On the validity of the local Fourier analysis

Local Fourier analysis (LFA) is a useful tool in predicting the convergence factors of geometric multigrid methods (GMG). As is well known, on rectangular domains with periodic boundary conditions this analysis gives the exact convergence factors of such methods. In this work, using the Fourier method, we extend these results by proving that such analysis yields the exact convergence factors for a wider class of problems

Fourier Analysis of Modified Nested Factorization Preconditioner for Three-Dimensional Isotropic Problems

For solving large sparse symmetric linear systems, arising from the discretization of elliptic problems, the preferred choice is the preconditioned con- jugate gradient method. The convergence rate of this method mainly depends on the condition number of the preconditioner chosen. Using Fourier analy- sis the condition number estimate of common preconditioning techniques for two dimensional elliptic problem has been studied by Chan and Elman [SIAM Rev., 31 (1989), pp. 20-49]. Nested Factorization(NF) is one of the powerful preconditioners for systems arising from discretization of elliptic or hyperbolic partial differential equations. The observed convergence behavior of NF is bet- ter compared to well known ILU(0) or modified ILU. In this paper we introduce Modified Nested Factorization(MNF) which is an improvement over NF. It is proved that condition number of modified NF is O(h−1 ). An optimal value of the parameter for the model problem is derived. The condition number of modified NF predicts the condition number of NF in limiting sense when the parameter is close to zero. Moreover it is proved that condition number of NF is atleast O(h−1 ). Numerical results justify Fourier analytic method by exhibiting remarkable similarity in spectrum of periodic and Dirichlet problems

Modified Tangential Frequency Filtering Decomposition and its Fourier Analysis

In this paper, a modified tangential frequency filtering decomposition (MTFFD) preconditioner is proposed. The optimal order of the modification and the optimal relaxation parameter are determined by Fourier analysis. With this choice of the optimal order of modification, the Fourier results show that the condition number of the preconditioned matrix is ${\cal O}(h^{-\frac{2}{3}})$, and the spectrum distribution of the preconditioned matrix can be predicted by the Fourier results. The performance of MTFFD is compared with tangential frequency filtering (TFFD) preconditioner on a variety of large sparse matrices arising from the discretization of PDEs with discontinuous coefficients. The numerical results show that the MTFFD preconditioner is much more efficient than the TFFD preconditioner

V-cycle optimal convergence for DCT-III matrices

The paper analyzes a two-grid and a multigrid method for matrices belonging to the DCT-III algebra and generated by a polynomial symbol. The aim is to prove that the convergence rate of the considered multigrid method (V-cycle) is constant independent of the size of the given matrix. Numerical examples from differential and integral equations are considered to illustrate the claimed convergence properties.Comment: 19 page

SEMIVARIOGRAM METHODS FOR MODELING WHITTLE-MATERN PRIORS IN BAYESIAN INVERSE PROBLEMS

We present a new technique, based on semivariogram methodology, for obtaining point estimates for use in prior modeling for solving Bayesian inverse problems. This method requires a connection between Gaussian processes with covariance operators defined by the Matérn covariance function and Gaussian processes with precision (inverse-covariance) operators defined by the Green’s functions of a class of elliptic stochastic partial differential equations (SPDEs). We present a detailed mathematical description of this connection. We will show that there is an equivalence between these two Gaussian processes when the domain is infinite which breaks down when the domain is finite due to the effect of boundary conditions on Green’s functions of PDEs. We show how this connection can be re-established using extended domains. We then introduce the semivariogram method for estimating the Mat ́ern covariance parameters, which specify the Gaussian prior needed for stabilizing the inverse problem. Results are extended from the isotropic case to the anisotropic case where the correlation length in one direction is larger than another. The situation where the correlation length is spatially depen- dent rather than constant will also be considered. Finally, we compare and contrast the semivariogram method with a fully-Bayesian approach of finding estimates for and quantifying uncertainty in the hyperparameters. We imple- ment each method in two-dimensional image inpainting test cases to show that it works on practical examples. The MATLAB code for all of these methods can be found here: https://github.com/rbrown53/DissertationCodes

A fast direct solver for nonlocal operators in wavelet coordinates

In this article, we consider fast direct solvers for nonlocal operators. The pivotal idea is to combine a wavelet representation of the system matrix, yielding a quasi-sparse matrix, with the nested dissection ordering scheme. The latter drastically reduces the fill-in during the factorization of the system matrix by means of a Cholesky decomposition or an LU decomposition, respectively. This way, we end up with the exact inverse of the compressed system matrix with only a moderate increase of the number of nonzero entries in the matrix. To illustrate the efficacy of the approach, we conduct numerical experiments for different highly relevant applications of nonlocal operators: We consider (i) the direct solution of boundary integral equations in three spatial dimensions, issuing from the polarizable continuum model, (ii) a parabolic problem for the fractional Laplacian in integral form and (iii) the fast simulation of Gaussian random fields

Fast Numerical Methods for Non-local Operators

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