A collocation/quadrature-based Sturm–Liouville problem solver

We present a computational method for solving a class of boundary-value problems in Sturm–Liouville form. The algorithms are based on global polynomial collocation methods and produce discrete representations of the eigenfunctions. Error control is performed by evaluating the eigenvalue problem residuals generated when the eigenfunctions are interpolated to a finer discretization grid; eigenfunctions that produce residuals exceeding an infinity-norm bound are discarded. Because the computational approach involves the generation of quadrature weights and arrays for discrete differentiation operations, our computational methods provide a convenient framework for solving boundary-value problems by eigenfunction expansion and other projection methods.     

On the implementation of ESIRK methods for stiff IVPs

The effective order singly-implicit methods (ESIRK) are designed for solving stiff IVPs. These generalizations of SIRK methods are shown to have some computational advantages over the classical SIRK methods by moving the abscissae inside the integration interval [6]. In this paper, we consider some of the important computational aspects associated with these methods. We show that the ESIRK methods can be implemented efficiently by the comparsion with the standard stiff solvers RADAU5 and LSODE.     

线性规划的单纯形法及其发展

本文给出了一种新的原对偶单纯形法,并通过它分析了隐藏在经典单纯形法中的对偶信息．我们重新评价经典单纯形法并详细讨论了它与现代单纯形法之间的联系．两个修改版本一并给出．新算法具有计算量小和实施简单等特点,计算效果也不错．初步数值实验表明现代单纯形法比经典方法具有明显的优越性．     

On a family of high-order iterative methods under gamma conditions with applications in denoising

We study a class of at least third order iterative methods for nonlinear equations on Banach spaces. A characterization of the convergence under Gamma-type conditions is presented. Though, in general, these methods are not very extended due to their computational costs, we can find examples in which they are competitive and even cheaper than other simpler methods. Indeed, we propose a new nonlinear mathematical model for the denoising of digital images, where the best method in the family has fourth order of convergence. Moreover, our family includes two-step Newton type methods with good numerical behavior in general. We center our analysis in both, analytic and computational, aspects.     

Stopping criteria for inner iterations in inexact potential reduction methods: a computational study

We focus on the use of adaptive stopping criteria in iterative methods for KKT systems that arise in Potential Reduction methods for quadratic programming. The aim of these criteria is to relate the accuracy in the solution of the KKT system to the quality of the current iterate, to get computational efficiency. We analyze a stopping criterion deriving from the convergence theory of inexact Potential Reduction methods and investigate the possibility of relaxing it in order to reduce as much as possible the overall computational cost. We also devise computational strategies to face a possible slowdown of convergence when an insufficient accuracy is required.     

Implementing and using high-order finite element methods

We present theoretical analyses of and detailed timings for two programs which use high-order finite element methods to solve the Navier- Strokes equations in two and three dimensions. The analyses show that algorithms popular in low-order finite element implementations are not always appropriate for high-order methods. The timings show that with the proper algorithms high-order finite element methods are viable for solving the Navier-Stokes equations. We show that it is more efficient, both in time and storage, not to precompute element matrices as the degree of approximating functions increases. We also study the cost of assembling the stiffness matrix versus directly evaluating bilinear forms in two and three dimensions. We show that it is more efficient not to assemble the full stiffness matrix for high-order methods in some cases. We consider the computational issues with regard to both Euclidean and isoparametric elements. We show that isoparametric elements may be used with higher-order elements without increasing the order of computational complexity.     

Strong convergence result for proximal split feasibility problem in Hilbert spaces

In this paper we describe and analyse new computational technique for solving proximal split feasibility problem (SFP) using a modified proximal split feasibility algorithm. The two convex and lower semi-continuous objective functions are assumed to be non-smooth. Some application to SFP are given. We demonstrate the computational efficiency of the proposed algorithm with nontrivial numerical experiments. We also compare our method with other relevant methods in the literature in terms of convergence, stability, efficiency and implementation with our illustrative numerical examples.     

