Showing papers in "International Journal of Computer Mathematics in 2007"
TL;DR: In this article, a meshless collocation method based on radial basis functions is proposed for solving the steady incompressible Navier-Stokes equations, which has the capability of solving the governing equations using scattered nodes in the domain.
Abstract: A meshless collocation method based on radial basis functions is proposed for solving the steady incompressible Navier-Stokes equations. This method has the capability of solving the governing equations using scattered nodes in the domain. We use the streamfunction formulation, and a trust-region method for solving the nonlinear problem. The no-slip boundary conditions are satisfied using a ghost node strategy. The efficiency of this method is demonstrated by solving three model problems: the driven cavity flows in square and rectangular domains and flow over a backward-facing step. The results obtained are in good agreement with benchmark solutions.
56 citations
TL;DR: A multilevel algorithm is proposed for efficiently solving the variational model for Poisson noise and it delivers the same numerical solution many orders of magnitude faster than the standard single-level method of coordinate descent time-marching.
Abstract: Many commonly used models for the fundamental image processing task of noise removal can deal with Gaussian white noise. However, such Gaussian models are not effective in restoring images with Poisson noise, which is ubiquitous in certain applications. Recently, Le-Chartrand-Asaki derived a new data-fitting term in the variational model for Poisson noise. This paper proposes a multilevel algorithm for efficiently solving this variational model. As expected of a multilevel method, it delivers the same numerical solution many orders of magnitude faster than the standard single-level method of coordinate descent time-marching. Supporting numerical experiments on 2D gray scale images are presented.
52 citations
TL;DR: The present analytical treatment has the potential to calculate the rate of flow, the resistive impedance and the wall shear stress without excessive computational complexity by exploiting the appropriate physiologically realistic prescribed conditions in nonuniform nonstaggered grids, and to estimate the effects of surface roughness as well as asymmetry of stenosis shape for both shear-thinning and shear -thickening models of Power-law fluid.
Abstract: The problem of non-Newtonian and nonlinear pulsatile flow through an irregularly stenosed arterial segment is solved numerically where the non-Newtonian rheology of the flowing blood is characterized by the generalized Power-law model where both the shear-thinning and shear-thickening models of the streaming blood are taken into account. The combined influence of an asymmetric shape and surface irregularities (roughness) of the constriction has been explored in a study of blood flow with 48% areal occlusion. The vascular wall deformability is taken to be anisotropic, linear, viscoelastic, incompressible circular cylindrical membrane shell. The effect of the surrounding connective tissues on the motion of the arterial wall is also paid due attention. Results are obtained for a smooth stenosis model and also for a stenosis model represented by the cosine curve. The present analytical treatment has the potential to calculate the rate of flow, the resistive impedance and the wall shear stress without excessive computational complexity by exploiting the appropriate physiologically realistic prescribed conditions in nonuniform nonstaggered grids, and to estimate the effects of surface roughness as well as asymmetry of stenosis shape for both shear-thinning and shear-thickening models of Power-law fluid, representing the streaming blood through graphical representations in order to validate the applicability of the present improved mathematical model.
48 citations
TL;DR: To compare the method developed in this paper with those developed by Inc and Evans, and Siddiqi and Twizell, two examples are considered and it is observed that the method is more efficient.
Abstract: The non-polynomial spline technique is used for the numerical solution of eighth-order linear special case boundary value problems. The method presented in this paper has also been proven to be second-order convergent. To compare the method developed in this paper with those developed by Inc and Evans, and Siddiqi and Twizell, two examples are considered and it is observed that our method is more efficient.
44 citations
TL;DR: A survey of singular perturbation methods for boundary value problems can be found in this paper, where a summary of the results of some recent methods is presented and this leads to conclusions and recommendations regarding methods to use.
