A Fast Discontinuous Galerkin Method for a Bond-Based Linear Peridynamic Model Discretized on a Locally Refined Composite Mesh

We develop a family of fast discontinuous Galerkin (DG) finite element methods for a bond-based linear peridynamic (PD) model in one space dimension. More precisely, we develop a preconditioned fast piecewise-constant DG scheme on a geometrically graded locally refined composite mesh which is suited for the scenario in which the jump discontinuity of the displacement field occurs at the one of the nodes in the original uniform partition. We also develop a preconditioned fast piecewise-linear DG scheme on a uniform mesh that has a second-order convergence rate when the jump discontinuity occurs at one of the computational nodes or has a convergence rate of one-half order otherwise. Motivated by these results, we develop a preconditioned fast hybrid DG scheme that is discretized on a locally uniformly refined composite mesh to numerically simulate the PD model where the jump discontinuity of the displacement field occurs inside a computational cell. We use a piecewise-constant DG scheme on a uniform mesh and a piecewise-linear DG scheme on a locally uniformly refined mesh of mesh size \(O(h^2)\), which has an overall convergence rate of O(h). Because of their nonlocal nature, numerical methods for PD models generate dense stiffness matrices which have \(O(N^2)\) memory requirement and \(O(N^3)\) computational complexity, where N is the number of computational nodes. In this paper, we explore the structure of the stiffness matrices to develop three preconditioned fast Krylov subspace iterative solvers for the DG method. Consequently, the methods have significantly reduced computational complexity and memory requirement. Numerical results show the utility of the numerical methods.     

A Static Analysis Method for a Classical Linear Logic Programming Language

In this paper, we propose a new static analysis method which is applicable for a classical linear logic programming language.Andreoli et al. proposed a static analysis method for the classical linear logic programming language LO, but their method did not cover multiplicative connectives which are important for a resource-sensitive feature of linear logic.Our method, in contrast, covers multiplicative conjunction in addition to multiplicative disjunction and linear implication. An abstract proof graph, an AND-OR graph representing all possible sequent proofs, is constructed from a given program and goal sequent. The graph can be repeatedly refined by propagating information to eliminate unprovable nodes from the graph.We applied our prototype analyzer for a sorting program written in Forum. The sorting program was improved about 1000 times faster than the ordinary program without analysis, for sorting 6 elements by using the analysis result.     

A Fast Finite Difference Method for Three-Dimensional Time-Dependent Space-Fractional Diffusion Equations with Fractional Derivative Boundary Conditions

We develop a fast finite difference method for time-dependent variable-coefficient space-fractional diffusion equations with fractional derivative boundary-value conditions in three dimensional spaces. Fractional differential operators appear in both of the equation and the boundary conditions. Because of the nonlocal nature of the fractional Neumann boundary operator, the internal and boundary nodes are strongly coupled together in the coupled linear system. The stability and convergence of the finite difference method are discussed. For the implementation, the development of the fast method is based upon a careful analysis and delicate decomposition of the structure of the coefficient matrix. The fast method has approximately linear computational complexity per Krylov subspace iteration and an optimal-order memory requirement. Numerical results are presented to show the utility of the method.     

线性连续系统模型离散化的递推算法

利用简单的数学工具实现了线性连续时间系统向离用时间系统的转换。算法既避免了矩阵的求逆运算，又可同时获得系统的脉冲传递函数，也可用于离散状态方程时的求解。仿真结果表明，所得结果对于连续时间系统的计算机仿真分析和控制是有效的。     

A Method for Fast Computation of the Fourier Transform over a Finite Field

We consider the problem of fast computation of the Fourier transform over a finite field by decomposing an arbitrary polynomial into a sum of linearized polynomials. Examples of algorithms for the Fourier transform with complexity less than that of the best known analogs are given.     

