Tridiagonal Systems Of Linear Equations Research Articles

Graded mesh B-spline collocation method for two parameters singularly perturbed boundary value problems

The solutions of two parameters singularly perturbed boundary value problems typically exhibit two boundary layers. Because of the presence of these layers standard numerical methods fail to give accurate approximations. This paper introduces a numerical treatment of a class of two parameters singularly perturbed boundary value problems whose solution exhibits boundary layer phenomena. A graded mesh is considered to resolve the boundary layers and collocation method with cubic B-splines on the graded mesh is proposed and analyzed. The proposed method leads to a tri-diagonal linear system of equations. The stability and parameters uniform convergence of the present method are examined. To verify the theoretical estimates and efficiency of the method several known test problems in the literature are considered. Comparisons to some existing results are made to show the better efficiency of the proposed method. Summing up:•The present method is found to be stable and parameters uniform convergent and the numerical results support the theoretical findings.•Experimental results show that the present method approximates the solution very well and has a rate of convergence of order two in the maximum norm.•Experimental results show that cubic B-spline collocation method on graded mesh is more efficient than cubic B-spline collocation method on Shishkin mesh and some other existing methods in the literature.

The N-shaped partition method: A novel parallel implementation of the Crank Nicolson algorithm

We develop an algorithm to solve tridiagonal systems of linear equations, which appear in implicit finite-difference schemes of partial differential equations (PDEs), being the time-dependent Schrödinger equation (TDSE) an ideal candidate to benefit from it. Our N-shaped partition method optimizes the implementation of the numerical calculation on parallel architectures, without memory size constraints. Specifically, we discuss the realization of our method on graphics processing units (GPUs) and the Message Passing Interface (MPI). In GPU implementations, our scheme is particularly advantageous for systems whose size exceeds the global memory of a single processor. Moreover, because of its lack of memory constraints and the generality of the algorithm, it is well-suited for mixed architectures, typically available in large high performance computing (HPC) centers. We also provide an analytical estimation of the optimal parameters to implement our algorithm, and test numerically the suitability of our formula in a GPU implementation. Our method will be helpful to tackle problems which require large spatial grids for which ab-initio studies might be otherwise prohibitive both because of large shared-memory requirements and computation times.

An efficient two-level factored method for advection-dispersion problem with spatio-temporal coefficients and source terms

<abstract><p>A two-level factored implicit scheme is considered for solving a two-dimensional unsteady advection-dispersion equation with spatio-temporal coefficients and source terms subjected to suitable initial and boundary conditions. The approach reduces multi-dimensional problems into pieces of one-dimensional subproblems and then solves tridiagonal systems of linear equations. The computational cost of the algorithm becomes cheaper and makes the method more attractive. Furthermore, the two-level approach is unconditionally stable, temporal second-order accurate and spatial fourth-order convergent. The developed numerical scheme is faster and more efficient than a broad range of methods widely studied in the literature for the considered initial-boundary value problem. The stability of the proposed procedure is analyzed in the $ L^{\infty}(t_{0}, T_{f}; L^{2}) $-norm whereas the convergence rate of the algorithm is numerically analyzed using the $ L^{2}(t_{0}, T_{f}; L^{2}) $-norm. Numerical examples are provided to verify the theoretical result.</p></abstract>

Least Square Method for inferring Phylogenic trees

Nowadays, an important problem in Biomathematics is how to efficiently compute the length for the phylogenic tree branches. The approach followed herein is to employ the distance matrices method which leads to least square problems. The related tree construction strategy then generates block tridiagonal systems of linear equations. Experiments in solving them with the conjugate gradient method coupled to different block preconditioners were performed. Preliminary results confirm a high performance for some preconditioning choices.

Numerical Solution of Two-Parameter Singularly Perturbed Convection-Diffusion Boundary Value Problems via Fourth Order Compact Finite Difference Method

In this study, we have developed a fourth order compact finite difference method for finding the numerical solution of two-parameter singularly perturbed convection-diffusion boundary value problems. We have used fourth order compact finite difference method on uniform mesh which provides a tridiagonal linear system of equations. The convergence analysis of the proposed method is established through a matrix analysis approach and it is proved that present method gives fourth order convergence results. Present method is implemented on two numerical examples for checking the efficiency and precision of the method. Numerical outcomes are exhibited which supports the theoretical outcomes. Numerical outcomes are compared with other existing methods and found that present method gives more accurate approximate solution as compare to the other existing methods.