Abstract: This survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of ideas and methods of singular perturbation theory. In continuation of a survey performed earlier, this paper limits its coverage to some standard numerical methods developed by numerous researchers between 2000 and 2005. A summary of the results of some recent methods is presented and this leads to conclusions and recommendations regarding methods to use on singular perturbation problems. Because of space constraints, we considered one-dimensional singularly perturbed boundary value problems only.
42 citations
TL;DR: A block triangular preconditioner based on an augmented Lagrangian formulation is shown to result in fast convergence of the GMRES iteration for a wide range of problem and algorithm parameters.
Abstract: We investigate the solution of linear systems of saddle point type with an indefinite (1, 1) block by preconditioned iterative methods. Our main focus is on block matrices arising from eigenvalue problems in incompressible fluid dynamics. A block triangular preconditioner based on an augmented Lagrangian formulation is shown to result in fast convergence of the GMRES iteration for a wide range of problem and algorithm parameters. Some theoretical estimates for the eigenvalues of the preconditioned matrices are given. Inexact variants of the preconditioner are also considered.
41 citations
TL;DR: The case study of the nonlinear Schrödinger equation is considered in detail, for which the previously known multisymplectic integrators are fully implicit and based on the (second order) box scheme.
Abstract: Although Runge-Kutta and partitioned Runge-Kutta methods are known to formally satisfy discrete multisymplectic conservation laws when applied to multi-Hamiltonian PDEs, they do not always lead to well-defined numerical methods. We consider the case study of the nonlinear Schrodinger equation in detail, for which the previously known multisymplectic integrators are fully implicit and based on the (second order) box scheme, and construct well-defined, explicit integrators, of various orders, with local discrete multisymplectic conservation laws, based on partitioned Runge-Kutta methods. We also show that two popular explicit splitting methods are multisymplectic.
40 citations
TL;DR: The experimental results show that QPSO outperforms the traditional PSOs and is a promising optimization algorithm for constrained NLP problems.
Abstract: In this paper, we focus on solving non-linear programming (NLP) problems using quantum-behaved particle swarm optimization (QPSO). After a brief introduction to the original particle swarm optimization (PSO), we describe the origin and development of QPSO, and the penalty function method for constrained NLP problems. The performance of QPSO is tested on some unconstrained and constrained benchmark functions and compared with PSO with inertia weight (PSO-In) and PSO with constriction factor (PSO-Co). The experimental results show that QPSO outperforms the traditional PSOs and is a promising optimization algorithm.
38 citations
TL;DR: The necessary and sufficient condition for the convergence of the GSOR-like method is derived and it is shown that when α is negative, the convergence domain for the parameter ω for the MSOR-Like method is larger than that for the SOR- like method.
Abstract: The SOR-like method with two real parameters ω and α is considered for solving the augmented system. The new method is called the modified SOR-like method (MSOR-like method). The MSOR-like method becomes the SOR-like method when α = 0. The functional equation relating the parameters and eigenvalues of the iteration matrix of the MSOR-like method is obtained. Hence the necessary and sufficient condition for the convergence of the GSOR-like method is derived. It is shown that when α is negative, the convergence domain for the parameter ω for the MSOR-like method is larger than that for the SOR-like method. Finally, a numerical computation based on a particular linear system is given which clearly shows that the MSOR-like method outperforms the SOR-like method.
35 citations
TL;DR: A mixed method is proposed for deriving reduced-order models of high-order linear time invariant systems using the combined advantages of eigen spectrum analysis and the Padé approximation technique.
Abstract: A mixed method is proposed for deriving reduced-order models of high-order linear time invariant systems using the combined advantages of eigen spectrum analysis and the Pade approximation technique. The denominator of the reduced-order model is found by eigen spectrum analysis, the dynamics of the numerator are chosen using the Pade approximation technique. This method guarantees stability of the reduced model if the original high-order system is stable. The method is illustrated by three numerical examples.
