scholarly journals Low memory requirement and computational time method for solving a class of integral equations

2011 ◽  
Vol 3 ◽  
pp. 104-109
Author(s):  
K. Maleknejad ◽  
M. Nosrati Sahlan ◽  
P. Torabi
Keyword(s):  
Integral Equations ◽  
Computational Time ◽  
Memory Requirement

A HIGH PERFORMANCE LARGE SPARSE SYMMETRIC SOLVER FOR THE MESHFREE GALERKIN METHOD

2008 ◽  
Vol 05 (04) ◽  
pp. 533-550 ◽  
Author(s):  
S. C. WU ◽  
H. O. ZHANG ◽  
C. ZHENG ◽  
J. H. ZHANG
Keyword(s):  
High Performance ◽  
Information Matrix ◽  
Computational Cost ◽  
Initial Boundary ◽  
Field Analysis ◽  
Iteration Step ◽  
Gradient Field ◽  
Computational Time ◽  
Preconditioned Conjugate Gradient ◽  
Memory Requirement

One main disadvantage of meshfree methods is that their memory requirement and computational cost are much higher than those of the usual finite element method (FEM). This paper presents an efficient and reliable solver for the large sparse symmetric positive definite (SPD) system resulting from the element-free Galerkin (EFG) approach. A compact mathematical model of heat transfer problems is first established using the EFG procedure. Based on the widely used Successive Over-Relaxation–Preconditioned Conjugate Gradient (SSOR–PCG) scheme, a novel solver named FastPCG is then proposed for solving the SPD linear system. To decrease the computational time in each iteration step, a new algorithm for realizing multiplication of the global stiffness matrix by a vector is presented for this solver. The global matrix and load vector are changed in accordance with a special rule and, in this way, a large account of calculation is avoided on the premise of not decreasing the solution's accuracy. In addition, a double data structure is designed to tackle frequent and unexpected operations of adding or removing nodes in problems of dynamic adaptive or moving high-gradient field analysis. An information matrix is also built to avoid drastic transformation of the coefficient matrix caused by the initial-boundary values. Numerical results show that the memory requirement of the FastPCG solver is only one-third of that of the well-developed AGGJE solver, and the computational cost is comparable with the traditional method with the increas of solution scale and order.

scholarly journals IMPLEMENTATION OF QUADRATIC AXIAL TRIAL FUNCTIONS IN THE HIGH-FIDELITY TRANSPORT CODE PROTEUS-MOC

2021 ◽  
Vol 247 ◽  
pp. 03015
Author(s):  
Guangchun Zhang ◽  
Won Sik Yang
Keyword(s):  
Linear Approximation ◽  
Quadratic Approximation ◽  
Linear Functions ◽  
Computational Time ◽  
High Fidelity ◽  
Test Results ◽  
Memory Requirement ◽  
Transport Code ◽  
Axial Variation ◽  
Memory Reduction

PROTEUS-MOC is a pin-resolved high-fidelity transport code, in which the axial variation of angular flux is represented in terms of orthogonal polynomials. Currently, PROTEUS-MOC employs linear functions and requires relatively fine axial meshes to achieve high accuracy, which increases the number of axial meshes and hence the memory requirement. In this study, aiming to reduce the memory requirement and potentially the computational time by allowing larger axial meshes, we have extended the PROTEUS-MOC transport solution method to quadratic trial functions. Preliminary tests for the performance of quadratic trial functions have been performed using the 3-D C5G7 benchmark problem. Test results showed that for the same axial mesh configuration with relatively large sizes, the quadratic approximation yields about 2 to 5 times more accurate pin powers than the linear approximation, depending on the degree of axial variation of angular fluxes. The quadratic approximation also allows the use of about 3 times coarser axial meshes than the linear approximation for comparable pin power accuracy, which consequently reduces the memory requirement by about 2 times. The memory reduction is not proportional because of the increased number of coefficients in each element from 2 to 3. However, the quadratic approximation did not reduce the computational time as expected because of the deteriorated performance of the pCMFD acceleration scheme due to large axial mesh sizes.

