817 research outputs found
Space Decompositions and Solvers for Discontinuous Galerkin Methods
We present a brief overview of the different domain and space decomposition
techniques that enter in developing and analyzing solvers for discontinuous
Galerkin methods. Emphasis is given to the novel and distinct features that
arise when considering DG discretizations over conforming methods. Connections
and differences with the conforming approaches are emphasized.Comment: 2 pages 2 figures no table
Euclid in a Taxicab: Sparse Blind Deconvolution with Smoothed l1/l2 Regularization
The l1/l2 ratio regularization function has shown good performance for
retrieving sparse signals in a number of recent works, in the context of blind
deconvolution. Indeed, it benefits from a scale invariance property much
desirable in the blind context. However, the l1/l2 function raises some
difficulties when solving the nonconvex and nonsmooth minimization problems
resulting from the use of such a penalty term in current restoration methods.
In this paper, we propose a new penalty based on a smooth approximation to the
l1/l2 function. In addition, we develop a proximal-based algorithm to solve
variational problems involving this function and we derive theoretical
convergence results. We demonstrate the effectiveness of our method through a
comparison with a recent alternating optimization strategy dealing with the
exact l1/l2 term, on an application to seismic data blind deconvolution.Comment: 5 page
Preconditioning Methods for Shift-Variant Image Reconstruction
Preconditioning methods can accelerate the convergence of gradient-based iterative methods for tomographic image reconstruction and image restoration. Circulant preconditioners have been used extensively for shift-invariant problems. Diagonal preconditioners offer some improvement in convergence rate, but do not incorporate the structure of the Hessian matrices in imaging problems. For inverse problems that are approximately shift-invariant (i.e. approximately block-Toeplitz or block-circulant Hessians), circulant or Fourier-based preconditioners can provide remarkable acceleration. However, in applications with nonuniform noise variance (such as arises from Poisson statistics in emission tomography and in quantum-limited optical imaging), the Hessian of the (penalized) weighted least-squares objective function is quite shift-variant, and the Fourier preconditioner performs poorly. Additional shift-variance is caused by edge-preserving regularization methods based on nonquadratic penalty functions. This paper describes new preconditioners that more accurately approximate the Hessian matrices of shift-variant imaging problems. Compared to diagonal or Fourier preconditioning, the new preconditioners lead to significantly faster convergence rates for the unconstrained conjugate-gradient (CG) iteration. Applications to position emission tomography (PET) illustrate the method.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85856/1/Fessler147.pd
Conjugate-Gradient Preconditioning Methods for Shift-Variant PET Image Reconstruction
Gradient-based iterative methods often converge slowly for tomographic image reconstruction and image restoration problems, but can be accelerated by suitable preconditioners. Diagonal preconditioners offer some improvement in convergence rate, but do not incorporate the structure of the Hessian matrices in imaging problems. Circulant preconditioners can provide remarkable acceleration for inverse problems that are approximately shift-invariant, i.e., for those with approximately block-Toeplitz or block-circulant Hessians. However, in applications with nonuniform noise variance, such as arises from Poisson statistics in emission tomography and in quantum-limited optical imaging, the Hessian of the weighted least-squares objective function is quite shift-variant, and circulant preconditioners perform poorly. Additional shift-variance is caused by edge-preserving regularization methods based on nonquadratic penalty functions. This paper describes new preconditioners that approximate more accurately the Hessian matrices of shift-variant imaging problems. Compared to diagonal or circulant preconditioning, the new preconditioners lead to significantly faster convergence rates for the unconstrained conjugate-gradient (CG) iteration. We also propose a new efficient method for the line-search step required by CG methods. Applications to positron emission tomography (PET) illustrate the method.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85979/1/Fessler85.pd
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