Inverse Eigenvalue Problems for Exploring the Dynamics of Systems Biology Models

This paper describes inverse eigenvalue problems that arise in studying qualitative dynamics in systems biology models. An algorithm based on lift-and-project iterations is proposed, where the lifting step entails solving a constrained matrix inverse eigenvalue problem. In particular, prior to carrying out the iterative steps, $a$-$priori$ bounds on the entries of the Jacobian matrix are computed by relying on the reaction network structure as well as the form of the rate law expressions for the model under consideration. Numerical results on a number of models show that the proposed algorithm can be used to computationally explore the possible dynamical scenarios while identifying the important mechanisms via the use of sparsity-promoting regularization.     

Hamilton–Jacobi Treatment of Constrained Systems with Second-Order Lagrangians

A new approach is examined in this paper for solving mechanical problems for both constrained and unconstrained systems with second-order Lagrangians, using the Hamilton–Jacobi formulation. The relevant Hamilton–Jacobi function is constructed first. This is then used to determine the solutions of the equations of motion for both systems.     

Dynamical consequences of a constraint on the Langevin thermostat in molecular cluster simulation

We investigate some unusual behaviour observed while performing molecular dynamics simulations of small molecular clusters using a constrained Langevin thermostat. Atoms appear to be thermalised to different temperatures that depend on their mass and on the total number of particles in the system. The deviation from the zeroth law of thermodynamics can be considerable for small systems of heavy and light particles. We trace this behaviour to the absence of thermal noise acting on the centre of mass of the system. This is demonstrated by solving the stochastic dynamics for the constrained thermostat and comparing the results with simulation data. By removing the constraint, the Langevin thermostat may be restored to its intended behaviour. We also investigate a Langevin thermostat constrained to have zero total force acting on its centre of mass, and find similar deficiencies.     

一种二维不稳定流形的新算法及其应用

提出了连续时间系统二维(不)稳定流形的一种新数值算法,不但可以快速地求得流形的直观图像,而且能够准确地获取流形上各点的位置、时间、轨道距离等丰富的信息,从而有利于人们从几何上去研究系统的全局行为,如边界特征、演化过程、奇异环等等.本算法首先通过解初值问题求出均匀分布的相邻轨道,然后连接这些轨道既得流形面.Lorenz系统原点的稳定流形的计算表明本算法快速有效.此外,通过试着寻找异宿轨道,还研究了一个三维神经网络中的混沌产生机理.     

An unsplit staggered mesh scheme for multidimensional magnetohydrodynamics

We introduce an unsplit staggered mesh scheme (USM) for multidimensional magnetohydrodynamics (MHD) that uses a constrained transport (CT) method with high-order Godunov fluxes and incorporates a new data reconstruction–evolution algorithm for second-order MHD interface states. In this new algorithm, the USM scheme includes so-called “multidimensional MHD terms”, proportional to $?·\mathbf{B}$, in a dimensionally-unsplit way in a single update. This data reconstruction–evolution step, extended from the corner transport upwind (CTU) approach of Colella, maintains in-plane dynamics very well, as shown by the advection of a very weak magnetic field loop in 2D. This data reconstruction–evolution algorithm is also of advantage in its consistency and simplicity when extended to 3D. The scheme maintains the $?·\mathbf{B}=0$ constraint by solving a set of discrete induction equations using the standard CT approach, where the accuracy of the computed electric field directly influences the quality of the magnetic field solution. We address the lack of proper dissipative behavior in the simple electric field averaging scheme and present a new modified electric field construction (MEC) that includes multidimensional derivative information and enhances solution accuracy. A series of comparison studies demonstrates the excellent performance of the full USM–MEC scheme for many stringent multidimensional MHD test problems chosen from the literature. The scheme is implemented and currently freely available in the University of Chicago ASC FLASH Center’s FLASH3 release.     

标准对力模型的新迭代方法(英文)

建立了一种求解标准对力模型的新迭代方法。该方法基于标准对力模型的多项式方案,为球形和形变系统提供了方便的初始值预测。特别是对于大尺寸系统,该方法将求解k对多项式的系统方程式简化为分步求解1对多项式系统的迭代过程,并通过快速Newton-Raphson以及Monte Carlo采样算法逐步提供初始值预测。通过扩展,本算法还可用于解决Gaudin型量子多体问题,例如考虑超过100条轨道、50对的大尺寸系统,以及超形变核、核裂变的研究中。A new iterative approach for solving the standard pairing problem is established based on polynomial approach. It provides an efficient way to derive the particle-number conserved pairing wave functions for both spherical and deformed systems, especially for large-size systems. The method reduces the complexity of solving a system for k-pairs polynomial equations into a system for one-pair polynomial equation, which can be efficiently implemented by the Newton-Raphson algorithm with a Monte Carlo sampling procedure for providing the initial guesses step by step. The present algorithm can also be used to solve a large class of Gaudin type quantum many-body problems as a more than 100 orbitals and 50 pairs system such as super-heavy nuclei and nuclear fission.     

