106 research outputs found
Preconditioners for ill-conditioned Toeplitz matrices
This paper is concerned with the solution of systems of linear equations ANχ
A preconditioned MINRES method for optimal control of wave equations and its asymptotic spectral distribution theory
In this work, we propose a novel preconditioned Krylov subspace method for
solving an optimal control problem of wave equations, after explicitly
identifying the asymptotic spectral distribution of the involved sequence of
linear coefficient matrices from the optimal control problem. Namely, we first
show that the all-at-once system stemming from the wave control problem is
associated to a structured coefficient matrix-sequence possessing an eigenvalue
distribution. Then, based on such a spectral distribution of which the symbol
is explicitly identified, we develop an ideal preconditioner and two
parallel-in-time preconditioners for the saddle point system composed of two
block Toeplitz matrices. For the ideal preconditioner, we show that the
eigenvalues of the preconditioned matrix-sequence all belong to the set
well separated from zero, leading to
mesh-independent convergence when the minimal residual method is employed. The
proposed {parallel-in-time} preconditioners can be implemented efficiently
using fast Fourier transforms or discrete sine transforms, and their
effectiveness is theoretically shown in the sense that the eigenvalues of the
preconditioned matrix-sequences are clustered around , which leads to
rapid convergence. When these parallel-in-time preconditioners are not fast
diagonalizable, we further propose modified versions which can be efficiently
inverted. Several numerical examples are reported to verify our derived
localization and spectral distribution result and to support the effectiveness
of our proposed preconditioners and the related advantages with respect to the
relevant literature
A block -circulant based preconditioned MINRES method for wave equations
In this work, we propose an absolute value block -circulant
preconditioner for the minimal residual (MINRES) method to solve an all-at-once
system arising from the discretization of wave equations. Since the original
block -circulant preconditioner shown successful by many recently is
non-Hermitian in general, it cannot be directly used as a preconditioner for
MINRES. Motivated by the absolute value block circulant preconditioner proposed
in [E. McDonald, J. Pestana, and A. Wathen. SIAM J. Sci. Comput.,
40(2):A1012-A1033, 2018], we propose an absolute value version of the block
-circulant preconditioner. Our proposed preconditioner is the first
Hermitian positive definite variant of the block -circulant
preconditioner, which fills the gap between block -circulant
preconditioning and the field of preconditioned MINRES solver. The
matrix-vector multiplication of the preconditioner can be fast implemented via
fast Fourier transforms. Theoretically, we show that for properly chosen
the MINRES solver with the proposed preconditioner has a linear
convergence rate independent of the matrix size. To the best of our knowledge,
this is the first attempt to generalize the original absolute value block
circulant preconditioner in the aspects of both theory and performance.
Numerical experiments are given to support the effectiveness of our
preconditioner, showing that the expected optimal convergence can be achieved
Spectral analysis for preconditioning of multi-dimensional Riesz fractional diffusion equations
In this paper, we analyze the spectra of the preconditioned matrices arising
from discretized multi-dimensional Riesz spatial fractional diffusion
equations. The finite difference method is employed to approximate the
multi-dimensional Riesz fractional derivatives, which will generate symmetric
positive definite ill-conditioned multi-level Toeplitz matrices. The
preconditioned conjugate gradient method with a preconditioner based on the
sine transform is employed to solve the resulting linear system. Theoretically,
we prove that the spectra of the preconditioned matrices are uniformly bounded
in the open interval (1/2,3/2) and thus the preconditioned conjugate gradient
method converges linearly. The proposed method can be extended to multi-level
Toeplitz matrices generated by functions with zeros of fractional order. Our
theoretical results fill in a vacancy in the literature. Numerical examples are
presented to demonstrate our new theoretical results in the literature and show
the convergence performance of the proposed preconditioner that is better than
other existing preconditioners
A note on parallel preconditioning for the all-at-once solution of Riesz fractional diffusion equations
The -step backwards difference formula (BDF) for solving the system of
ODEs can result in a kind of all-at-once linear systems, which are solved via
the parallel-in-time preconditioned Krylov subspace solvers (see McDonald,
Pestana, and Wathen [SIAM J. Sci. Comput., 40(2) (2018): A1012-A1033] and Lin
and Ng [arXiv:2002.01108, 17 pages]. However, these studies ignored that the
-step BDF () is not selfstarting, when they are exploited to solve
time-dependent PDEs. In this note, we focus on the 2-step BDF which is often
superior to the trapezoidal rule for solving the Riesz fractional diffusion
equations, but its resultant all-at-once discretized system is a block
triangular Toeplitz system with a low-rank perturbation. Meanwhile, we first
give an estimation of the condition number of the all-at-once systems and then
adapt the previous work to construct two block circulant (BC) preconditioners.
Both the invertibility of these two BC preconditioners and the eigenvalue
distributions of preconditioned matrices are discussed in details. The
efficient implementation of these BC preconditioners is also presented
especially for handling the computation of dense structured Jacobi matrices.
Finally, numerical experiments involving both the one- and two-dimensional
Riesz fractional diffusion equations are reported to support our theoretical
findings.Comment: 18 pages. 2 figures. 6 Table. Tech. Rep.: Institute of Mathematics,
Southwestern University of Finance and Economics. Revised-1: refine/shorten
the contexts and correct some typos; Revised-2: correct some reference
Some fast algorithms in signal and image processing.
Kwok-po Ng.Thesis (Ph.D.)--Chinese University of Hong Kong, 1995.Includes bibliographical references (leaves 138-139).AbstractsSummaryIntroduction --- p.1Summary of the papers A-F --- p.2Paper A --- p.15Paper B --- p.36Paper C --- p.63Paper D --- p.87Paper E --- p.109Paper F --- p.12
FAST PRECONDITIONERS FOR TOTAL VARIATION DEBLURRING WITH ANTIREFLECTIVE BOUNDARY CONDITIONS
Natural Science Foundation of Fujian Province of China for Distinguished Young Scholars [2010J06002]; NCET; U. Insubria; MIUR [20083KLJEZ]In recent works several authors have proposed the use of precise boundary conditions (BCs) for blurring models, and they proved that the resulting choice (Neumann or reflective, antireflective) leads to fast algorithms both for deblurring and for detecting the regularization parameters in presence of noise. When considering a symmetric point spread function, the crucial fact is that such BCs are related to fast trigonometric transforms. In this paper we combine the use of precise BCs with the total variation (TV) approach in order to preserve the jumps of the given signal (edges of the given image) as much as possible. We consider a classic fixed point method with a preconditioned Krylov method (usually the conjugate gradient method) for the inner iteration. Based on fast trigonometric transforms, we propose some preconditioning strategies that are suitable for reflective and antireflective BCs. A theoretical analysis motivates the choice of our preconditioners, and an extensive numerical experimentation is reported and critically discussed. Numerical tests show that the TV regularization with antireflective BCs implies not only a reduced analytical error, but also a lower computational cost of the whole restoration procedure over the other BCs
TR-2008005: Weakly Random Additive Preconditioning for Matrix Computations
Our weakly random additive preconditioners facilitate the solution of linear systems of equa-tions and other fundamental matrix computations. Compared to the popular SVD-based multiplicative preconditioners, these preconditioners are generated more readily and for a much wider class of input matrices. Furthermore they better preserve matrix structure and sparseness and have a wider range of applications, in particular to linear systems with rectangular coefficient matrices. We study the generation of such preconditioners and their impact on conditioning of the input matrix. Our analysis and experiments show the power of our approach even where w
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