A Comparative Numerical Study of Parabolic Partial Integro-Differential Equation Arising from Convection-Diffusion

This article studies the development of two numerical techniques for solving convection-diffusion type partial integro-differential equation (PIDE) with a weakly singular kernel. Cubic trigonometric B-spline (CTBS) functions are used for interpolation in both methods. The first method is CTBS based collocation method which reduces the PIDE to an algebraic tridiagonal system of linear equations. The other method is CTBS based differential quadrature method which converts the PIDE to a system of ODEs by computing spatial derivatives as weighted sum of function values. An efficient tridiagonal solver is used for the solution of the linear system obtained in the first method as well as for determination of weighting coefficients in the second method. An explicit scheme is employed as time integrator to solve the system of ODEs obtained in the second method. The methods are tested with three nonhomogeneous problems for their validation. Stability, computational efficiency and numerical convergence of the methods are analyzed. Comparison of errors in approximations produced by the present methods versus different values of discretization parameters and convection-diffusion coefficients are made. Convection and diffusion dominant cases are discussed in terms of Peclet number. The results are also compared with cubic B-spline collocation method.

A fourth-order two-level factored implicit scheme for solving two-dimensional unsteady transport equation with time-dependent dispersion coefficients

This paper analyzes a two-level factored implicit scheme in a numerical solution of two-dimensional unsteady advection-diffusion equation with time dependent dispersion coefficients subjects to initial and boundary conditions. The proposed approach is fast and efficient: unconditionally stable, second order accurate in time, spatial fourth order convergent and it requires less computing time. In fact, the two-level factored technique reduces to solve a tridiagonal system of linear equations at each calculation step. This reduces the computational cost of the algorithm. The analysis of the stability of the numerical scheme considers the -norm while the error estimates and convergence rate use L 2-norm. A wide set of numerical evidences are presented and discussed.

Shifted Cubic Spline Wavelets with Two Vanishing Moments on the Interval and a Splitting Algorithm

This paper deals with the use of the first two vanishing moments for constructing cubic spline-wavelets orthogonal to polynomials of the first degree. A decrease in the supports of these wavelets is shown in comparison with the classical semiorthogonal wavelets. For splines with homogeneous Dirichlet boundary conditions of the second order, an algorithm of the shifted wavelet transform is obtained in the form of a solution of a tridiagonal system of linear equations with a strict diagonal dominance

Planar interpolation by second order spiral splines

Second order spiral splines are C2 unit-speed planar curves that can be used to interpolate a finite list of points in the Euclidean plane. A fast algorithm is given for interpolation when the data comes from a strictly convex planar curve. The method uses a pair of tridiagonal systems of linear equations to find an approximate interpolant. Then the approximation is used with standard software to construct an exact interpolant.•The data should be planar and strictly convex.•The method is robust and extremely fast.

Semi-implicit level set formulation for lithographic source and mask optimization.

The convergence of lithographic source and mask optimization (SMO) has been plagued by the prohibitive time-step dictated by the stability of the explicit Euler-forward scheme in the gradient-based optimization procedure. As a remedy, we solve the distance level-set regularized reformulation of the SMO by discretizing the stability-relevant terms in an implicit manner and apply operator splitting to separately update source and mask patterns in coordinate dimensions by solving the tridiagonal systems of linear equations using the Thomas method, combining stability and simplicity. Simulation results merit the superiority of the proposed SMO approach with improved convergence by overcoming the stability constraints of the Courant-Friedrichs-Lewy (CFL) condition.

RETRACTED: New algorithms for numerically solving a class of bordered tridiagonal systems of linear equations

This article has been retracted: please see Elsevier Policy on Article Withdrawal ( https://www.elsevier.com/about/our-business/policies/article-withdrawal ). This article has been retracted at the request of the Editors-in-Chief. After a thorough investigation, the Editors have concluded that the acceptance of this article was based upon the positive advice of at least one illegitimate reviewer report. The report was submitted from an email account which was provided to the journal as a suggested reviewer during the submission of the article. Although purportedly a real reviewer account, the Editors have concluded that this was not of an appropriate, independent reviewer. This manipulation of the peer-review process represents a clear violation of the fundamentals of peer review, our publishing policies, and publishing ethics standards. Apologies are offered to the reviewers whose identities were assumed and to the readers of the journal that this deception was not detected during the submission process.