The economized monic Chebyshev polynomials for solving weakly singular Fredholm integral equations of the first kind

2018 ◽  
Vol 13 (01) ◽  
pp. 2050030 ◽  
Author(s):  
E. S. Shoukralla ◽  
M. A. Markos
Keyword(s):  
Integral Equations ◽  
Chebyshev Polynomials ◽  
Unknown Function ◽  
Asymptotic Expression ◽  
High Order Accuracy ◽  
Computational Time ◽  
Fredholm Integral Equations ◽  
Algebraic Equations ◽  
Weakly Singular ◽  
Parametric Functions

This paper presents a numerical method for solving a certain class of Fredholm integral equations of the first kind, whose unknown function is singular at the end-points of the integration domain, and has a weakly singular logarithmic kernel with analytical treatments of the singularity. To achieve this goal, the kernel is parametrized, and the unknown function is assumed to be in the form of a product of two functions; the first is a badly-behaved known function, while the other is a regular unknown function. These two functions are approximated by using the economized monic Chebyshev polynomials of the same degree, while the given potential function is approximated by monic Chebyshev polynomials of the same degree. Further, the two parametric functions associated to the parametrized kernel are expanded into Taylor polynomials of the first degree about the singular parameter, and an asymptotic expression is created, so that the obtained improper integrals of the integral operator become convergent integrals. Thus, and after using a set of collocation points, the required numerical solution is found to be equivalent to the solution of a linear system of algebraic equations. From the illustrated example, it turns out that the proposed method minimizes the computational time and gives a high order accuracy.

scholarly journals A Robust and Accurate Quasi-Monte Carlo Algorithm for Estimating Eigenvalue of Homogeneous Integral Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
F. Mehrdoust ◽  
B. Fathi Vajargah ◽  
E. Radmoghaddam
Keyword(s):  
Monte Carlo ◽  
Integral Equations ◽  
Numerical Algorithm ◽  
Variance Reduction ◽  
Numerical Results ◽  
Monte Carlo Algorithm ◽  
Computational Time ◽  
Reduction Procedure ◽  
Quasi Monte Carlo

We present an efficient numerical algorithm for computing the eigenvalue of the linear homogeneous integral equations. The proposed algorithm is based on antithetic Monte Carlo algorithm and a low-discrepancy sequence, namely, Faure sequence. To reduce the computational time we reduce the variance by using the antithetic variance reduction procedure. Numerical results show that our scheme is robust and accurate.

Symmetric Multistep Methods Revisited: II. Numerical Experiments

1999 ◽  
Vol 173 ◽  
pp. 309-314 ◽  
Author(s):  
T. Fukushima
Keyword(s):  
Multistep Methods ◽  
Computational Time ◽  
Integration Time ◽  
Implicit Methods ◽  
Symmetric Methods ◽  
Celestial Bodies ◽  
The Stability ◽  
Predictor Corrector ◽  
Symmetric Multistep Methods ◽  
Integration Errors

AbstractBy using the stability condition and general formulas developed by Fukushima (1998 = Paper I) we discovered that, just as in the case of the explicit symmetric multistep methods (Quinlan and Tremaine, 1990), when integrating orbital motions of celestial bodies, the implicit symmetric multistep methods used in the predictor-corrector manner lead to integration errors in position which grow linearly with the integration time if the stepsizes adopted are sufficiently small and if the number of corrections is sufficiently large, say two or three. We confirmed also that the symmetric methods (explicit or implicit) would produce the stepsize-dependent instabilities/resonances, which was discovered by A. Toomre in 1991 and confirmed by G.D. Quinlan for some high order explicit methods. Although the implicit methods require twice or more computational time for the same stepsize than the explicit symmetric ones do, they seem to be preferable since they reduce these undesirable features significantly.

A Comparative Study of Ant and Genetic Algorithms in Digital Mammography

2018 ◽  
pp. 194-200
Author(s):  
J. Magelin Mary ◽  
Chitra K. ◽  
Y. Arockia Suganthi
Keyword(s):  
Genetic Algorithm ◽  
Image Processing ◽  
Medical Image ◽  
Medical Image Processing ◽  
Processing Technique ◽  
Input Image ◽  
Image Processing Technique ◽  
Computational Time ◽  
Image Mining ◽  
The Individual

Image processing technique in general, involves the application of signal processing on the input image for isolating the individual color plane of an image. It plays an important role in the image analysis and computer version. This paper compares the efficiency of two approaches in the area of finding breast cancer in medical image processing. The fundamental target is to apply an image mining in the area of medical image handling utilizing grouping guideline created by genetic algorithm. The parameter using extracted border, the border pixels are considered as population strings to genetic algorithm and Ant Colony Optimization, to find out the optimum value from the border pixels. We likewise look at cost of ACO and GA also, endeavors to discover which one gives the better solution to identify an affected area in medical image based on computational time.

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