2,171 research outputs found
Preconditioning complex symmetric linear systems
A new polynomial preconditioner for symmetric complex linear systems based on
Hermitian and skew-Hermitian splitting (HSS) for complex symmetric linear
systems is herein presented. It applies to Conjugate Orthogonal Conjugate
Gradient (COCG) or Conjugate Orthogonal Conjugate Residual (COCR) iterative
solvers and does not require any estimation of the spectrum of the coefficient
matrix. An upper bound of the condition number of the preconditioned linear
system is provided. Moreover, to reduce the computational cost, an inexact
variant based on incomplete Cholesky decomposition or orthogonal polynomials is
proposed. Numerical results show that the present preconditioner and its
inexact variant are efficient and robust solvers for this class of linear
systems. A stability analysis of the method completes the description of the
preconditioner.Comment: 26 pages, 4 figures, 4 table
Convergence rates of Kernel Conjugate Gradient for random design regression
We prove statistical rates of convergence for kernel-based least squares
regression from i.i.d. data using a conjugate gradient algorithm, where
regularization against overfitting is obtained by early stopping. This method
is related to Kernel Partial Least Squares, a regression method that combines
supervised dimensionality reduction with least squares projection. Following
the setting introduced in earlier related literature, we study so-called "fast
convergence rates" depending on the regularity of the target regression
function (measured by a source condition in terms of the kernel integral
operator) and on the effective dimensionality of the data mapped into the
kernel space. We obtain upper bounds, essentially matching known minimax lower
bounds, for the (prediction) norm as well as for the stronger
Hilbert norm, if the true regression function belongs to the reproducing kernel
Hilbert space. If the latter assumption is not fulfilled, we obtain similar
convergence rates for appropriate norms, provided additional unlabeled data are
available
Fast convergence of trust-regions for non-isolated minima via analysis of CG on indefinite matrices
Trust-region methods (TR) can converge quadratically to minima where the
Hessian is positive definite. However, if the minima are not isolated, then the
Hessian there cannot be positive definite. The weaker
Polyak\unicode{x2013}{\L}ojasiewicz (P{\L}) condition is compatible with
non-isolated minima, and it is enough for many algorithms to preserve good
local behavior. Yet, TR with an subproblem solver lacks even
basic features such as a capture theorem under P{\L}.
In practice, a popular subproblem solver is the truncated
conjugate gradient method (tCG). Empirically, TR-tCG exhibits super-linear
convergence under P{\L}. We confirm this theoretically.
The main mathematical obstacle is that, under P{\L}, at points arbitrarily
close to minima, the Hessian has vanishingly small, possibly negative
eigenvalues. Thus, tCG is applied to ill-conditioned, indefinite systems. Yet,
the core theory underlying tCG is that of CG, which assumes a positive definite
operator. Accordingly, we develop new tools to analyze the dynamics of CG in
the presence of small eigenvalues of any sign, for the regime of interest to
TR-tCG
Application of wavelets to singular integral scattering equations
The use of orthonormal wavelet basis functions for solving singular integral
scattering equations is investigated. It is shown that these basis functions
lead to sparse matrix equations which can be solved by iterative techniques.
The scaling properties of wavelets are used to derive an efficient method for
evaluating the singular integrals. The accuracy and efficiency of the wavelet
transforms is demonstrated by solving the two-body T-matrix equation without
partial wave projection. The resulting matrix equation which is characteristic
of multiparticle integral scattering equations is found to provide an efficient
method for obtaining accurate approximate solutions to the integral equation.
These results indicate that wavelet transforms may provide a useful tool for
studying few-body systems.Comment: 11 pages, 4 figure
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