34 citations
TL;DR: In the finite dimensional linear algebra setting, however, all norms are equivalent and little attention is often paid to the norms used as discussed by the authors, and it is known that stopping an iteration which is rapidly converging in a highly scaled norm at some tolerance level may still give a poor answer.
Abstract: The convergence of numerical approximations to the solutions of differential equations is a key aspect of numerical analysis and scientific computing. Iterative solution methods for the systems of linear(ized) equations which often result are also underpinned by analyses of convergence. In the function space setting, it is widely appreciated that there are appropriate ways in which to assess convergence and it is well-known that different norms are not equivalent. In the finite dimensional linear algebra setting, however, all norms are equivalent and little attention is often paid to the norms used. In this paper, we highlight this consideration in the context of preconditioning for minimum residual methods (MINRES and GMRES/GCR/ORTHOMIN) and argue that even in the linear algebra setting there is a 'right' norm in which to consider convergence: stopping an iteration which is rapidly converging in an irrelevant or highly scaled norm at some tolerance level may still give a poor answer.
TL;DR: A new method for defining preconditioners for the iterative solution of a system of linear equations is introduced and by this MSPAI (modified SPAI) probing approach the authors can improve any given preconditionser with respect to this probing subspace.
Abstract: In this paper we introduce a new method for defining preconditioners for the iterative solution of a system of linear equations. By generalizing the class of modified preconditioners (e.g. MILU), the interface probing, and the class of preconditioners related to the Frobenius norm minimization (e.g. FSAI, SPAI) we develop a toolbox for computing preconditioners that are improved relative to a given small probing subspace. Furthermore, by this MSPAI (modified SPAI) probing approach we can improve any given preconditioner with respect to this probing subspace. All the computations are embarrassingly parallel. Additionally, for symmetric linear system we introduce new techniques for symmetrizing preconditioners. Many numerical examples, e.g. from PDE applications such as domain decomposition and Stokes problem, show that these new preconditioners often lead to faster convergence and smaller condition numbers.
TL;DR: In this article, rank-constrained Hermitian nonnegative-definite least squares solutions to the matrix equation AXAH=B have been investigated, under the conditions that B is hermitian and nonnegativedefinite.
Abstract: In the literature, rank-constrained Hermitian nonnegative-definite solutions to the matrix equation AXAH=B have been investigated, under the conditions that B is Hermitian and nonnegative-definite, and the matrix equation is consistent. In this paper, we discuss rank-constrained Hermitian nonnegative-definite least squares solutions to this matrix equation, in which the above conditions may not hold. We derive the rank range and expression of these least squares solutions. Therefore, the results obtained in this paper generalize those in the literature.
TL;DR: In this paper, an original algebraic technique based on the computation of small patches is presented for the Helmholtz equation, not directly linked to the continuous equations of the problem, nor to the numerical scheme.
Abstract: Recent work has shown that designing absorbing boundary conditions through algebraic approaches may be a nice alternative to the continuous approaches based on a Fourier analysis. In this paper, an original algebraic technique based on the computation of small patches is presented for the Helmholtz equation. This new technique is not directly linked to the continuous equations of the problem, nor to the numerical scheme. These properties make this technique very convenient to implement in a domain decomposition context. The proposed algebraic absorbing boundary conditions are used in a non-overlapping domain decomposition method and are defined on the interface between the subdomains. An additional coarse grid correction is then applied to ensure full scalability of the domain decomposition method upon the number of subdomains. This coarse grid correction involves trigonometric functions defined on the interface between the subdomains. Numerical experiments are presented and illustrate the robustness and parallel efficiency of the proposed method for acoustics applications.
TL;DR: In this paper, a uniform finite difference method on an S-mesh (Shishkin type mesh) for a singularly perturbed semilinear one-dimensional convection-diffusion three-point boundary value problem with zeroth-order reduced equation is considered.
