17 research outputs found
Multilevel Approach For Signal Restoration Problems With Toeplitz Matrices
We present a multilevel method for discrete ill-posed problems arising from the discretization of Fredholm integral equations of the first kind. In this method, we use the Haar wavelet transform to define restriction and prolongation operators within a multigrid-type iteration. The choice of the Haar wavelet operator has the advantage of preserving matrix structure, such as Toeplitz, between grids, which can be exploited to obtain faster solvers on each level where an edge-preserving Tikhonov regularization is applied. Finally, we present results that indicate the promise of this approach for restoration of signals and images with edges
MINRES-QLP: a Krylov subspace method for indefinite or singular symmetric systems
CG, SYMMLQ, and MINRES are Krylov subspace methods for solving symmetric
systems of linear equations. When these methods are applied to an incompatible
system (that is, a singular symmetric least-squares problem), CG could break
down and SYMMLQ's solution could explode, while MINRES would give a
least-squares solution but not necessarily the minimum-length (pseudoinverse)
solution. This understanding motivates us to design a MINRES-like algorithm to
compute minimum-length solutions to singular symmetric systems.
MINRES uses QR factors of the tridiagonal matrix from the Lanczos process
(where R is upper-tridiagonal). MINRES-QLP uses a QLP decomposition (where
rotations on the right reduce R to lower-tridiagonal form). On ill-conditioned
systems (singular or not), MINRES-QLP can give more accurate solutions than
MINRES. We derive preconditioned MINRES-QLP, new stopping rules, and better
estimates of the solution and residual norms, the matrix norm, and the
condition number.Comment: 26 pages, 6 figure
Symmetrization Techniques in Image Deblurring
This paper presents a couple of preconditioning techniques that can be used
to enhance the performance of iterative regularization methods applied to image
deblurring problems with a variety of point spread functions (PSFs) and
boundary conditions. More precisely, we first consider the anti-identity
preconditioner, which symmetrizes the coefficient matrix associated to problems
with zero boundary conditions, allowing the use of MINRES as a regularization
method. When considering more sophisticated boundary conditions and strongly
nonsymmetric PSFs, the anti-identity preconditioner improves the performance of
GMRES. We then consider both stationary and iteration-dependent regularizing
circulant preconditioners that, applied in connection with the anti-identity
matrix and both standard and flexible Krylov subspaces, speed up the
iterations. A theoretical result about the clustering of the eigenvalues of the
preconditioned matrices is proved in a special case. The results of many
numerical experiments are reported to show the effectiveness of the new
preconditioning techniques, including when considering the deblurring of sparse
images
Subspace Recycling for Sequences of Shifted Systems with Applications in Image Recovery
For many applications involving a sequence of linear systems with slowly
changing system matrices, subspace recycling, which exploits relationships
among systems and reuses search space information, can achieve huge gains in
iterations across the total number of linear system solves in the sequence.
However, for general (i.e., non-identity) shifted systems with the shift value
varying over a wide range, the properties of the linear systems vary widely as
well, which makes recycling less effective. If such a sequence of systems is
embedded in a nonlinear iteration, the problem is compounded, and special
approaches are needed to use recycling effectively.
In this paper, we develop new, more efficient, Krylov subspace recycling
approaches for large-scale image reconstruction and restoration techniques that
employ a nonlinear iteration to compute a suitable regularization matrix. For
each new regularization matrix, we need to solve regularized linear systems,
, for a sequence of regularization parameters,
, to find the optimally regularized solution that, in turn, will
be used to update the regularization matrix.
In this paper, we analyze system and solution characteristics to choose
appropriate techniques to solve each system rapidly. Specifically, we use an
inner-outer recycling approach with a larger, principal recycle space for each
nonlinear step and smaller recycle spaces for each shift. We propose an
efficient way to obtain good initial guesses from the principle recycle space
and smaller shift-specific recycle spaces that lead to fast convergence. Our
method is substantially reduces the total number of matrix-vector products that
would arise in a naive approach. Our approach is more generally applicable to
sequences of shifted systems where the matrices in the sum are positive
semi-definite