An N log N approximation based on the natural organization of biomolecules for speeding up the computation of long range interactions

Presented here is a method, the hierarchical charge partitioning (HCP) approximation, for speeding up computation of pairwise electrostatic interactions in biomolecular systems. The approximation is based on multiple levels of natural partitioning of biomolecular structures into a hierarchical set of its constituent structural components. The charge distribution in each component is systematically approximated by a small number of point charges, which, for the highest level component, are much fewer than the number of atoms in the component. For short distances from the point of interest, the HCP uses the full set of atomic charges available. For long‐distance interactions, the approximate charge distributions with smaller sets of charges are used instead. For a structure consisting of N charges, the computational cost of computing the pairwise interactions via the HCP scales as O(N log N), under assumptions about the structural organization of biomolecular structures generally consistent with reality. A proof‐of‐concept implementation of the HCP shows that for large structures it can lead to speed‐up factors of up to several orders of magnitude relative to the exact pairwise O(N²) all‐atom computation used as a reference. For structures with more than 2000–3000 atoms the relative accuracy of the HCP (relative root‐mean‐square force error per atom), approaches the accuracy of the particle mesh Ewald (PME) method with parameter settings typical for biomolecular simulations. When averaged over a set of 600 representative biomolecular structures, the relative accuracies of the two methods are roughly equal. The HCP is also significantly more accurate than the spherical cutoff method. The HCP has been implemented in the freely available nucleic acids builder (NAB) molecular dynamics (MD) package in Amber tools. A 10 ns simulation of a small protein indicates that the HCP based MD simulation is stable, and that it can be faster than the spherical cutoff method. A critical benefit of the HCP approximation is that it is algorithmically very simple, and unlike the PME, the HCP is straightforward to use with implicit solvent models. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2010     

Assessment of linear finite‐difference Poisson–Boltzmann solvers

CPU time and memory usage are two vital issues that any numerical solvers for the Poisson–Boltzmann equation have to face in biomolecular applications. In this study, we systematically analyzed the CPU time and memory usage of five commonly used finite‐difference solvers with a large and diversified set of biomolecular structures. Our comparative analysis shows that modified incomplete Cholesky conjugate gradient and geometric multigrid are the most efficient in the diversified test set. For the two efficient solvers, our test shows that their CPU times increase approximately linearly with the numbers of grids. Their CPU times also increase almost linearly with the negative logarithm of the convergence criterion at very similar rate. Our comparison further shows that geometric multigrid performs better in the large set of tested biomolecules. However, modified incomplete Cholesky conjugate gradient is superior to geometric multigrid in molecular dynamics simulations of tested molecules. We also investigated other significant components in numerical solutions of the Poisson–Boltzmann equation. It turns out that the time‐limiting step is the free boundary condition setup for the linear systems for the selected proteins if the electrostatic focusing is not used. Thus, development of future numerical solvers for the Poisson–Boltzmann equation should balance all aspects of the numerical procedures in realistic biomolecular applications. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

The energy conserving particle-in-cell method

A new particle-in-cell (PIC) method, that conserves energy exactly, is presented. The particle equations of motion and the Maxwell’s equations are differenced implicitly in time by the midpoint rule and solved concurrently by a Jacobian-free Newton Krylov (JFNK) solver. Several tests show that the finite grid instability is eliminated in energy conserving PIC simulations, and the method correctly describes the two-stream and Weibel instabilities, conserving exactly the total energy. The computational time of the energy conserving PIC method increases linearly with the number of particles, and it is rather insensitive to the number of grid points and time step. The kinetic enslavement technique can be effectively used to reduce the problem matrix size and the number of JFNK solver iterations.     

