iosr-Jm   Volume-2 ~ Issue-2

Abstract: This paper presents fast iterative algorithms for solution of PDEs arisen from minimization of multiplicative noise removal model [14]. This model may be regarded as an improved version of the Total Variation (TV) de-noising models. For the TV and the multiplicative noise removal models, their associated Euler-Lagrange equations are highly nonlinear Partial Differential Equations (PDEs). For this model a very slow explicit time marching method has been reported. The main contribution we present in this paper is the implementation of the fixed point, semi-implicit and additive operator splitting schemes which do not yield good results. Consequently a fast and efficient multi-grid method with AOS as smoother is developed. Numerical experiments are presented to show the good performance of the fast multi-grid algorithm.
Key words. Synthetic Aperture Radar (SAR), Total Variation (TV)-based noise reduction, AOS (Additive Operator Splitting), Multi-Grid (MG), BV-Bounded Variation.

[1] G. Aubert and J. F. Aujol, A variational approach to removing multiplicative noise,SIAM Journal on Applied Mathematics 68 (2008), no. 4, 925–946.
[2] G. Aubert and P. Kornprobst, Mathematical problems in image processing of applied mathematical sciences, Springer, Berlin, Germany 147 (2002).
[3] N. Badshah and K. Chen, Multigrid method for the chan-vese model in variational segmentation, Communications in Computational Physics 4 (2008), no. 2, 294–316.
[4] N. Badshah and K. Chen, On two multi-grid algorithms for modelling variational multi-phase image segmentation, IEEE transactions on image Processing 18 (2009),no. 5, 1097–1106.7 Conclusion 13
[5] C.B. Burkhardt, Speckle in ultrasound b-mode scans,, IEEE Trans. Ultrasonic, 25 (1978), no. 1, 1–6.
[6] V. Vaselles G. Sapiro C. Ballester, M. Bertalmio and J. Verera, Filling in by joing interpolation of vector fields and grey levels, IMA Technical Report, university of Minnesota 69 (2002), no. 7, 131–147.
[7] Y. Liping C. Sheng, Y. Xin and S. Kun, Total variation based speckle reduction using multigrid algorithm for ultrasound images,, Springer-Verlag Berlin Heidelberg 36 (2005), no. 17, 245–252.
[8] T. F. Chan and K. Chen, On a nonlinear multi-grid algorithm with primal relaxation for the image total variation minimization, SIAM J. Sci. Comput. 20 (2006), no. 13, 387–411.
[9] R. Deriche, Fast algorithms for low-level vision, IEEE Transactions pattern Anal. Mach. Intell. 12 (1990), no. 9, 78–87.
[10] X. Zeng F. Tian Z. Li G, Liu and K. Chaibou, Speckle reduction by adaptive window anisotropic diffusion, signal processing 89 (2009), no. 11, 233–243.

Abstract : In this paper, the challenging difficulties encountered in solving non-polynomial variable coefficients differential equations by the use ofChebyshev expansion method is resolved. In such problems, where f(x) is non-polynomial, there exists the need to express products like f(x)T (x) r in series of Chebyshev polynomials for easy comparison of both sides of the differential equation. Numerical experiments is carried out on notable functions and the results are presented.
Keyword: Chebyshev polynomials, taylor's series expansion, non-polynomial variable coeefficients, .

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Abstract : We explore the use of Legendre polynomials of the first kind in solving constant coefficients , nonhomogenous differential equations. To achieve this, trial solution is formulated with the use of Legendre polynomials as basis functions. We thereafter apply direct and indirect comparison techniques to reduce the entire problem whether initial or boundary value problems into a system of algebraic equations. Numerical examples are given to illustrate the efficiency and good performance of these methods.
Keywords: Algebraic equations, Direct comparison, Indirect comparison , Legendre polynomials.

[1] Mason, J.C. and Handscomb, D.C. (2003): Chebyshev Polynomials, Rhapman & Hall – CRC, Roca Raton, London, New York, Washington D.C
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Abstract: In this paper, it is shown that neither of the iterative methods always converges. That is, it is possible to apply the Jacobi method or the Gauss-Seidel method to a system of linear equations and obtain a divergent sequence of approximations. In such cases, it is said that the method diverges.So for convergence, the Diagonal Dominance of the matrix is necessary condition before applying any iterative methods. Moreover, also discussed about the error reduction factor in each iteration in Jacobi and Gauss-Seidel method.
Keywords: Jacobi Method, Gauss-Seidel Method, Convergence and Divergence, Diagonal Dominance, Reduction of Error.

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