Nonlinear total variation based noise removal algorithms
Proceedings of the eleventh annual international conference of the Center for Nonlinear Studies on Experimental mathematics : computational issues in nonlinear science: computational issues in nonlinear science
Matrix computations (3rd ed.)
Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion
Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion
Iterative Regularization and MINRES
SIAM Journal on Matrix Analysis and Applications
LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares
ACM Transactions on Mathematical Software (TOMS)
A multigrid tutorial (2nd ed.)
A multigrid tutorial (2nd ed.)
Computational Methods for Inverse Problems
Computational Methods for Inverse Problems
V-cycle Optimal Convergence for Certain (Multilevel) Structured Linear Systems
SIAM Journal on Matrix Analysis and Applications
On the Regularizing Power of Multigrid-type Algorithms
SIAM Journal on Scientific Computing
A V-cycle Multigrid for multilevel matrix algebras: proof of optimality
Numerische Mathematik
A Projection-Based Approach to General-Form Tikhonov Regularization
SIAM Journal on Scientific Computing
Cascadic multilevel methods for ill-posed problems
Journal of Computational and Applied Mathematics
Cascadic Multiresolution Methods for Image Deblurring
SIAM Journal on Imaging Sciences
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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.