A Neural-based Model for Fast Continuous and Global Robot Location

One of the problems in the field of mobile robotics is the estimation of the robot position in an environment. This paper proposes a model for estimating a confidence interval of the robot position in order to compare it with the estimation made by a dead-reckoning system. Both estimations are fused using heuristic rules. The positioning model is very valuable in estimating the current robot position with or without knowledge about the previous positions. Furthermore, it is possible to define the degree of knowledge of the robot previous position, making it possible to adapt the estimation by varying this knowledge degree. This model is based on a one-pass neural network which adapts itself in real time and learns about the relationship between the measurements from sensors and the robot position.     

A Fast Sweeping Method for Static Convex Hamilton–Jacobi Equations

We develop a fast sweeping method for static Hamilton–Jacobi equations with convex Hamiltonians. Local solvers and fast sweeping strategies apply to structured and unstructured meshes. With causality correctly enforced during sweepings numerical evidence indicates that the fast sweeping method converges in a finite number of iterations independent of mesh size. Numerical examples validate both the accuracy and the efficiency of the new method. In memory of Xu-Dong Liu.     

一种基于故障模型的代码静态测试方法研究

软件测试是排除软件故障,提高软件质量和可靠性的重要手段。基于故障模型的软件测试是软件编码阶段的主流测试方法之一。基于故障模型的代码静态测试技术具有测试效率高、对逻辑复杂故障测试效果好等特点。鉴于此,本文采取一种特殊的静态分析技术来实现对代码的测试。首先讨论传统软件测试方法的缺点和局限性,给出基于故障模型的静态测试方法的优越性;然后在分析过程中,综合应用抽象语法树和控制流图,提出一种基于故障模型的软件测试方法。依据该算法开发自动化测试工具,给出实验结果和对比分析,并指出下一步的研究方向。     

A Fast Nonlinear Model Identification Method

The identification of nonlinear dynamic systems using linear-in-the-parameters models is studied. A fast recursive algorithm (FRA) is proposed to select both the model structure and to estimate the model parameters. Unlike orthogonal least squares (OLS) method, FRA solves the least-squares problem recursively over the model order without requiring matrix decomposition. The computational complexity of both algorithms is analyzed, along with their numerical stability. The new method is shown to require much less computational effort and is also numerically more stable than OLS.     

A $$C^0$$ Linear Finite Element Method for Biharmonic Problems

In this paper, a \(C^0\) linear finite element method for biharmonic equations is constructed and analyzed. In our construction, the popular post-processing gradient recovery operators are used to calculate approximately the second order partial derivatives of a \(C^0\) linear finite element function which do not exist in traditional meaning. The proposed scheme is straightforward and simple. More importantly, it is shown that the numerical solution of the proposed method converges to the exact one with optimal orders both under \(L^2\) and discrete \(H^2\) norms, while the recovered numerical gradient converges to the exact one with a superconvergence order. Some novel properties of gradient recovery operators are discovered in the analysis of our method. In several numerical experiments, our theoretical findings are verified and a comparison of the proposed method with the nonconforming Morley element and \(C^0\) interior penalty method is given.     

A Fast Method for Finding the Basis of Non-negative Solutions to a Linear Diophantine Equation

We present a complete characterization of the set of minimal solutions of a single linear Diophantine equation in three unknowns over the natural numbers. This characterization, for which we give a geometric interpretation, is based on well-known properties of congruences and we use it as the foundation of direct algorithms for solving this particular kind of equation. These direct algorithms and an enumeration procedure are then put together to build an algorithm for solving the general case of a Diophantine equation over the naturals. We also put forth a statistical method for comparing algorithms for solving Diophantine equations which is more sound than comparisons based on times observed for small sets of equations. From an extensive comparison with algorithms described by other authors it becomes clear that our algorithm is the fastest known to date for a class of equations. Typically the equations in this class have a small number of unknowns in one side, the maximum value for their coefficients being greater than 3.     

A Two-Grid Block-Centered Finite Difference Method for the Nonlinear Time-Fractional Parabolic Equation

In this article, a two-grid block-centered finite difference scheme is introduced and analyzed to solve the nonlinear time-fractional parabolic equation. This method is considered where the nonlinear problem is solved only on a coarse grid of size H and a linear problem is solved on a fine grid of size h. Stability results are proven rigorously. Error estimates are established on non-uniform rectangular grid which show that the discrete \(L^{\infty }(L^2)\) and \(L^2(H^1)\) errors are \(O(\triangle t^{2-\alpha }+h^2+H^3)\). Finally, some numerical experiments are presented to show the efficiency of the two-grid method and verify that the convergence rates are in agreement with the theoretical analysis.     

