Auxiliary linear multistep methods: implicit

A class of stable implicit 2-step methods of high order for the solution of ordinary differential equations have been developed. The methods use slopes at some auxiliary points within a step. The methods with more number of auxiliary points have shown better stability characteristics. The efficiency of these methods has been established by comparing numerical results with those of 4-step Adams predictor-corrector method and 2-step Butcher's hybrid method.     

Stability of linear multistep methods for delay integro-differential equations

This paper is concerned with the numerical solution of delay integro-differential equations. The adaptation of linear multistep methods is considered. The emphasis is on the linear stability of numerical methods. It is shown that every A-stable, strongly 0-stable linear multistep method of Pouzet type can preserve the delay-independent stability of the underlying linear systems. In addition, some delay-dependent stability conditions for the stability of numerical methods are also given.     

A fixed-point approach for nonlinear discrete boundary value problems

Existence results are established for second-order discrete boundary value problems.     

Preconditioning in parallel Runge-Kutta methods for stiff initial value problems

From a theoretical point of view, Runge-Kutta methods of collocation type belong to the most attractive step-by-step methods for integrating stiff problems. These methods combine excellent stability features with the property of superconvergence at the step points. Like the initial-value problem itself, they only need the given initial value without requiring additional starting values, and therefore, are a natural discretization of the initial-value problem. On the other hand, from a practical point of view, these methods have the drawback of requiring in each step the solution of a system of equations of dimension sd, s and d being the number of stages and the dimension of the initial-value problem, respectively. In contrast, linear multistep methods, the main competitor of Runge-Kutta methods, require the solution of systems of dimension d. However, parallel computers have changed the scene and have motivated us to design parallel iteration methods for solving the implicit systems in such a way that the resulting methods become efficient step-by-step methods for integrating stiff initial-value problems.     

The iterated defect correction methods for singular two-point boundary value problems

In this paper, the iterated defect correction (IDeC) techniques based on the centered Euler method for the equivalent first order system of the singular two-point boundary value problem in linear case (x ^α y′(x))′ = f(x), y(0) = a,y(1) = b, where 0 < α < 1 are considered. By using the asymptotic expansion of the global error, it is analyzed that the IDeC methods improved the approximate results by means of IDeC steps and the degree of the interpolating polynomials used. Some numerical examples from the literature are given in illustration of this theory.     

A note on the satisfaction of the boundary conditions for Chebyshev collocation methods in rectangular domains

The way boundary conditions are imposed when applying Chebyshev collocation methods to Poisson and biharmonic-type problems in rectangular domains is investigated. It is shown that careful selection of the number of collocation points leads to a linear system ofn linearly independent equations inn unknowns.     

Parallel algorithms for the solution of certain large sparse linear systems

A couple of approximate inversion techniques are presented which provide a parallel enhancement to several iterative methods for solving linear systems arising from the discretization of boundary value problems. In particular, the Jacobi, Gauss‐Seidel, and successive overrelaxation methods can be improved substantially in a parallel environment by the extensions considered. A special case convergence proof is presented. The use of our approximate inverses with the preconditioned conjugate gradient method is examined and comparisons are made with some recently proposed algorithms in this area that also employ approximate inverses. The methods considered are compared under sequential and parallel hardware assumptions.     

On modified Runge-Kutta trees and methods

Modified Runge-Kutta (mRK) methods can have interesting properties as their coefficients may depend on the step length. By a simple perturbation of very few coefficients we may produce various function-fitted methods and avoid the overload of evaluating all the coefficients in every step. It is known that, for Runge-Kutta methods, each order condition corresponds to a rooted tree. When we expand this theory to the case of mRK methods, some of the rooted trees produce additional trees, called mRK rooted trees, and so additional conditions of order. In this work we present the relative theory including a theorem for the generating function of these additional mRK trees and explain the procedure to determine the extra algebraic equations of condition generated for a major subcategory of these methods. Moreover, efficient symbolic codes are provided for the enumeration of the trees and the generation of the additional order conditions. Finally, phase-lag and phase-fitted properties are analyzed for this case and specific phase-fitted pairs of orders 8(6) and 6(5) are presented and tested.     

A fourth order multiderivative method for linear second order boundary value problems

Fourth order methods are developed and analysed for the numerical solution of linear second order boundary value problems.

The methods are developed by replacing the exponential terms in a three-point recurrence relation by Padé approximants.

The derivations of second order and sixth order methods from the recurrence relation are outlined briefly.

One method is tested on two problems from the literature, one of which is mildly nonlinear.     

A new approach for multistep numerical methods in several frequencies for perturbed oscillators

This article presents a new multi-step numerical method based on φ-function series and designed to integrate forced oscillators with precision. The new algorithm retains the good properties of the MDF_pPC methods while presenting the advantages of greater precision and that of integrating the non-perturbed problem without any discretization error. In addition, this new method permits a single formulation to be obtained from the MDF_pPC schemes independently of the parity of the number of steps, which facilitates the design of a computational algorithm thus permitting improved implementation in a computer.The construction of a new method for accurately integrating the homogenous problem is necessary if a method is sought which would be comparable to the methods based on Scheifele G-function series, very often used when problems of satellite orbital dynamics need to be resolved without discretization error.Greater precision compared to the MDF_pPC methods and other known integrators is demonstrated by overcoming stiff and highly oscillatory problems with the new method and comparing approximations obtained with those calculated by means of other integrators.     

