An accelerated subspace iteration method

The subspace iteration method for solving symmetric eigenproblems in computational mechanics is considered. Effective procedures for accelerating the convergence of the basic subspace iteration method are presented. The accelerated subspace iteration method has been implemented and the results of some demonstrative sample solutions are presented and discussed.     

Preconditioned HSS iteration method and its non-alternating variant for continuous Sylvester equations

Bai (2010) proposed an efficient Hermitian and skew-Hermitian splitting (HSS) iteration method for solving a broad class of large sparse continuous Sylvester equations. To further improve the efficiency of the HSS method, in this paper we present a preconditioned HSS (PHSS) iteration method and its non-alternating variant (NPHSS) for this matrix equation. The convergence properties of the PHSS and NPHSS methods are studied in depth and the quasi-optimal values of the iteration parameters for the two methods are also derived. Moreover, to reduce the computational cost, we establish the inexact variants of the two iteration methods. Numerical experiments illustrate the efficiency and robustness of the two iteration methods and their inexact variants.     

A new alternating direction method for solving variational inequalities

We introduce a modified alternating direction method for structured monotone variational inequalities by performing an additional projection step at each iteration and another optimal step length is employed to reach substantial progress in each iteration. This method only needs functional values for given variables in the solution process and avoids the task of estimating the co-coercive modulus. All the computing process are easily implemented and the global convergence is also presented under mild assumptions. Some preliminary computational results are given.     

Numerical methods for the QCD overlap operator: III. Nested iterations

The numerical and computational aspects of chiral fermions in lattice quantum chromodynamics are extremely demanding. In the overlap framework, the computation of the fermion propagator leads to a nested iteration where the matrix vector multiplications in each step of an outer iteration have to be accomplished by an inner iteration; the latter approximates the product of the sign function of the hermitian Wilson fermion matrix with a vector.In this paper we investigate aspects of this nested paradigm. We examine several Krylov subspace methods to be used as an outer iteration for both propagator computations and the Hybrid Monte-Carlo scheme. We establish criteria on the accuracy of the inner iteration which allow to preserve an a priori given precision for the overall computation. It will turn out that the accuracy of the sign function can be relaxed as the outer iteration proceeds. Furthermore, we consider preconditioning strategies, where the preconditioner is built upon an inaccurate approximation to the sign function. Relaxation combined with preconditioning allows for considerable savings in computational efforts up to a factor of 4 as our numerical experiments illustrate. We also discuss the possibility of projecting the squared overlap operator into one chiral sector.     

Solving linear and nonlinear initial value problems by the projected differential transform method

In this work, we propose a novel computational algorithm for solving linear and nonlinear initial value problems by using the modified version of differential transform method (DTM), which is called the projected differential transform method (PDTM). The PDTM can be easily applied to the initial value problems with less computational work. For the several illustrative examples, the computational results are compared with those obtained by many other methods; the Adomian decomposition, the variational iteration and the spline method. For all examples, the PDTM provides exact solutions. It has been shown that the PDTM is a reliable algorithm in obtaining analytic as well as approximate solution for the initial value problems.     

Splitting iteration method for simple singular points and simple bifurcation points

A splitting iteration method is introduced to compute the simple singular points and the simple bifurcation points of nonlinear problems. It needs little computational work and converges with an adjustable rate. Numerical examples are presented.     

Three-dimensional eigenvalue analysis of the Helmholtz equation by multiple reciprocity boundary element method

The eigenvalue of the three-dimensional Helmholtz equation is determined efficiently by extending the previously developed method for the two-dimensional problem. Boundary integral equation is formulated in the realm of the multiple reciprocity method, using higher order fundamental solutions for the Laplace equation; yielding polynomial coefficient matrices in terms of unknown wavenumber (eigenvalue). The Newton iteration method with the help of LU decomposition is employed to search eigenvalue, which can reduce the computational task significantly.     

An accelerated first-order method for solving SOS relaxations of unconstrained polynomial optimization problems

Our interest lies in solving sum of squares (SOS) relaxations of large-scale unconstrained polynomial optimization problems. Because interior-point methods for solving these problems are severely limited by the large-scale, we are motivated to explore efficient implementations of an accelerated first-order method to solve this class of problems. By exploiting special structural properties of this problem class, we greatly reduce the computational cost of the first-order method at each iteration. We report promising computational results as well as a curious observation about the behaviour of the first-order method for the SOS relaxations of the unconstrained polynomial optimization problem.     