基于矩阵模型表示的有限自动机极小化方法

论文基于有限自动机的矩阵模型犤1犦,并以矩阵理论和布尔代数为工具,给出了一种有限自动机极小化的新方法。该方法不仅有利于算法设计和计算机自动处理,也表明了矩阵模型方法在有限自动机应用研究中的重要作用。     

A Static Output Feedback Control for Linear Systems

Necessary and sufficient condition for the stabilizability of linear discrete systems by a static output feedback control are formulated and used in designing algorithms for robust stabilizing controls and controls that ensure concurrent stabilization of a family of discrete systems. Algorithms are implemented by MATLAB and SCILAB with LMISOLVER and SeDuMi interfaces. The results are generalized to random-structure systems.     

泰勒有限差分方法数值求解常微分方程

本文提出一种新的更高精度的泰勒有限差分公式并且应用于求解常微分方程。这种应用泰勒有限差分公式来求解常微分方程的方法称为泰勒有限差分方法。此外,出于比较的目的,使用欧拉方法求解常微分方程的算法流程也被提及,并且在MATLAB软件平台进行了两组对比性的数值实验。两组对比性的数值实验结果均表明,使用泰勒有限差分方法求解常微分方程的精度要比使用欧拉方法求解常微分方程的精度更高。后续可以应用该方法去开发常微分方程数值求解器的软件。     

A Compact Higher Order Finite Difference Method for the Incompressible Navier–Stokes Equations

A numerical method based on compact fourth order finite difference approximations is used for the solution of the incompressible Navier–Stokes equations. Our method is implemented for two dimensional, curvilinear coordinates on orthogonal, staggered grids. Two numerical experiments confirm the theoretically expected order of accuracy.     

A Finite Difference Method and Analysis for 2D Nonlinear Poisson–Boltzmann Equations

A fast finite difference method based on the monotone iterative method and the fast Poisson solver on irregular domains for a 2D nonlinear Poisson–Boltzmann equation is proposed and analyzed in this paper. Each iteration of the monotone method involves the solution of a linear equation in an exterior domain with an arbitrary interior boundary. A fast immersed interface method for generalized Helmholtz equations on exterior irregular domains is used to solve the linear equation. The monotone iterative method leads to a sequence which converges monotonically from either above or below to a unique solution of the problem. This monotone convergence guarantees the existence and uniqueness of a solution as well as the convergence of the finite difference solution to the continuous solution. A comparison of the numerical results against the exact solution in an example indicates that our method is second order accurate. We also compare our results with available data in the literature to validate the numerical method. Our method is efficient in terms of accuracy, speed, and flexibility in dealing with the geometry of the domain     

A Well-balanced Finite Volume-Augmented Lagrangian Method for an Integrated Herschel-Bulkley Model

We are interested in the derivation of an integrated Herschel-Bulkley model for shallow flows, as well as in the design of a numerical algorithm to solve the resulting equations. The goal is to simulate the evolution of thin sheet of viscoplastic materials on inclined planes and, in particular, to be able to compute the evolution from dynamic to stationary states. The model involves a variational inequality and it is valid from null to moderate slopes. The proposed numerical scheme is well balanced and involves a coupling between a duality technique (to treat plasticity), a fixed point method (to handle the power law) and a finite volume discretization. Several numerical tests are done, including a comparison with an analytical solution, to confirm the well balanced property and the ability to cope with the various rheological regimes associated with the Herschel-Bulkley constitutive law.     

A Chebyshev Finite Difference Method For Solving A Class Of Optimal Control Problems

A new Chebyshev finite difference method for solving class of optimal control problem is proposed. The algorithm is based on Chebyshev approximations of the derivatives arising in system dynamics. In the performance index, we use Chebyshev approximations for integration. The numerical examples illustrate the robustness, accuracy and efficiency of the proposed technique.