Asymptotic Convergence of Conjugate Gradient Methods for the Partial Symmetric Eigenproblem

Recently an efficient method (DACG) for the partial solution of the symmetric generalized eigenproblem A x = δB x has been developed, based on the conjugate gradient (CG) minimization of the Rayleigh quotient over successive deflated subspaces of decreasing size. The present paper provides a numerical analysis of the asymptotic convergence rate ρ_j of DACG in the calculation of the eigenpair λ_j, u _j, when the scheme is preconditioned with A⁻¹. It is shown that, when the search direction are A-conjugate, ρ_j is well approximated by 4/ξ_j, where ξ_j is the Hessian condition number of a Rayleigh quotient defined in appropriate oblique complements of the space spanned by the leftmost eigenvectors u ₁, u ₂,…, u _j−1 already calculated. It is also shown that 1/ξ_j is equal to the relative separation between the eigenvalue λ_j currently sought and the next higher one λ_j+1 and the next higher one λ_{j + 1}. A modification of DACG (MDACG) is studied, which involves a new set of CG search directions which are made M-conjugate, with M-conjugate, with M-conjugate, with M a matrix approximating the Hessian. By distinction, MDACG has an asymptotic rate of convergence which appears to be inversely proportional to the square root of ξ_j, in complete agreement with the theoretical results known for the CG solution to linear systems. © 1997 by John Wiley & Sons, Ltd.     

A dynamical systems model of unorganized segregation

We consider Schelling’s bounded neighborhood model (BNM) of unorganized segregation, from the perspective of modern dynamical systems theory. We carry out a complete quantitative analysis of the system for linear tolerance schedules. We derive a fully predictive model and associate each term with a social meaning. We recover and generalize Schelling’s qualitative results.

For the case of unlimited population movement, we derive exact formulae for regions in parameter space where stable integrated population mixes can occur, and show how neighborhood tipping can be explained in terms of basins of attraction.

When population movement is limited, we derive exact criteria for the occurrence of new population mixes.

For nonlinear tolerance schedules, we illustrate our approach with numerical simulations.     

Feedback minimization of first-passage failure of quasi non-integrable Hamiltonian systems

The non-linear stochastic optimal control of quasi non-integrable Hamiltonian systems for minimizing their first-passage failure is investigated. A controlled quasi non-integrable Hamiltonian system is reduced to an one-dimensional controlled diffusion process of averaged Hamiltonian by using the stochastic averaging method for quasi non-integrable Hamiltonian systems. The dynamical programming equations and their associated boundary and final time conditions for the problems of maximization of reliability and of maximization of mean first-passage time are formulated. The optimal control law is derived from the dynamical programming equations and the control constraints. The dynamical programming equations for maximum reliability problem and for maximum mean first-passage time problem are finalized and their relationships to the backward Kolmogorov equation for the reliability function and the Pontryagin equation for mean first-passage time, respectively, are pointed out. The boundary condition at zero Hamiltonian is discussed. Two examples are worked out to illustrate the application and effectiveness of the proposed procedure.     

THE APPLICATION OF RANDOM MATRICES IN MATHEMATICAL PHYSICS

In this paper, we introduce the application of random matrices in mathematical physics including Riemann-Hilbert problem, nuclear physics, big data, image processing, compressed sensing and so on. We start with the Riemann-Hilbert problem and state the relation between the probability distribution of nontrivial zeros and the eigenvalues of the random matrices. Through the random matrices theory, we derive the distribution of Neutron width and probability density between energy levels. In addition, the application of random matrices in quantum chromo dynamics and two dimensional Einstein gravity equations is also present in this paper.     

Harnack’s inequality for a class of non-divergent equations in the Heisenberg group

We prove an invariant Harnack’s inequality for operators in non-divergence form structured on Heisenberg vector fields when the coe?cient matrix is uniformly positive definite, continuous, and symplectic. The method consists in constructing appropriate barriers to obtain pointwise-to-measure estimates for supersolutions in small balls, and then invoking the axiomatic approach developed by Di Fazio, Gutiérrez, and Lanconelli to obtain Harnack’s inequality.     

建立基于力的变截面梁单元质量矩阵的新方法

基于位移的有限梁单元中三次Hermite插值函数不能有效地描述变截面梁单元内部位移变化,只能通过加密网格增加单元数解决,会造成计算量增大。基于力的有限梁单元由于使用的力插值函数不受截面形状变化的影响,在处理变截面梁时有很大优势,可以得到精确的位移插值函数,利用较少的单元可以达到很高的精度,解决了基于位移的有限梁单元在处理变截面梁时的不足。本文得到了考虑剪切变形的位移插值函数和考虑转动惯量的一致质量矩阵。利用算例验证了本文理论的正确性和高效性。