Numerical modeling of air mass motion by the mesh-free method

The Navier-Stokes equation, applied to the calculation of wind velocity without accounting for the turbulent motion of the atmosphere, is considered in this work. The main flow characteristics were computed with the use of the Lagrange discrete vortex method for finding the solution of the Poisson equation under the Dirichlet and Neumann boundary conditions. To do this, two mesh-free methods: the element-free Galerkin (EFG) and the Finite Pointset (FP) methods, as well as the modification of the latter, have been analyzed. It is shown that the computation speed of the EFG method is higher than of the FP-method. It is determined that a serious disadvantage of the FP-method is its low rate convergence, while the computational complexity of each iteration is reasonable. The use of the modified FP-method has shown its computational speed to be comparable with that of the EFG method, although the advantage of the FP-method is not obvious when the size of the problem increases.     

A nonlinear interval-based optimization method with local-densifying approximation technique

In this paper, a new method is proposed to promote the efficiency and accuracy of nonlinear interval-based programming (NIP) based on approximation models and a local-densifying method. In conventional NIP methods, searching for the response bounds of objective and constraints are required at each iteration step, which forms a nested optimization and leads to extremely low efficiency. In order to reduce the computational cost, approximation models based on radial basis functions (RBF) are used to replace the actual computational models. A local-densifying method is suggested to guarantee the accuracy of the approximation models by reconstructing them with densified samples in iterations. Thus, through a sequence of optimization processes, an optimal result with fine accuracy can be finally achieved. Two numerical examples are used to test the effectiveness of the present method, and it is then applied to a practical engineering problem.     

An augmented Lagrangian relaxation for analytical target cascading using the alternating direction method of multipliers

Analytical target cascading is a method for design optimization of hierarchical, multilevel systems. A quadratic penalty relaxation of the system consistency constraints is used to ensure subproblem feasibility. A typical nested solution strategy consists of inner and outer loops. In the inner loop, the coupled subproblems are solved iteratively with fixed penalty weights. After convergence of the inner loop, the outer loop updates the penalty weights. The article presents an augmented Lagrangian relaxation that reduces the computational cost associated with ill-conditioning of subproblems in the inner loop. The alternating direction method of multipliers is used to update penalty parameters after a single inner loop iteration, so that subproblems need to be solved only once. Experiments with four examples show that computational costs are decreased by orders of magnitude ranging between 10 and 1000.     

Enhanced parallel cat swarm optimization based on the Taguchi method

In this paper, we present an enhanced parallel cat swarm optimization (EPCSO) method for solving numerical optimization problems. The parallel cat swarm optimization (PCSO) method is an optimization algorithm designed to solve numerical optimization problems under the conditions of a small population size and a few iteration numbers. The Taguchi method is widely used in the industry for optimizing the product and the process conditions. By adopting the Taguchi method into the tracing mode process of the PCSO method, we propose the EPCSO method with better accuracy and less computational time. In this paper, five test functions are used to evaluate the accuracy of the proposed EPCSO method. The experimental results show that the proposed EPCSO method gets higher accuracies than the existing PSO-based methods and requires less computational time than the PCSO method. We also apply the proposed method to solve the aircraft schedule recovery problem. The experimental results show that the proposed EPCSO method can provide the optimum recovered aircraft schedule in a very short time. The proposed EPCSO method gets the same recovery schedule having the same total delay time, the same delayed flight numbers and the same number of long delay flights as the Liu, Chen, and Chou method (2009). The optimal solutions can be found by the proposed EPCSO method in a very short time.     

On the multi-grid method applied to difference equations

Multi-grid methods are characterized by the simultaneous use of additional auxiliary grids corresponding to coarser step widths. Contrary to usual iterative methods the speed of convergence is very fast and does not tend to one if the step size approaches zero. The computational amount of one iteration is proportional toN, the number of grid points. Thus, a solution with accuracy ? requires 0 (|log ?|N) operations. In this paper we apply a multi-grid method to Helmholtz's equation (Dirichlet boundary data) in a general region and to a differential equation with variable coefficients subject to arbitrary boundary conditions.     

A full multigrid method for the Steklov eigenvalue problem

ABSTRACT

This paper introduces a kind of parallel multigrid method for solving Steklov eigenvalue problem based on the multilevel correction method. Instead of the common costly way of directly solving the Steklov eigenvalue problem on some fine space, the new method contains some boundary value problems on a series of multilevel finite element spaces and some steps of solving Steklov eigenvalue problems on a very low dimensional space. The linear boundary value problems are solved by some multigrid iteration steps. We will prove that the computational work of this new scheme is truly optimal, the same as solving the corresponding linear boundary value problem. Besides, this multigrid scheme has a good scalability by using parallel computing technique. Some numerical experiments are presented to validate our theoretical analysis.     