A posteriori error bound methods for the inclusion of polynomial zeros

Using Carstensen's results from 1991 we state a theorem concerning the localization of polynomial zeros and derive two a posteriori error bound methods with the convergence order 3 and 4. These methods possess useful property of inclusion methods to produce disks containing all simple zeros of a polynomial. We establish computationally verifiable initial conditions that guarantee the convergence of these methods. Some computational aspects and the possibility of implementation on parallel computers are considered, including two numerical examples. A comparison of a posteriori error bound methods with the corresponding circular interval methods, regarding the computational costs and sizes of produced inclusion disks, were given.     

Detecting structures in differential algebraic equations: Computational aspects

Differential algebraic equations (DAEs) are often automatically generated, in particular, by coupling different tools. These DAEs are unstructured in the sense that they do not reveal their mathematical structure a priori. In view of a reliable treatment of those DAEs, their mathematical structure should be uncovered and monitored also by computational methods. We discuss several computational aspects of the tractability index concept.     

Nonlinear eigenvalue problems with symmetry

In this paper we use the conjugate gradient predictor corrector method (CGPCM) in the context of continuation methods. By exploiting symmetry in certain nonlinear eigenvalue problems, we can decompose the centered difference discretization matrices into small ones and reduce computational cost. We use the cyclic group of order two to divide the system into two smaller systems. We reduce the cost and the computational time by combining CGPCM with the idea of the exploiting symmetries. Theoretical and numerical results are presented. Conclusions are given.     

A proximal point algorithm for finding a common zero of a finite family of maximal monotone operators in the presence of computational errors

In a Hilbert space, we study the convergence of a proximal point method to a common zero of a finite family of maximal monotone operators under the presence of computational errors. Most results known in the literature establish the convergence of proximal point methods, when computational errors are summable. In the present paper, the convergence of the method is established for nonsummable computational errors. We show that the proximal point method generates a good approximate solution, if the sequence of computational errors is bounded from above by a constant.     

Numerical solution of linear Volterra integral equations of the second kind with sharp gradients

Collocation methods are a well-developed approach for the numerical solution of smooth and weakly singular Volterra integral equations. In this paper, we extend these methods through the use of partitioned quadrature based on the qualocation framework, to allow the efficient numerical solution of linear, scalar Volterra integral equations of the second kind with smooth kernels containing sharp gradients. In this case, the standard collocation methods may lose computational efficiency despite the smoothness of the kernel. We illustrate how the qualocation framework can allow one to focus computational effort where necessary through improved quadrature approximations, while keeping the solution approximation fixed. The computational performance improvement introduced by our new method is examined through several test examples. The final example we consider is the original problem that motivated this work: the problem of calculating the probability density associated with a continuous-time random walk in three dimensions that may be killed at a fixed lattice site. To demonstrate how separating the solution approximation from quadrature approximation may improve computational performance, we also compare our new method to several existing Gregory, Sinc, and global spectral methods, where quadrature approximation and solution approximation are coupled.     

Maximal Monotone Operators and the Proximal Point Algorithm in the Presence of Computational Errors

In a finite-dimensional Euclidean space, we study the convergence of a proximal point method to a solution of the inclusion induced by a maximal monotone operator, under the presence of computational errors. Most results known in the literature establish the convergence of proximal point methods, when computational errors are summable. In the present paper, the convergence of the method is established for nonsummable computational errors. We show that the proximal point method generates a good approximate solution, if the sequence of computational errors is bounded from above by a constant.     

Numerical discretization-based kernel type estimation methods for ordinary differential equation models

We consider the problem of parameter estimation in both linear and nonlinear ordinary differential equation(ODE) models. Nonlinear ODE models are widely used in applications. But their analytic solutions are usually not available. Thus regular methods usually depend on repetitive use of numerical solutions which bring huge computational cost. We proposed a new two-stage approach which includes a smoothing method(kernel smoothing or local polynomial fitting) in the first stage, and a numerical discretization method(Eulers discretization method, the trapezoidal discretization method,or the Runge–Kutta discretization method) in the second stage. Through numerical simulations, we find the proposed method gains a proper balance between estimation accuracy and computational cost.Asymptotic properties are also presented, which show the consistency and asymptotic normality of estimators under some mild conditions. The proposed method is compared to existing methods in term of accuracy and computational cost. The simulation results show that the estimators with local linear smoothing in the first stage and trapezoidal discretization in the second stage have the lowest average relative errors. We apply the proposed method to HIV dynamics data to illustrate the practicability of the estimator.     