Abstract: We consider a uniform finite difference method on an S-mesh (Shishkin type mesh) for a singularly perturbed semilinear one-dimensional convection-diffusion three-point boundary value problem with zeroth-order reduced equation. We show that the method is first-order convergent in the discrete maximum norm, independently of the perturbation parameter except for a logarithmic factor. An effective iterative algorithm for solving the non-linear difference problem and some numerical results are presented.
TL;DR: This paper provides models for the treatment of ordinal data in cost efficiency analysis that have multiplier forms with additional weight restrictions and are based on the weighted enumeration of the number of inputs/outputs of each unit.
Abstract: Standard Data Envelopment Analysis models obtain the cost efficiency of units when the data are known exactly, but these models fail to evaluate the units in the presence of ordinal data. Therefore, this paper provides models for the treatment of ordinal data in cost efficiency analysis. The models have multiplier forms with additional weight restrictions. The main idea in constructing these models is based on the weighted enumeration of the number of inputs/outputs of each unit which are categorized on the same scale rate. Some techniques to reduce the complexity of the models are introduced.
TL;DR: In this article, a two-dimensional integral equation in the domain of interest can be transformed into an algebraic equation, and the coefficients W(k, h) for k, h = 1, 2,... are determined.
Abstract: The differential transformation method provides an iterative procedure to obtain the spectrum of analytic solutions. In this paper we extend the differential transformation approach for the solution of two-dimensional integral equations. We give some basic properties and a new differential transformation-type method for the solution of linear and nonlinear two-dimensional Volterra integral equations. By extension of the operations, a two-dimensional integral equation in the domain of interest can be transformed into an algebraic equation in the domain K, H. We show that, after transforming the original equation into an algebraic equation, the coefficients W(k, h) for k, h=1, 2,... are determined and then, by substituting the values of W(k, h) in the transformed equation, the closed form solution of the original equation can be obtained. The reliability and efficiency of the proposed scheme are demonstrated by numerical experiments.
TL;DR: An uncertain discrete-time stochastic system is employed to represent a model for gene regulatory networks from time series data to design a linear filter such that, for the admissible bounded uncertainties, the filtering error system is Schur stable and the individual error variance is less than a prespecified upper bound.
Abstract: In this paper, an uncertain discrete-time stochastic system is employed to represent a model for gene regulatory networks from time series data. A robust variance-constrained filtering problem is investigated for a gene expression model with stochastic disturbances and norm-bounded parameter uncertainties, where the stochastic perturbation is in the form of a scalar Gaussian white noise with constant variance and the parameter uncertainties enter both the system matrix and the output matrix. The purpose of the addressed robust filtering problem is to design a linear filter such that, for the admissible bounded uncertainties, the filtering error system is Schur stable and the individual error variance is less than a prespecified upper bound. By using the linear matrix inequality (LMI) technique, sufficient conditions are first derived for ensuring the desired filtering performance for the gene expression model. Then the filter gain is characterized in terms of the solution to a set of LMIs, which can easily be solved by using available software packages. A simulation example is exploited for a gene expression model in order to demonstrate the effectiveness of the proposed design procedures.
TL;DR: The model shows how efficient and cost-effective strategies can be obtained by acting on the incidence of pre-diabetes and/or controlling the evolution to the stages of diabetes without and with complications.
Abstract: The incidence and prevalence of diabetes are increasing all over the world. Complications of diabetes constitute a burden for individuals and the whole of society. In the present paper, ordinary differential equations and numerical approximations are used to monitor the size of populations of pre-diabetes, and diabetes with and without complications. Different scenarios are discussed according to a set of parameters, and the dynamical evolution of the population from the stage of diabetes to the stage of diabetes with complications is illustrated. The model shows how efficient and cost-effective strategies can be obtained by acting on the incidence of pre-diabetes and/or controlling the evolution to the stages of diabetes without and with complications.
