263,267 research outputs found
Some matrix nearness problems suggested by Tikhonov regularization
The numerical solution of linear discrete ill-posed problems typically
requires regularization, i.e., replacement of the available ill-conditioned
problem by a nearby better conditioned one. The most popular regularization
methods for problems of small to moderate size are Tikhonov regularization and
truncated singular value decomposition (TSVD). By considering matrix nearness
problems related to Tikhonov regularization, several novel regularization
methods are derived. These methods share properties with both Tikhonov
regularization and TSVD, and can give approximate solutions of higher quality
than either one of these methods
Non-perturbative regularization and renormalization: simple examples from non-relativistic quantum mechanics
We examine several zero-range potentials in non-relativistic quantum
mechanics. The study of such potentials requires regularization and
renormalization. We contrast physical results obtained using dimensional
regularization and cutoff schemes and show explicitly that in certain cases
dimensional regularization fails to reproduce the results obtained using cutoff
regularization. First we consider a delta-function potential in arbitrary space
dimensions. Using cutoff regularization we show that for the
renormalized scattering amplitude is trivial. In contrast, dimensional
regularization can yield a nontrivial scattering amplitude for odd dimensions
greater than or equal to five. We also consider a potential consisting of a
delta function plus the derivative-squared of a delta function in three
dimensions. We show that the renormalized scattering amplitudes obtained using
the two regularization schemes are different. Moreover we find that in the
cutoff-regulated calculation the effective range is necessarily negative in the
limit that the cutoff is taken to infinity. In contrast, in dimensional
regularization the effective range is unconstrained. We discuss how these
discrepancies arise from the dimensional regularization prescription that all
power-law divergences vanish. We argue that these results demonstrate that
dimensional regularization can fail in a non-perturbative setting.Comment: 19 pages, LaTeX, uses epsf.te
Fractional regularization matrices for linear discrete ill-posed problems
The numerical solution of linear discrete ill-posed problems typically requires regularization. Two of the most popular regularization methods are due to Tikhonov and Lavrentiev. These methods require the choice of a regularization matrix. Common choices include the identity matrix and finite difference approximations of a derivative operator. It is the purpose of the present paper to explore the use of fractional powers of the matrices {Mathematical expression} (for Tikhonov regularization) and A (for Lavrentiev regularization) as regularization matrices, where A is the matrix that defines the linear discrete ill-posed problem. Both small- and large-scale problems are considered. © 2013 Springer Science+Business Media Dordrecht
Regularization matrices determined by matrix nearness problems
This paper is concerned with the solution of large-scale linear discrete
ill-posed problems with error-contaminated data. Tikhonov regularization is a
popular approach to determine meaningful approximate solutions of such
problems. The choice of regularization matrix in Tikhonov regularization may
significantly affect the quality of the computed approximate solution. This
matrix should be chosen to promote the recovery of known important features of
the desired solution, such as smoothness and monotonicity. We describe a novel
approach to determine regularization matrices with desired properties by
solving a matrix nearness problem. The constructed regularization matrix is the
closest matrix in the Frobenius norm with a prescribed null space to a given
matrix. Numerical examples illustrate the performance of the regularization
matrices so obtained
- …