FOURTH-ORDER COMPACT SCHEME FOR OPTION PRICING UNDER THE MERTON’S AND KOU’S JUMP-DIFFUSION MODELS

In this paper, a compact scheme with three time levels is proposed to solve the partial integro-differential equation that governs the option prices in jump-diffusion models. In the proposed compact scheme, the second derivative approximation of the unknowns is approximated using the value of these unknowns and their first derivative approximations, thereby allowing us to obtain a tridiagonal system of linear equations for a fully discrete problem. Moreover, the consistency and stability of the proposed compact scheme are proved. Owing to the low regularity of typical initial conditions, a smoothing operator is employed to ensure the fourth-order convergence rate. Numerical illustrations concerning the pricing of European options under the Merton’s and Kou’s jump-diffusion models are presented to validate the theoretical results.

High Resolution Finite Volume Scheme Based on the Quintic Spline Reconstruction on Non-uniform Grids

This paper presents a compact quintic spline reconstruction for finite volume method on non-uniform structured grids. The primitive function of a dependent variable is reconstructed by a piece-wise quintic polynomial by requiring the derivatives up to fourth order being continuous at the cell interfaces. This procedure results in a block tridiagonal system of linear equations which can be solved efficiently by incorporating certain boundary closure relations. There are some distinct features in the quintic spline reconstruction. Firstly, the reconstruction stencil is compact; Secondly, the reconstruction can be applied on arbitrary non-uniform grids; and finally, the reconstruction is continuous at cell interface without the need of a Riemann solver. To stabilize the scheme, the sixth order artificial viscosity is introduced. The quintic spline reconstruction achieves sixth-order accuracy on uniform grids without artificial viscosity and fifth-order accuracy on both uniform and non-uniform grids when artificial viscosity is added. The splined scheme is blended with shock-capturing WENO scheme to suppress non-physical oscillations near discontinuities. Numerical results demonstrate the accuracy, robustness and efficiency of the proposed scheme.

Fourth-Order Compact Finite Difference Scheme for American Option Pricing Under Regime-Switching Jump-Diffusion Models

In this paper, a fourth-order compact finite difference scheme is proposed for the solution of partial integro-differential equation arising in option pricing under jump-diffusion models. In proposed compact scheme, second derivative approximations of the unknowns are eliminated with the unknowns itself and their first derivative approximations while retaining the fourth order accuracy and tri-diagonal nature of the scheme. We use the proposed compact scheme with three time levels along with operator splitting technique to solve the linear complementary problem which arises in American option pricing under regime-switching jump-diffusion models. Moreover, it is shown that the proposed scheme leads to a tri-diagonal system of linear equations and fourth order accuracy is obtained. Since initial condition is not smooth enough for jump-diffusion model, we employ the smoothing operators to ensure high-order convergence rate. Numerical examples for American option pricing under Merton and Kou regime-switching jump-diffusion models are given.

Computationally inexpensive and revised normalized weighting factor method for segregated solvers

ABSTRACTA new methodology for the implementation of high-resolution (HR) scheme is being presented. The proposed method is a revision of the existing normalized weighting factor (NWF) method and henceforth, it is termed as revised normalized weighting factor (RNWF) method. Unlike the previous method which is penta-diagonal in nature, the RNWF method generates tri-diagonal system of linear equations and is expected to be computationally inexpensive and robust to use. Further to assess the performance of this method, iterations required for the convergence of HR schemes and to configure the CPU clock time, pure advection tests (viz. Step, Double-Step and Sine) were performed. These results were then compared with the existing methods such as, deferred correction (DC) and NWF which suggested that the proposed RNWF method is 4 times faster than DC and 1.3 times faster than NWF on uniform grids. To further evaluate the performance of this method, assessment tests for incompressible, newtonian fluid using the segregated solver (SIMPLE algorithm) are also detailed. Results suggest that RNWF method delivers computationally inexpensive and convergent numerical solutions when compared to the existing methods. Moreover, this method outperforms the NWF as it has been successfully applied for segregated solvers.

