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

Two sides tangential filtering decomposition

AbstractIn this paper we study a class of preconditioners that satisfy the so-called left and/or right filtering conditions. For practical applications, we use a multiplicative combination of filtering based preconditioners with the classical ILU(0) preconditioner, which is known to be efficient. Although the left filtering condition has a more sound theoretical motivation than the right one, extensive tests on convection–diffusion equations with heterogeneous and anisotropic diffusion tensors reveal that satisfying left or right filtering conditions lead to comparable results. On the filtering vector, these numerical tests reveal that e=[1,…,1]T is a reasonable choice, which is effective and can avoid the preprocessing needed in other methods to build the filtering vector. Numerical tests show that the composite preconditioners are rather robust and efficient for these problems with strongly varying coefficients

Spectral methods for CFD

One of the objectives of these notes is to provide a basic introduction to spectral methods with a particular emphasis on applications to computational fluid dynamics. Another objective is to summarize some of the most important developments in spectral methods in the last two years. The fundamentals of spectral methods for simple problems will be covered in depth, and the essential elements of several fluid dynamical applications will be sketched

Domain decomposition preconditioning for the Helmholtz equation: a coarse space based on local Dirichlet-to-Neumann maps

In this thesis, we present a two-level domain decomposition method for the iterative solution of the heterogeneous Helmholtz equation. The Helmholtz equation governs wave propagation and scattering phenomena arising in a wide range of engineering applications. Its discretization with piecewise linear finite elements results in typically large, ill-conditioned, indefinite, and non- Hermitian linear systems of equations, for which standard iterative and direct methods encounter convergence problems. Therefore, especially designed methods are needed. The inherently parallel domain decomposition methods constitute a promising class of preconditioners, as they subdivide the large problems into smaller subproblems and are hence able to cope with many degrees of freedom. An essential element of these methods is a good coarse space. Here, the Helmholtz equation presents a particular challenge, as even slight deviations from the optimal choice can be fatal. We develop a coarse space that is based on local eigenproblems involving the Dirichlet-to-Neumann operator. Our construction is completely automatic, ensuring good convergence rates without the need for parameter tuning. Moreover, it naturally respects local variations in the wave number and is hence suited also for heterogeneous Helmholtz problems. Apart from the question of how to design the coarse space, we also investigate the question of how to incorporate the coarse space into the method. Also here the fact that the stiffness matrix is non-Hermitian and indefinite constitutes a major challenge. The resulting method is parallel by design and its efficiency is investigated for two- and three-dimensional homogeneous and heterogeneous numerical examples

Accelerating Cosmic Microwave Background map-making procedure through preconditioning

Estimation of the sky signal from sequences of time ordered data is one of the key steps in Cosmic Microwave Background (CMB) data analysis, commonly referred to as the map-making problem. Some of the most popular and general methods proposed for this problem involve solving generalised least squares (GLS) equations with non-diagonal noise weights given by a block-diagonal matrix with Toeplitz blocks. In this work we study new map-making solvers potentially suitable for applications to the largest anticipated data sets. They are based on iterative conjugate gradient (CG) approaches enhanced with novel, parallel, two-level preconditioners. We apply the proposed solvers to examples of simulated non-polarised and polarised CMB observations, and a set of idealised scanning strategies with sky coverage ranging from nearly a full sky down to small sky patches. We discuss in detail their implementation for massively parallel computational platforms and their performance for a broad range of parameters characterising the simulated data sets. We find that our best new solver can outperform carefully-optimised standard solvers used today by a factor of as much as 5 in terms of the convergence rate and a factor of up to $4$ in terms of the time to solution, and to do so without significantly increasing the memory consumption and the volume of inter-processor communication. The performance of the new algorithms is also found to be more stable and robust, and less dependent on specific characteristics of the analysed data set. We therefore conclude that the proposed approaches are well suited to address successfully challenges posed by new and forthcoming CMB data sets.Comment: 19 pages // Final version submitted to A&

Accelerating Cosmic Microwave Background map-making procedure through preconditioning

Estimation of the sky signal from sequences of time ordered data is one of the key steps in Cosmic Microwave Background (CMB) data analysis, commonly referred to as the map-making problem. Some of the most popular and general methods proposed for this problem involve solving generalised least squares (GLS) equations with non-diagonal noise weights given by a block-diagonal matrix with Toeplitz blocks. In this work we study new map-making solvers potentially suitable for applications to the largest anticipated data sets. They are based on iterative conjugate gradient (CG) approaches enhanced with novel, parallel, two-level preconditioners. We apply the proposed solvers to examples of simulated non-polarised and polarised CMB observations, and a set of idealised scanning strategies with sky coverage ranging from nearly a full sky down to small sky patches. We discuss in detail their implementation for massively parallel computational platforms and their performance for a broad range of parameters characterising the simulated data sets. We find that our best new solver can outperform carefully-optimised standard solvers used today by a factor of as much as 5 in terms of the convergence rate and a factor of up to $4$ in terms of the time to solution, and to do so without significantly increasing the memory consumption and the volume of inter-processor communication. The performance of the new algorithms is also found to be more stable and robust, and less dependent on specific characteristics of the analysed data set. We therefore conclude that the proposed approaches are well suited to address successfully challenges posed by new and forthcoming CMB data sets.Comment: 19 pages // Final version submitted to A&

Parallel Controllability Methods For the Helmholtz Equation

The Helmholtz equation is notoriously difficult to solve with standard numerical methods, increasingly so, in fact, at higher frequencies. Controllability methods instead transform the problem back to the time-domain, where they seek the time-harmonic solution of the corresponding time-dependent wave equation. Two different approaches are considered here based either on the first or second-order formulation of the wave equation. Both are extended to general boundary-value problems governed by the Helmholtz equation and lead to robust and inherently parallel algorithms. Numerical results illustrate the accuracy, convergence and strong scalability of controllability methods for the solution of high frequency Helmholtz equations with up to a billion unknowns on massively parallel architectures

Fourier domain preconditioned conjugate gradient algorithm for atmospheric tomography

By 'atmospheric tomography' we mean the estimation of a layered atmospheric turbulence profile from measurements of the pupil-plane phase (or phase gradients) corresponding to several different guide star directions. We introduce what we believe to be a new Fourier domain preconditioned conjugate gradient (FD-PCG) algorithm for atmospheric tomography, and we compare its performance against an existing multigrid preconditioned conjugate gradient (MG-PCG) approach. Numerical results indicate that on conventional serial computers, FD-PCG is as accurate and robust as MG-PCG, but it is from one to two orders of magnitude faster for atmospheric tomography on 30 m class telescopes. Simulations are carried out for both natural guide stars and for a combination of finite-altitude laser guide stars and natural guide stars to resolve tip-tilt uncertainty