Foreign Ownership Restrictions: A Numerical Approach

In this paper, we analyze the reason behind the use of foreign ownership restrictions on inward Foreign Direct Investment (FDI). We extend the results developed in Karabay (2005) by changing the condition on share distribution in the model. Due to this change, we are able to analyze the political economy aspect of this restrictive policy, i.e., we can study the effect of the host government’s welfare preference on the optimal foreign ownership restriction. Since the analytical solution to the optimal share restriction policy cannot be specified in general, we use a numerical approach based on collocation to approximate the solution to the problem. Within this framework, under certain conditions, it turns out that the rent extraction-efficiency trade-off is sharper the less the host government favors the local firm. We show that not only economic factors but also political factors play an important role in the determination of the foreign ownership restrictions. The views expressed in this paper are those of the authors and are not necessarily reflective of views of the Central Bank of the Republic of Turkey. All errors are our own.     

Convergence of relaxation methods for two-point boundary-value linear problems

The solution of linear differential problems, with explicit two-point boundary conditions, can sometimes be obtained by a relaxation method of computation. This paper shows that the convergence of iterations is linked to the spectral radius value of an integral operator. The equivalence between eigenvalues research and critical lengths calculation of a differential system is demonstrated. In this context, we present a case of optimal control law calculation of a linear system. The efficiency of this preliminary convergence calculation is illustrated by a numerical example.     

Pseudospectral integration matrix and boundary value problems

This paper presents a new spectral successive integration matrix. This matrix is used to construct a Chebyshev expansion method for the solution of boundary value problems. The method employs the pseudospectral approximation of the highest-order derivative to generate an approximation to the lower-order derivatives. Application to the linear stability problem for plane Poiseuille flow is presented. The present numerical results are in satisfactory agreement with the exact solutions.     

Stability analysis of extended block boundary value methods for linear neutral delay integro-differential equations

This paper is concerned with the stability of extended block boundary value methods (B₂VMs) for the linear neutral delay integro-differential-algebraic equations (NDIDAEs) and the linear neutral delay integro-differential equations (NDIDEs). It is proved that every A-stable B₂VM can preserve the asymptotic stability of the exact solution of NDIDAEs under some certain conditions. A necessary and sufficient condition of the B₂VMs to be asymptotically stable for NDIDEs is also obtained. A few numerical experiments confirm the expected results.     

The effect of global extrapolation on the phase-lag of symmetric methods for solving periodic initial value problems

The effects are discussed of two-grid global extrapolation procedures on the phase-lags of convergent numerical methods for solving periodic initial value problems.

The procedure is tested on two methods applied to two problems from the literature, one nonlinear the other linear, and the effects of the extrapolation are examined by comparing corresponding zeros of the waves generated by the theoretical and computed solutions.

Extensions to three- and four-grid extrapolation procedures are outlined in an Appendix.     

The precise finite strip method

This paper introduces a new development for the finite strip method. The precise integration method along the space coordinate is combined with the semi-analytical analysis of prismatic domain structures and then a novel precise finite strip method is proposed. By using such a novel semi-analytical method, the structure, which is as usual discretized along the transverse direction in a prismatic domain, will end up as a two point boundary value problem in the longitudinal direction. For the derived ODEs with constant coefficient of the two point boundary value problems, the displacement function is computed by integration instead of being given by traditional series. The numerical result of the present method will be very precise, almost up to the computer precision. Thus, the precise finite strip method can deal with various local effect problems which cannot be solved by many traditional methods.     

The ‘Idiot’ crash quadratic penalty algorithm for linear programming and its application to linearizations of quadratic assignment problems

ABSTRACT

We provide the first meaningful documentation and analysis of the ‘Idiot’ crash implemented by Forrest in Clp that aims to obtain an approximate solution to linear programming (LP) problems for warm-starting the primal simplex method. The underlying algorithm is a penalty method with naive approximate minimization in each iteration. During initial iterations an approach similar to augmented Lagrangian is used. Later the technique corresponds closely to a classical quadratic penalty method. We discuss the extent to which it can be used to obtain fast approximate solutions of LP problems, in particular when applied to linearizations of quadratic assignment problems.     

Solving and factoring boundary problems for linear ordinary differential equations in differential algebras

We present a new approach for expressing and solving boundary problems for linear ordinary differential equations in the language of differential algebras. Starting from an algebra with a derivation and integration operator, we construct an algebra of linear integro-differential operators that is expressive enough for specifying regular boundary problems with arbitrary Stieltjes boundary conditions as well as their solution operators.     

The Omni-family of all-purpose access methods: a simple and effective way to make similarity search more efficient

Similarity search operations require executing expensive algorithms, and although broadly useful in many new applications, they rely on specific structures not yet supported by commercial DBMS. In this paper we discuss the new Omni-technique, which allows to build a variety of dynamic Metric Access Methods based on a number of selected objects from the dataset, used as global reference objects. We call them as the Omni-family of metric access methods. This technique enables building similarity search operations on top of existing structures, significantly improving their performance, regarding the number of disk access and distance calculations. Additionally, our methods scale up well, exhibiting sub-linear behavior with growing database size.     

Kernel methods in system identification,machine learning and function estimation: A survey

Most of the currently used techniques for linear system identification are based on classical estimation paradigms coming from mathematical statistics. In particular, maximum likelihood and prediction error methods represent the mainstream approaches to identification of linear dynamic systems, with a long history of theoretical and algorithmic contributions. Parallel to this, in the machine learning community alternative techniques have been developed. Until recently, there has been little contact between these two worlds. The first aim of this survey is to make accessible to the control community the key mathematical tools and concepts as well as the computational aspects underpinning these learning techniques. In particular, we focus on kernel-based regularization and its connections with reproducing kernel Hilbert spaces and Bayesian estimation of Gaussian processes. The second aim is to demonstrate that learning techniques tailored to the specific features of dynamic systems may outperform conventional parametric approaches for identification of stable linear systems.