Shape Preserving Interpolation UsingC2Rational Cubic Spline

This paper discusses the construction of newC2rational cubic spline interpolant with cubic numerator and quadratic denominator. The idea has been extended to shape preserving interpolation for positive data using the constructed rational cubic spline interpolation. The rational cubic spline has three parametersαi,βi, andγi. The sufficient conditions for the positivity are derived on one parameterγiwhile the other two parametersαiandβiare free parameters that can be used to change the final shape of the resulting interpolating curves. This will enable the user to produce many varieties of the positive interpolating curves. Cubic spline interpolation withC2continuity is not able to preserve the shape of the positive data. Notably our scheme is easy to use and does not require knots insertion andC2continuity can be achieved by solving tridiagonal systems of linear equations for the unknown first derivativesdi,i=1,…,n-1. Comparisons with existing schemes also have been done in detail. From all presented numerical results the newC2rational cubic spline gives very smooth interpolating curves compared to some established rational cubic schemes. An error analysis when the function to be interpolated isft∈C3t0,tnis also investigated in detail.

A highly scalable parallel algorithm for solving Toeplitz tridiagonal systems of linear equations

Based on a modification of the dichotomy algorithm, we propose a novel parallel procedure for solving tridiagonal systems of equations with Toeplitz matrices. Taking the structure of the Toeplitz matrices, we may substantially reduce the number of the preliminary calculations of the dichotomy algorithm, which makes possible to efficiently solve systems of linear equations with both one and several right-hand sides. On examples of solving the 2D Poisson equation by the variable separation method and the 3D Poisson equation by a combination of the alternating direction implicit and the variable separation methods we show that the computation accuracy is comparable with the sequential version of the Thomas method, the dependence of the speedup on the number of processors being almost linear. The proposed modification is aimed at parallel implementation of a broad class of numerical methods including the Toeplitz tridiagonal matrices inversion.

A convergence analysis for a sweeping preconditioner for block tridiagonal systems of linear equations

SummaryWe study sweeping preconditioners for symmetric and positive definite block tridiagonal systems of linear equations. The algorithm provides an approximate inverse that can be used directly or in a preconditioned iterative scheme. These algorithms are based on replacing the Schur complements appearing in a block Gaussian elimination direct solve by hierarchical matrix approximations with reduced off‐diagonal ranks. This involves developing low rank hierarchical approximations to inverses. We first provide a convergence analysis for the algorithm for reduced rank hierarchical inverse approximation. These results are then used to prove convergence and preconditioning estimates for the resulting sweeping preconditioner. Copyright © 2014 John Wiley & Sons, Ltd.

An Optimized FFT-Based Direct Poisson Solver on CUDA GPUs

A highly multithreaded FFT-based direct Poisson solver that makes effective use of the capabilities of the current NVIDIA graphics processing units (GPUs) is presented. Our algorithms carefully manage the multiple layers of the memory hierarchy of the GPUs such that almost all the global memory accesses are coalesced into 128-byte device memory transactions, and all computations are carried out directly on the registers. A new strategy to interleave the FFT computation along each dimension with other computations is used to minimize the total number of accesses to the 3D grid. We illustrate the performance of our algorithms on the NVIDIA Tesla and Fermi architectures for a wide range of grid sizes, up to the largest size that can fit on the device memory ($(512\times 512\times 512)$ on the Tesla C1060/C2050 and $(512\times 256\times 256)$ on the GeForce GTX 280/480). We achieve up to 140 GFLOPS and a bandwidth of 70 GB/s on the Tesla C1060, and up to 375 GFLOPS with a bandwidth of 120GB/s on the GTX 480. The performance of our algorithms is superior to what can be achieved using the CUDA FFT library in combination with well-known parallel algorithms for solving tridiagonal linear systems of equations.

Algorithms for Solving Doubly Bordered Tridiagonal Linear Systems

In this paper we focus on solving doubly bordered tridiagonal linear systems of equations via transformation. It investigates numeric and symbolic algorithms for solving such systems. The computational cost of the algorithms are given. A MAPLE procedure based on these algorithms is listed. Some illustrative examples are introduced.