SHAKE parallelization

SHAKE is a widely used algorithm to impose general holonomic constraints during molecular simulations. By imposing constraints on stiff degrees of freedom that require integration with small time steps (without the constraints) we are able to calculate trajectories with time steps larger by approximately a factor of two. The larger time step makes it possible to run longer simulations. Another approach to extend the scope of Molecular Dynamics is parallelization. Parallelization speeds up the calculation of the forces between the atoms and makes it possible to compute longer trajectories with better statistics for thermodynamic and kinetic averages. A combination of SHAKE and parallelism is therefore highly desired. Unfortunately, the most widely used SHAKE algorithm (of bond relaxation) is inappropriate for parallelization and alternatives are needed. The alternatives must minimize communication, lead to good load balancing, and offer significantly better performance than the bond relaxation approach. The algorithm should also scale with the number of processors. We describe the theory behind different implementations of constrained dynamics on parallel systems, and their implementation on common architectures.     

Sparsity constrained deconvolution approaches for acoustic source mapping

Using microphone arrays for estimating source locations and strengths has become common practice in aeroacoustic applications. The classical delay-and-sum approach suffers from low resolution and high sidelobes and the resulting beamforming maps are difficult to interpret. The deconvolution approach for the mapping of acoustic sources (DAMAS) deconvolution algorithm recovers the actual source levels from the contaminated delay-and-sum results by defining an inverse problem that can be represented as a linear system of equations. In this paper, the deconvolution problem is carried onto the sparse signal representation area and a sparsity constrained deconvolution approach (SC-DAMAS) is presented for solving the DAMAS inverse problem. A sparsity preserving covariance matrix fitting approach (CMF) is also presented to overcome the drawbacks of the DAMAS inverse problem. The proposed algorithms are convex optimization problems. Our simulations show that CMF and SC-DAMAS outperform DAMAS and as the noise in the measurements increases, CMF works better than both DAMAS and SC-DAMAS. It is observed that the proposed algorithms converge faster than DAMAS. A modification to SC-DAMAS is also provided which makes it significantly faster than DAMAS and CMF. For the correlated source case, the CMF-C algorithm is proposed and compared with DAMAS-C. Improvements in performance are obtained similar to the uncorrelated case.     

高阶非完整约束系统嵌入变分恒等式的积分变分原理

本文从高阶非完整系统嵌入变分恒等式的积分变分原理出发, 根据三种不等价条件变分的选取, 得到了高阶非完整系统的三类不等价动力学模型, 即高阶非完整约束系统的vakonomic方程、Lagrange-d'Alembert 方程和一种新的动力学方程. 当高阶非完整约束方程退化为一阶非完整约束时, 利用此理论可以得到一般非完整系统的vakonomic模型、Chetaev模型和一种新的动力学模型. 最后借助于应用实例验证了结论的正确性. 关键词： 高阶非完整约束 变分恒等式 条件变分 vakonomic动力学     

Simulating nanoscale dielectric response

We introduce a constrained energy functional to describe dielectric response. We demonstrate that the local functional is a generalization of the long-ranged Marcus energy. Our reformulation is used to implement a cluster Monte Carlo algorithm for the simulation of dielectric media. The algorithm avoids solving the Poisson equation and remains efficient in the presence of spatial heterogeneity, nonlinearity, and scale dependent dielectric properties.     

New approach to spin dynamics simulation in magnetic system

New approach to spin dynamics simulation in magnetic system is presented. In the approach, we substitute new algorithm for Landau-Lifshitz-Gilbert dynamics, which enable us to achieve the final stable state of magnetic system much faster than traditional spin dynamics simulation. A square-shaped sample with 32×32×4 size under different conditions is calculated. Our results show the new approach can largely reduce the computational time.     

New computer program for three-dimensional relativistic hydrodynamical model

An effective computer program for three-dimensional relativistic hydrodynamical model has been developed. It implements a new approach to the early hot phase of relativistic heavy-ion collisions. The computer program simulates time-space evolution of nuclear matter in terms of ideal-fluid dynamics. Equations of motions of hydrodynamics are solved making use of finite difference methods. Commonly-used algorithms of numerical relativistic hydrodynamics RHLLE and MUSTA-FORCE have been applied in simulations. To speed-up calculations, parallel processing has been made available for solving hydrodynamical equations. The test results of simulations for 3D, 2D and Bjorken expansion are reported in this paper. As a next step we plan to implement the hadronization algorithm by implementing the continuous particle emission for freeze-out and comparing it with Cooper-Frye formula.     

