Search results for " convergence."

Helmholtz equation in unbounded domains: some convergence results for a constrained optimization problem

2016

We consider a constrained optimization problem arising from the study of the Helmholtz equation in unbounded domains. The optimization problem provides an approximation of the solution in a bounded computational domain. In this paper we prove some estimates on the rate of convergence to the exact solution.

On the Convergence of Formal Integrals in Finite Time

1982

Consider a differential system: x = f (x) + e g(x), \(x \in {R^n}.\). Let h(x) = ho(x) + eh1 (x)... a “third” integral. For finite time t, I obtain an eo such that the series h(x) converges if e > eo. When t tends to infinite, eo tends to zero.

Approximation method for computationally expensive nonconvex multiobjective optimization problems

2012

Variational Henstock integrability of Banach space valued functions

2016

We study the integrability of Banach space valued strongly measurable functions defined on $[0,1]$. In the case of functions $f$ given by $\sum \nolimits _{n=1}^{\infty } x_n\chi _{E_n}$, where $x_n $ are points of a Banach space and the sets $E_n$ are Lebesgue measurable and pairwise disjoint subsets of $[0,1]$, there are well known characterizations for Bochner and Pettis integrability of $f$. The function $f$ is Bochner integrable if and only if the series $\sum \nolimits _{n=1}^{\infty }x_n|E_n|$ is absolutely convergent. Unconditional convergence of the series is equivalent to Pettis integrability of $f$. In this paper we give some conditions for variational Henstock integrability of a…

On the equivalence between the Scheduled Relaxation Jacobi method and Richardson's non-stationary method

2017

The Scheduled Relaxation Jacobi (SRJ) method is an extension of the classical Jacobi iterative method to solve linear systems of equations ($Au=b$) associated with elliptic problems. It inherits its robustness and accelerates its convergence rate computing a set of $P$ relaxation factors that result from a minimization problem. In a typical SRJ scheme, the former set of factors is employed in cycles of $M$ consecutive iterations until a prescribed tolerance is reached. We present the analytic form for the optimal set of relaxation factors for the case in which all of them are different, and find that the resulting algorithm is equivalent to a non-stationary generalized Richardson's method. …

Multiphoton-ionization transition amplitudes and the Keldysh approximation.

1989

The Keldysh approximation to treat the multiphoton ionization of atoms is reconsidered. It is shown that, if one consistently uses the hypothesis under which the approximation should be valid (essentially, that of a weak, short-range binding potential), a Keldysh-like term results as an approximation to the first term of a uniformly convergent series in powers of the binding potential. No cancellation occurs when higher-order terms are taken into account. This result allows one to consider the Keldysh approximation as a well-defined theoretical model, without implying, however, that it is adequate to describe multiphoton ionization of real atoms.

Assessment of the accuracy of coupled cluster perturbation theory for open-shell systems. I. Triples expansions

2016

The accuracy at which total energies of open-shell atoms and organic radicals may be calculated is assessed for selected coupled cluster perturbative triples expansions, all of which augment the coupled cluster singles and doubles (CCSD) energy by a non-iterative correction for the effect of triple excitations. Namely, the second- through sixth-order models of the recently proposed CCSD(T-n) triples series [J. Chem. Phys. 140, 064108 (2014)] are compared to the acclaimed CCSD(T) model for both unrestricted as well as restricted open-shell Hartree-Fock (UHF/ROHF) reference determinants. By comparing UHF- and ROHF-based statistical results for a test set of 18 modest-sized open-shell species …

Relaxation of periodic and nonstandard growth integrals by means of two-scale convergence

2019

An integral representation result is obtained for the variational limit of the family functionals $\int_{\Omega}f\left(\frac{x}{\varepsilon}, Du\right)dx$, as $\varepsilon \to 0$, when the integrand $f = f (x,v)$ is a Carath\'eodory function, periodic in $x$, convex in $v$ and with nonstandard growth.

Effective theory for low-energy nuclear energy density functionals

2012

We introduce a new class of effective interactions to be used within the energy-density-functional approaches. They are based on regularized zero-range interactions and constitute a consistent application of the effective-theory methodology to low-energy phenomena in nuclei. They allow for defining the order of expansion in terms of the order of derivatives acting on the finite-range potential. Numerical calculations show a rapid convergence of the expansion and independence of results of the regularization scale.

Comparison of two non-primitive methods for path integral simulations: Higher-order corrections vs. an effective propagator approach

2002

Two methods are compared that are used in path integral simulations. Both methods aim to achieve faster convergence to the quantum limit than the so-called primitive algorithm (PA). One method, originally proposed by Takahashi and Imada, is based on a higher-order approximation (HOA) of the quantum mechanical density operator. The other method is based upon an effective propagator (EPr). This propagator is constructed such that it produces correctly one and two-particle imaginary time correlation functions in the limit of small densities even for finite Trotter numbers P. We discuss the conceptual differences between both methods and compare the convergence rate of both approaches. While th…