37,467 research outputs found
An Accelerated Conjugate Gradient Algorithm to Compute Low-Lying Eigenvalues --- a Study for the Dirac Operator in SU(2) Lattice QCD
The low-lying eigenvalues of a (sparse) hermitian matrix can be computed with
controlled numerical errors by a conjugate gradient (CG) method. This CG
algorithm is accelerated by alternating it with exact diagonalisations in the
subspace spanned by the numerically computed eigenvectors. We study this
combined algorithm in case of the Dirac operator with (dynamical) Wilson
fermions in four-dimensional \SUtwo gauge fields. The algorithm is
numerically very stable and can be parallelized in an efficient way. On
lattices of sizes an acceleration of the pure CG method by a factor
of~ is found.Comment: 25 pages, uuencoded tar-compressed .ps-fil
Accelerating two projection methods via perturbations with application to Intensity-Modulated Radiation Therapy
Constrained convex optimization problems arise naturally in many real-world
applications. One strategy to solve them in an approximate way is to translate
them into a sequence of convex feasibility problems via the recently developed
level set scheme and then solve each feasibility problem using projection
methods. However, if the problem is ill-conditioned, projection methods often
show zigzagging behavior and therefore converge slowly.
To address this issue, we exploit the bounded perturbation resilience of the
projection methods and introduce two new perturbations which avoid zigzagging
behavior. The first perturbation is in the spirit of -step methods and uses
gradient information from previous iterates. The second uses the approach of
surrogate constraint methods combined with relaxed, averaged projections.
We apply two different projection methods in the unperturbed version, as well
as the two perturbed versions, to linear feasibility problems along with
nonlinear optimization problems arising from intensity-modulated radiation
therapy (IMRT) treatment planning. We demonstrate that for all the considered
problems the perturbations can significantly accelerate the convergence of the
projection methods and hence the overall procedure of the level set scheme. For
the IMRT optimization problems the perturbed projection methods found an
approximate solution up to 4 times faster than the unperturbed methods while at
the same time achieving objective function values which were 0.5 to 5.1% lower.Comment: Accepted for publication in Applied Mathematics & Optimizatio
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