TL;DR: This paper studies the approximation by splitting techniques of the ordinary differential equation U˙+A U+B U=0, U(0)=U 0 with A and B two matrices to prove some error estimates for two general matrices and in the stiff case, where the estimates are independent of U 0 and the commutator between A and A.
Abstract: In this paper we study the approximation by splitting techniques of the ordinary differential equation U+A U+B U=0, U(0)=U0 with A and B two matrices. We assume that we have a stiff problem in the sense that A is ill-conditionned and U0 is a vector which is the discretization of a function with a very high derivative. This situation may appear for example when we study the discretization of a partial differential equation. We prove some error estimates for two general matrices and in the stiff case, where the estimates are independent of U0 and the commutator between A and B.
TL;DR: A time-splitting spectral method for the generalized Gross–Pitaevskii equations, which describe the dynamics of spinor F=1 Bose–Einstein condensates at a very low temperature, which is explicit, unconditionally stable, and of spectral accuracy in space.
Abstract: We propose a time-splitting spectral method for the generalized Gross-Pitaevskii equations, which describe the dynamics of spinor F=1 Bose-Einstein condensates at a very low temperature. The new numerical method is explicit, unconditionally stable, and of spectral accuracy in space. Moreover, it conserves the position densities at the discretized level. We apply the method for studying both the dynamic generation of vortices and vortex lattice dynamics for the spinor F=1 Bose-Einstein condensates held in an Ioffe-Pritchard magnetic field.
TL;DR: A novel pattern recognition method for selecting co-expressed genes based on rate of change and modulation status of gene expression at each time interval is proposed and a quality index based on the semantic similarity in gene annotations to assess the likelihood of a cluster being a co-regulated group is developed.
Abstract: Identification of co-expressed genes sharing similar biological behaviours is an essential step in functional genomics. Traditional clustering techniques are generally based on overall similarity of expression levels and often generate clusters with mixed profile patterns. A novel pattern recognition method for selecting co-expressed genes based on rate of change and modulation status of gene expression at each time interval is proposed in this paper. This method is capable of identifying gene clusters consisting of highly similar shapes of expression profiles and modulation patterns. Furthermore, we develop a quality index based on the semantic similarity in gene annotations to assess the likelihood of a cluster being a co-regulated group. The effectiveness of the proposed methodology is demonstrated by applying it to the well-known yeast sporulation dataset and an in-house cancer genomics dataset.
TL;DR: This work tries to solve a fuzzy system of linear equations having fuzzy coefficients and crisp variables using a polynomial parametric form of fuzzy numbers.
Abstract: A systems of linear equations are used in many fields of science and industry, such as control theory and image processing, and solving a fuzzy linear system of equations is now a necessity. In this work we try to solve a fuzzy system of linear equations having fuzzy coefficients and crisp variables using a polynomial parametric form of fuzzy numbers.
TL;DR: A class of singularly perturbed two point boundary value problems of convection-diffusion type for third-order ordinary differential equations (ODEs) with a small positive parameter multi-plying the highest derivative and a discontinuous source term is considered.
Abstract: A class of singularly perturbed two point boundary value problems (BVPs) of convection-diffusion type for third-order ordinary differential equations (ODEs) with a small positive parameter (e) multi-plying the highest derivative and a discontinuous source term is considered. The BVP is reduced to a weakly coupled system consisting of one first-order ordinary differential equation with a suitable initial condition and one second-order singularly perturbed ODE subject to boundary conditions. In order to solve this system, a computational method is suggested. In the proposed method we first find a zero-order asymptotic expansion approximation of the solution of the weakly coupled system. Then the system is decoupled by replacing the first component of the solution by its zero-order asymptotic expansion approximation in the second equation. Then the second equation is solved by a finite difference method on a Shishkin mesh (a fitted mesh method). Examples are provided to illustrate the method.
TL;DR: In this paper, the solution of a generalized Hirota-Satsuma Korteweg-de Vires (KdV) equation using a pseudospectral method is presented.