用于过失测量数据侦破与校正的改进MT-NT算法

介绍了一种用于过失误差侦破和校正改进的MT-NT算法。改进后的算法采用逐次侦破、校正的策略,有效地解决了在侦破过失误差过程中出现的系数矩阵降秩问题,减少了运算量,增加了信息的可用性和完整性。给出了该算法的框图及步骤,并采用面向对象的方法和C 十语言编制出了过程测量数据校正软件。经过实例验证,该算法可有效侦破测量数据中的过失误差,避免了在运算过程中出现的系数矩阵降秩问题,具有一定的实用性。     

A simple self-adaptive alternating direction method for linear variational inequality problems

In this study, we propose a new alternating direction method for solving linear variational variational inequality problems (LVIP). It is simple in the sense that, at each iteration, it needs only to perform a projection onto a simple set and some matrix–vector multiplications. The simplicity of the solution method makes it attractive for solving large-scale problems. To further improve its efficiency, we devise a self-adaptive strategy for choosing the necessary parameters of the solution procedure. We prove the global convergence of this new method under some mild conditions. Finally, some computational results are reported to demonstrate the properties and efficiency of the method.     

基于逆迭代的增量LLE算法

Locally Linear Embedding（LLE）算法是一种很好的流形学习算法,但是它只能以批处理的方式进行,只要有新的样本加入,就必须重作该算法的全部内容。而原来的运算结果被全部丢弃。提出了一种基于逆迭代的增量LLE算法,实现了流形的增量学习。在Swiss roll和S-curve数据库上的实验表明,该算法与LLE算法所计算出的投影值误差小于0.001%,运行的耗时少,具有很好的应用价值。     

A numerical method for determining monotonicity and convergence rate in iterative learning control

In iterative learning control (ILC), a lifted system representation is often used for design and analysis to determine the convergence rate of the learning algorithm. Computation of the convergence rate in the lifted setting requires construction of large N×N matrices, where N is the number of data points in an iteration. The convergence rate computation is O(N²) and is typically limited to short iteration lengths because of computational memory constraints. As an alternative approach, the implicitly restarted Arnoldi/Lanczos method (IRLM) can be used to calculate the ILC convergence rate with calculations of O(N). In this article, we show that the convergence rate calculation using IRLM can be performed using dynamic simulations rather than matrices, thereby eliminating the need for large matrix construction. In addition to faster computation, IRLM enables the calculation of the ILC convergence rate for long iteration lengths. To illustrate generality, this method is presented for multi-input multi-output, linear time-varying discrete-time systems.     

An efficient swap algorithm for the lattice Boltzmann method

During the last decade, the lattice-Boltzmann method (LBM) as a valuable tool in computational fluid dynamics has been increasingly acknowledged. The widespread application of LBM is partly due to the simplicity of its coding. The most well-known algorithms for the implementation of the standard lattice-Boltzmann equation (LBE) are the two-lattice and two-step algorithms. However, implementations of the two-lattice or the two-step algorithm suffer from high memory consumption or poor computational performance, respectively. Ultimately, the computing resources available decide which of the two disadvantages is more critical. Here we introduce a new algorithm, called the swap algorithm, for the implementation of LBE. Simulation results demonstrate that implementations based on the swap algorithm can achieve high computational performance and have very low memory consumption. Furthermore, we show how the performance of its implementations can be further improved by code optimization.     

A novel weighted least squares PET image reconstruction method using adaptive variable index sets

In this paper, we present a novel image reconstruction method based on weighted least squares (WLS) objective function for positron emission tomography (PET). Unlike a usual WLS algorithm, the proposed method, which we call it SA-WLS, combines the SAGE algorithm with WLS algorithm. It minimized the WLS objective function using single coordinate descent (SCD) method in a sequence of small “hidden” data spaces (HDS). Although SA-WLS used a strategy to update parameter sequentially just like common SCD method, the use of these small HDS makes it converge much faster and produce the reconstructed images with greater contrast and detail than the usual WLS method. In order to decrease further the actual CPU time per iteration, the adaptive variable index sets were introduced to modify SA-WLS (MSA-WLS). Instead of optimizing each pixel, this MSA-WLS method sequentially optimizes many pixels located in an index set at one time. The index sets were automatically modified during each iteration step. MSA-WLS gathers the virtue of simultaneously and sequentially updating the parameters so that it achieves a good compromise between the convergence rate and the computational cost in PET reconstruction problem. Details of these algorithms were presented and the performances were evaluated by a simulated head phantom.