we consider MFBD problems that are parameterized by a large number of variables. The formulas required for efficient implementation are carefully derived using the spectral decomposition and by exploiting properties of conjugate symmetric vectors. In addition, a new approach is proposed to provide a mathematical decoupling of the optimization problem, leading to a block structure of the Jacobian matrix. An application in astronomical imaging is considered, and numerical experiments illustrate the effectiveness of our approach.     

Finite element solution of the Kohn-Sham equations

Many properties of condensed matter as for example electric conductivity, magnetism as well as the mechanical response upon external excitations are determined by the electronic structure of the material. Its evaluation represents a coupled, quantum mechanical many body problem, consisting of the positively charged atomic cores and the negatively charged, fermionic electrons. By the Hohenberg-Kohn theorem a method was established to calculate the ground state electron density through a density functional. Then the Kohn-Sham equations resemble a nonlinear, single electron problem with an effective potential, accounting for the Coulomb interactions between the particles as well as for quantum mechanical effects. They can be solved within a self consistent field procedure. In this work a real space formulation for the calculation of the electronic structure in the context of density functional theory is shown. In a first step a finite element based solution algorithm for the Kohn-Sham equations is developed, based upon which the electron density is obtained in a non-periodic setting. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analytical modeling of active structural acoustic control

Since the reduction of disturbing noise is highly important in industrial and civil engineering, much research and development has been done in this field over the past decades. The application of smart structures provides a way to reduce unwanted noise radiation. A smart structure is an integrated system consisting of the passive base structure attached with actuators and sensors. Piezoelectric ceramics are widely used as sensors and actuators, because they can easily be bonded on or imbedded into conventional structures. An often used concept for noise reduction is active structural acoustic control (ASAC). This concept means that the noise radiation will be reduced by minimizing or changing the vibrational behavior of a structure using additional actuator forces. The development and industrial use of smart structures for ASAC requires efficient and reliable simulation and design tools. Usually the design process of smart structures is based on numerical methods. In order to verify numerical techniques, the paper presents an analytical reference solution for an ASAC system. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)