Abstract: In this paper the solution of a generalized Hirota-Satsuma Korteweg-de Vires (KdV) equation using a pseudospectral method is presented. To reduce roundoff error we use some preconditionings. Firstly, we discretize the equation in space to obtain a system of time-dependent ordinary differential equations. Secondly, we solve the obtained system of ordinary differential equations using the fourth-order Runge-Kutta method. The method works very well because the absolute errors are very small.
TL;DR: In this article, a Taylor method is developed for finding the approximate solution of high-order linear Fredholm integro-differential equations in the most general form under the mixed conditions, where the problem is defined on the interval [-1, 1] and the solution is obtained in terms of Taylor polynomials about the origin.
Abstract: A Taylor method is developed for finding the approximate solution of high-order linear Fredholm integro-differential equations in the most general form under the mixed conditions. The problem is defined on the interval [-1, 1] and the solution is obtained in terms of Taylor polynomials about the origin. Transforming the interval [a, b] to the interval [-1, 1], a problem defined on [a, b] can also be solved using this method. Numerical examples are presented to illustrate the accuracy of the method.
TL;DR: In this article, the stability analysis of BDF Slowest-first multirate timeintegration methods applied to the transient analysis of circuit models is investigated and it appears that these methods are indeed stable if the subsystems are stable and weakly coupled.
Abstract: This paper deals with the stability analysis of BDF Slowest-first multirate time-integration methods applied to the transient analysis of circuit models. From an asymptotic analysis it appears that these methods are indeed stable if the subsystems are stable and weakly coupled.
TL;DR: The purpose of the present paper is to introduce another line of reasoning concerning the accuracy of operator splitting and to obtain a bound on the difference between the fully coupled implicit discrete solutions and the solutions obtained by applying operator splitting to these discrete equations.
Abstract: Nonlinear systems of time dependent partial differential equations of real life phenomena tend to be very complicated to solve numerically. One common approach to solve such problems is by applying operator splitting which introduces subproblems that are easier to handle. The solutions of the subsystems are usually glued together by either the Godunov method (first-order) or the Strang method (second-order). However, the accuracy of such an approach may be very hard to analyse because of the complexity of the equations involved. The purpose of the present paper is to introduce another line of reasoning concerning the accuracy of operator splitting. Let us assume that a fully coupled and implicit discretization of the complete system has been developed. Under appropriate conditions on the continuous problem, such discretizations provide reasonable and convergent approximations. As in the continuous case, operator splitting can be utilized to obtain tractable algebraic subsystems. The problem we address in the present paper is to obtain a bound on the difference between the fully coupled implicit discrete solutions, uh, and the solutions, uh, s, obtained by applying operator splitting to these discrete equations. Suppose we know that uh converges to the analytical solution u as the grid is properly refined and, applying the results from this paper, that uh, s converges toward uh under grid refinement. Then, by the triangle inequality, also the splitting approximation uh, s converges toward the analytical solution u and convergence is thus obtained for an approximation that, from a practical point of view, is easier to compute.
TL;DR: A new spectral successive integration matrix is used to construct a Chebyshev expansion method for the solution of boundary value problems and application to the linear stability problem for plane Poiseuille flow is presented.
Abstract: 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.
TL;DR: It is proved that the solution u 0(x) exists and is unique, and the error estimation for an elliptic membrane is given, which is smaller than the error with other models.
Abstract: In this paper, we split the 3-D linearly elastic shell problem into a 2-D problem by using the approach of formal asymptotic expansion. The approximate solution UKT(x, ξ) for the 3-D linearly elastic shell consists of [image omitted] , where u0, u1, u2 are independent of ξ. The leading term u0 satisfies a 2-D elliptic boundary value problem, and other terms u1, u2 will be derived using the algebraic expression for u0 without solving PDEs. We prove that the solution u0(x) exists and is unique, and give the error estimation for an elliptic membrane, which is smaller than the error with other models.