Current Issue : January - March Volume : 2013 Issue Number : 1 Articles : 4 Articles
The performance of the multigrid method and the effect of different grid levels on the convergence rate are evaluated. The two-\r\n, three-, and four-level V-cycle multigrid methods with the Gauss-Seidel iterative solver are employed for this purpose. The\r\nnumerical solution of the one-dimensional Laplace equation with the Dirichlet boundary conditions is obtained using these\r\nmethods. For the Laplace equation, a two-frequency function involving high- and low-frequency components is defined. It is\r\nobserved that, however, the GS method can smooth out the high-frequency error components properly, but because the difference\r\nscheme for Laplace equation is remarkably concise, in the fine grids, a very large number of iterations are needed for extending\r\nthe boundary conditions into the domain. Furthermore, the obtained results reveal that the number of necessary iterations for\r\nconvergence is reduced considerably by employing the two-level multigrid algorithm. But increasing the number of levels of\r\nalgorithm does not have any significant effect on the convergence rate in this study....
Computational complexity is examined using the principle of increasing entropy. To consider computation as a physical process\r\nfrom an initial instance to the final acceptance is motivated because information requires physical representations and because\r\nmany natural processes complete in nondeterministic polynomial time (NP). The irreversible process with three or more degrees of\r\nfreedom is found intractable when, in terms of physics, flows of energy are inseparable from their driving forces. In computational\r\nterms, when solving a problem in the class NP, decisions among alternatives will affect subsequently available sets of decisions.\r\nThus the state space of a nondeterministic finite automaton is evolving due to the computation itself, hence it cannot be efficiently\r\ncontracted using a deterministic finite automaton. Conversely when solving problems in the class P, the set of states does not\r\ndepend on computational history, hence it can be efficiently contracted to the accepting state by a deterministic sequence of\r\ndissipative transformations. Thus it is concluded that the state set of class P is inherently smaller than the state set of class NP.\r\nSince the computational time needed to contract a given set is proportional to dissipation, the computational complexity class P\r\nis a proper (strict) subset of NP....
An analytic approximation to the solution of wave equation is studied. Wave equation is in radial form with indicated initial\r\nand boundary conditions, by variational iteration method it has been used to derive this approximation and some examples are\r\npresented to show the simplicity and efficiency of the method....
We introduce multiscale wavelet kernels to kernel principal component analysis (KPCA) to narrow down the search of parameters\r\nrequired in the calculation of a kernel matrix. This new methodology incorporatesmultiscale methods into KPCA for transforming\r\nmultiscale data. In order to illustrate application of our proposed method and to investigate the robustness of the wavelet kernel in\r\nKPCA under different levels of the signal to noise ratio and different types of wavelet kernel, we study a set of two-class clustered\r\nsimulation data. We show that WKPCA is an effective feature extraction method for transforming a variety of multidimensional\r\nclustered data into data with a higher level of linearity among the data attributes. That brings an improvement in the accuracy of\r\nsimple linear classifiers. Based on the analysis of the simulation data sets, we observe that multiscale translation invariant wavelet\r\nkernels for KPCA has an enhanced performance in feature extraction. The application of the proposed method to real data is also\r\naddressed....
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