A genetic algorithm for the unrelated parallel machine scheduling problem with sequence dependent setup times

In this work a genetic algorithm is presented for the unrelated parallel machine scheduling problem in which machine and job sequence dependent setup times are considered. The proposed genetic algorithm includes a fast local search and a local search enhanced crossover operator. Two versions of the algorithm are obtained after extensive calibrations using the Design of Experiments (DOE) approach. We review, evaluate and compare the proposed algorithm against the best methods known from the literature. We also develop a benchmark of small and large instances to carry out the computational experiments. After an exhaustive computational and statistical analysis we can conclude that the proposed method shows an excellent performance overcoming the rest of the evaluated methods in a comprehensive benchmark set of instances.     

Zero-clusters of polynomials: Best approach in supercomputing era

Computing a zero-cluster of a polynomial sufficiently accurately within the available precision of computation has been an important issue from time immemorial. All the deterministic numerical methods so far known to us produce varying degree of errors. Often the errors are so dominant that the distinction between two zeros in the cluster becomes meaningfully difficult. Multiple zeros on the other hand can be more easily tackled and do not pose any serious computational problem. We discuss here the limits of both deterministic and randomized methods for zero-clusters and propose a simple exhaustive search algorithm that would obtain the zeros in a real/complex zero-cluster in a reasonable time. We present the computational error and computational/time complexity of this algorithm focusing on the fact that no measuring device can usually measure a quantity with an accuracy greater than 0.005%. We stress the fact that no other algorithm can perform better than the proposed algorithm in an ultra-high speed computing environment for most real-world problems.     

Regularization parameter selection and an efficient algorithm for total variation-regularized positron emission tomography

In positron emission tomography, image data corresponds to measurements of emitted photons from a radioactive tracer in the subject. Such count data is typically modeled using a Poisson random variable, leading to the use of the negative-log Poisson likelihood fit-to-data function. Regularization is needed, however, in order to guarantee reconstructions with minimal artifacts. Given that tracer densities are primarily smoothly varying, but also contain sharp jumps (or edges), total variation regularization is a natural choice. However, the resulting computational problem is quite challenging. In this paper, we present an efficient computational method for this problem. Convergence of the method has been shown for quadratic regularization functions and here convergence is shown for total variation regularization. We also present three regularization parameter choice methods for use on total variation-regularized negative-log Poisson likelihood problems. We test the computational and regularization parameter selection methods on two synthetic data sets.     

Nonlinear force density method for the form-finding of minimal surface membrane structures

We develop an alternative approach for the form-finding of the minimal surface membranes (including cable membranes) using discrete models and nonlinear force density method. Two directed weighted graphs with 3 and 4-sided regional cycles, corresponding to triangular and quadrilateral finite element meshes are introduced as computational models for the form-finding problem. The triangular graph model is closely related to the triangular computational models available in the literature whilst the quadrilateral graph uses a novel averaging approach for the form-finding of membrane structures within the context of nonlinear force density method. The viability of the mentioned discrete models for form-finding are studied through two solution methods including a fixed-point iteration method and the Newton–Raphson method with backtracking. We suggest a hybrid version of these methods as an effective solution strategy. Examples of the formation of certain well-known minimal surfaces are presented whilst the results obtained are compared and contrasted with analytical solutions in order to verify the accuracy and viability of the suggested methods.     

Convergence Aspects of Step-Parallel Iteration of Runge-Kutta Methods for Delay Differential Equations

Implicit Runge-Kutta methods are known as highly accurate and stable methods for solving differential equations. However, the iteration technique used to solve implicit Runge-Kutta methods requires a lot of computational efforts. To lessen the computational effort, one can iterate simultaneously at a number of points along the t-axis. In this paper, we extend the PDIRK (Parallel Diagonal Iterated Runge-Kutta) methods to delay differential equations (DDEs). We give the region of convergence and analyze the speed of convergence in three parts for the P-stability region of the Runge-Kutta corrector. It is proved that PDIRK methods to DDEs are efficient, and the diagonal matrix D of the PDIRK methods for DDES can be selected in the same way as for ordinary differential equations (ODEs).