Failsafe flux limiting and constrained data projections for equations of gas dynamics

A new approach to flux limiting for systems of conservation laws is presented. The Galerkin finite element discretization/L² projection is equipped with a failsafe mechanism that prevents the birth and growth of spurious local extrema. Within the framework of a synchronized flux-corrected transport (FCT) algorithm, the velocity and pressure fields are constrained using node-by-node transformations from the conservative to the primitive variables. An additional correction step is included to ensure that all the quantities of interest (density, velocity, pressure) are bounded by the physically admissible low-order values. The result is a conservative and bounded scheme with low numerical diffusion. The new failsafe FCT limiter is integrated into a high-resolution finite element scheme for the Euler equations of gas dynamics. Also, bounded L² projection operators for conservative interpolation/initialization are designed. The performance of the proposed limiting strategy and the need for a posteriori control of flux-corrected solutions are illustrated by numerical examples.     

一个新的求解点堆中子动力学方程组的数值方法

使用中子密度一阶泰劳多项式分段近似技术,给出一个新的求解点堆中子动力学方程组的数值方法并采用全隐格式以克服方程组的刚性,同时确保解的必要精度。数值结果表明:在隐式一阶多项式近似下,对合适的反应性输入能够取得足够精确的结果。当反应性给定时,对于求解反应堆动力学问题,能给出一个简法的计算过程。     

The CP^1 nonlinear sigma model with ChernSimons term in the Faddeev-Jachiw quantization formalism

Using the Faddeev-Jackiw （FJ） quantization method, this paper treats the CP^1nonlinear sigma model with ChernSimons term. The generalized FJ brackets are obtained in the framework of this quantization method, which agree with the results obtained by using the Dirac＇s method.     

Algebraic dynamics solutions and algebraic dynamics algorithm for nonlinear partial differential evolution equations of dynamical systems

Using functional derivative technique in quantum field theory, the algebraic dynamics approach for solution of ordinary differential evolution equations was generalized to treat partial differential evolution equations. The partial differential evolution equations were lifted to the corresponding functional partial differential equations in functional space by introducing the time translation operator. The functional partial differential evolution equations were solved by algebraic dynamics. The algebraic dynamics solutions are analytical in Taylor series in terms of both initial functions and time. Based on the exact analytical solutions, a new numerical algorithm—algebraic dynamics algorithm was proposed for partial differential evolution equations. The difficulty of and the way out for the algorithm were discussed. The application of the approach to and computer numerical experiments on the nonlinear Burgers equation and meteorological advection equation indicate that the algebraic dynamics approach and algebraic dynamics algorithm are effective to the solution of nonlinear partial differential evolution equations both analytically and numerically. Supported by the National Natural Science Foundation of China (Grant Nos. 10375039, 10775100 and 90503008), the Doctoral Program Foundation of the Ministry of Education of China, and the Center of Nuclear Physics of HIRFL of China     

Quasi-physical Algorithm for Protein Folding in an Off-Lattice Model

We study a three-dimensional off-lattice protein folding model, which involves two species of residues interacting through Lennard-Jones potentials. By incorporating an extra energy contribution into the original potential function, we replace the original constrained problem with an unconstrained minimization of a mixed potential function. As such an efficient quasi-physical algorithm for solving the protein folding problem is presented. We apply the proposed algorithm to sequences with up to 55 residues and compare the computational results with the putative lowest energy found by several of the most famous algorithms, showing the advantages of our method. The dynamic behavior of the quasi-physlcal algorithm is also discussed.     

Constrained molecular dynamics in the isothermal-isobaric ensemble and its adaptation for adiabatic free energy dynamics

The implementation of holonomic constraints within measure-preserving integrators for molecular dynamics simulations in the isothermal-isobaric ensemble is considered. We review the basic methodology of generating measure-preserving integrators for the microcanonical, canonical, and isothermal-isobaric ensembles and proceed to show how the standard SHAKE and RATTLE algorithms must be modified for the isothermal-isobaric ensemble. Comparison is made between constrained and unconstrained simulations employing multiple time scale integration techniques. Finally, we describe a temperature accelerated version of the isothermal-isobaric molecular dynamics approach, in which the cell matrix is adiabatically decoupled from the particles and maintained at a high temperature as a means of exploring polymorphism in molecular crystals. We demonstrate that constraints can be easily adapted for this new approach and, again, we compare the performace of this temperature-accelerated scheme with and without bond constraints.     

Visual enhancement based on salient region detection and layered difference representation

We propose a novel image enhancement method based on salient region detection and a layered difference representation of 2D histograms. We first obtain the visual salient region corresponding to maximal human attention using saliency filters. Then, we obtain a difference vector for the visual salient region by solving a constrained optimization problem of the layered difference representation at a specified layer. Finally, the new difference vector and the difference vector of the original image are aggregated to enhance the salient region and protect other regions from overstretching or brightness shift. Experimental results including comparisons with other methods show that our proposed algorithm produces more suitable enhanced images compared with the results of existing algorithms.