277 research outputs found
Jordan canonical form of the Google matrix: A potential contribution to the PageRank computation
We consider the web hyperlink matrix used by Google for computing the PageRank whose form is given by A(c) = [cP + (1 - c)E]T, where P is a row stochastic matrix, E is a row stochastic rank one matrix, and c 08 [0,1]. We determine the analytic expression of the Jordan form of A (c) and, in particular, a rational formula for the PageRank in terms of c. The use of extrapolation procedures is very promising for the efficient computation of the PageRank when c is close or equal to 1
On the regularizing power of multigrid-type algorithms
We consider the deblurring problem of noisy and blurred images in
the case of known space invariant point spread functions with four
choices of boundary conditions. We combine an algebraic multigrid
previously defined ad hoc for structured matrices related to space
invariant operators (Toeplitz, circulants, trigonometric matrix
algebras, etc.) and the classical geometric multigrid studied in
the partial differential equations context. The resulting
technique is parameterized in order to have more degrees of
freedom: a simple choice of the parameters allows us to devise a
quite powerful regularizing method. It defines an iterative
regularizing method where the smoother itself has to be an
iterative regularizing method (e.g., conjugate gradient, Landweber,
conjugate gradient for normal equations, etc.).
More precisely, with respect to the smoother, the regularization
properties are improved and the total complexity is lower.
Furthermore, in several cases, when it is directly applied to the
system , the quality of the restored image is
comparable with that of all the best known techniques for the
normal equations , but the related
convergence is substantially faster. Finally, the associated
curves of the relative errors versus the iteration numbers are
``flatter'' with respect to the smoother
(the estimation of the stop iteration is less crucial).
Therefore, we
can choose multigrid procedures which are much more efficient than
classical techniques without losing accuracy in the restored image
(as often occurs when using preconditioning). Several numerical
experiments show the effectiveness of our proposals
On the asymptotic spectrum of finite element matrix sequences
We derive a new formula for the asymptotic eigenvalue distribution of stiffness matrices obtained by applying Fi finite elements with standard mesh refinement to the semielliptic PDE of second order in divergence form -\u25bd(\u39a\u25bdTu) = f on \u3a9, u = g on 02\u3a9. Here \u3a9 82 \u211d2, and K is supposed to be piecewise continuous and point wise symmetric semipositive definite. The symbol describing this asymptotic eigenvalue distribution depends on the PDE, but also both on the numerical scheme for approaching the underlying bilinear form and on the geometry of triangulation of the domain. Our work is motivated by recent results on the superlinear convergence behavior of the conjugate gradient method, which requires the knowledge of such asymptotic eigenvalue distributions for sequences of matrices depending on a discretization parameter h when h \u2192 0. We compare our findings with similar results for the finite difference method which were published in recent years. In particular we observe that our sequence of stiffness matrices is part of the class of generalized locally Toeplitz sequences for which many theoretical tools are available. This enables us to derive some results on the conditioning and preconditioning of such stiffness matrices
AMG preconditioning for nonlinear degenerate parabolic equations on nonuniform grids with application to monument degradation
Analysis of Multigrid Preconditioning for Implicit PDE Solvers for Degenerate Parabolic Equations
Abstract. In this paper an implicit numerical method designed for nonlinear degenerate parabolic equations is proposed. A convergence analysis and the study of the related computa-tional cost are provided. In fact, due to the nonlinear nature of the underlying mathematical model, the use of a fixed point scheme is required. The chosen scheme is the Newton method and its con-vergence is proven under mild assumptions. Every step of the Newton method implies the solution of large, locally structured, linear systems. A special effort is devoted to the spectral analysis of the relevant matrices and to the design of appropriate multigrid preconditioned Krylov methods. Numerical experiments for the validation of our analysis complement this contribution
Fast non-Hermitian Toeplitz eigenvalue computations, joining matrix-less algorithms and FDE approximation matrices
The present work is devoted to the eigenvalue asymptotic expansion of the
Toeplitz matrix whose generating function is complex valued and
has a power singularity at one point. As a consequence, is
non-Hermitian and we know that the eigenvalue computation is a non-trivial task
in the non-Hermitian setting for large sizes. We follow the work of Bogoya,
B\"ottcher, Grudsky, and Maximenko and deduce a complete asymptotic expansion
for the eigenvalues. After that, we apply matrix-less algorithms, in the spirit
of the work by Ekstr\"om, Furci, Garoni, Serra-Capizzano et al, for computing
those eigenvalues. Since the inner and extreme eigenvalues have different
asymptotic behaviors, we worked on them independently, and combined the results
to produce a high precision global numerical and matrix-less algorithm.
The numerical results are very precise and the computational cost of the
proposed algorithms is independent of the size of the considered matrices for
each eigenvalue, which implies a linear cost when all the spectrum is computed.
From the viewpoint of real world applications, we emphasize that the matrix
class under consideration includes the matrices stemming from the numerical
approximation of fractional diffusion equations. In the final conclusion
section a concise discussion on the matter and few open problems are presented.Comment: 21 page
Superoptimal Preconditioned Conjugate Gradient Iteration for Image Deblurring
We study the superoptimal Frobenius operators in the two-level circulant algebra. We consider two specific viewpoints: ( 1) the regularizing properties in imaging and ( 2) the computational effort in connection with the preconditioned conjugate gradient method. Some numerical experiments illustrating the effectiveness of the proposed technique are given and discussed
A Numerical Simulation for Darcy-Forchheimer Flow of Nanofluid by a Rotating Disk With Partial Slip Effects
This study examines Darcy-Forchheimer 3D nanoliquid flow caused by a rotating disk with heat generation/absorption. The impacts of Brownian motion and thermophoretic are considered. Velocity, concentration, and thermal slips at the surface of the rotating disk are considered. The change from the non-linear partial differential framework to the non-linear ordinary differential framework is accomplished by utilizing appropriate variables. A shooting technique is utilized to develop a numerical solution of the resulting framework. Graphs have been sketched to examine how the concentration and temperature fields are affected by several pertinent flow parameters. Skin friction and local Sherwood and Nusselt numbers are additionally plotted and analyzed. Furthermore, the concentration and temperature fields are enhanced for larger values of the thermophoresis parameter
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