一类求解单调变分不等式的隐式方法

1．引言变分不等式是一个非常有趣。非常困难的数学问题［"］．它具有广泛的应用（例如，数学规划中的许多基本问题都可以归结为一个变分不等式问题），因而得到深入的研究并有了不少算法［1，2，5－8，17－21］．对线性单调变分不等式，我们最近提出了一系列投影收缩算法Ig－13］．本文考虑求解单调变分不等式其中0CW是一闭凸集，F是从正p到自身的一个单调算子，一即有我们用比（·）表示到0上的投影．求解单调变分不等式的一个简单方法是基本投影法［1，6］，它的迭代式为然而，如果F不是仿射函数，只有当F一致强单调且LIPSChitZ连续…     

Traveling wave solutions of a model of skin pattern formation in a singular case

We study traveling wave solutions to a system of four non‐linear partial differential equations, which arise in a tissue interaction model for skin morphogenesis. Under the assumption that the strength of attachment of the epidermis to the basal lamina is sufficiently large, we prove the existence and uniqueness (up to a translation) of traveling wave solutions connecting two stationary states of the system with the dermis and epidermis cell densities being positive. We discuss the problem of the minimal wave speed. Copyright © 2010 John Wiley & Sons, Ltd.     

Strong Convergence Theorems for the Generalized Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in the Intermediate Sense in Hilbert Spaces

Abstract

The aim of this paper is to introduce the generalized viscosity implicit rules of one asymptotically nonexpansive mapping in the intermediate sense in Hilbert spaces. We obtain some strong convergence theorems under certain assumptions imposed on the parameters. We also give a numerical example to support our main results. The results obtained in this paper improve and extend many recent ones in this culture.     

Dynamics of implicit operations and tameness of pseudovarieties of groups

This work gives a new approach to the construction of implicit operations. By considering ``higher-dimensional' spaces of implicit operations and implicit operators between them, the projection of idempotents back to one-dimensional spaces produces implicit operations with interesting properties. Besides providing a wealth of examples of implicit operations which can be obtained by these means, it is shown how they can be used to deduce from results of Ribes and Zalesski, Margolis, Sapir and Weil, and Steinberg that the pseudovariety of -groups is tame. More generally, for a recursively enumerable extension closed pseudovariety of groups , if it can be decided whether a finitely generated subgroup of the free group with the pro- topology is dense, then is tame.

     

基于混合网格Navier-Stokes方程的并行隐式计算方法研究

针对结构网格很难处理复杂外形和非结构网格无法计算具有边界层的粘性流动的缺点,发展了基于混合网格格点的隐式算法,成功地解决了在工程应用中难于处理的复杂外形粘性流场计算和效率问题。同时针对大规模的工程问题,发展了基于MPI通信技术的染色分层通讯并行计算方法。其中空间离散采用基于Roe格式发展的三阶迎风HLLEW(Harten-Lax-Van Leer-Einfeldt-Wada)或AUSM格式,湍流模型采用k??两方程湍流模型,时间推进考虑到LU-SGS并行等效较困难则采用基于DP-LUR(Data-Parallel Lower-Upper Relaxation)格式的隐式算法,计算CFL数可取到105量级,从2个到128个CPU的并行加速效率都保持在90%以上,大大提高了计算效率。算例对标模M6机翼模型流场进行计算,验证了方法的可靠性;然后对标模DLR-F6翼身组合体进行混合网格粘性与无粘计算结果进行比较,进一步验证混合网格方法;最后计算了DLR-WBNP外挂发动机翼身组合体模型,准确模拟了外挂和超临界机翼的相互干扰流动问题,采用4 CPU 16 CORE到24 CPU 96 CORE,2000步计算时间都不超...     

广义复合隐迭代格式逼近非扩张映像公共不动点

提出并使用如下广义复合隐迭代格式逼近非扩张映像族{Ti}Ni=1公共不动点:{xn=αnxn-1 (1-αn)Tnyn,yn=rnxn snxn-1 tnTnxn wnTnxn-1,rn sn tn wn=1,{αn},{rn},{sn},{tn},{wn}∈[0,1],这里Tn=TnmodN.该文提出的广义复合隐迭代格式包含了目前多种迭代格式,因此,所得强弱收敛定理推广及发展了Mann,Ishikawa,XuandOri,等许多作者的结果.     

抛物型方程的一个高精度隐格式

利用待定参数法,对一维抛物型方程构造出了一个截断误差为Ｏ（△ｘ＾４＋△ｘ＾４）的隐式差分格式,格式的稳定性条件为ｒ＝ａ△ｔ／△ｘ＾２≤１／√２,可用追赶法求解。