The effect of a penalty term involving higher order derivatives on the distribution of phases in an elastic medium with a two‐well elastic potential

We consider the problem of minimizing 0<p<1, h∈?, σ>0, among functions u:?^d?Ω→?^d, u_∣?Ω=0, and measurable characteristic functions χ:Ω→?. Here ?⁺_h, ?^?, denote quadratic potentials defined on the space of all symmetric d×d matrices, h is the minimum energy of ?⁺_h and ε(u) denotes the symmetric gradient of the displacement field. An equilibrium state û, χ?, of I [·,·,h, σ] is termed one‐phase if χ?≡0 or χ?≡1, two‐phase otherwise. We investigate the way in which the distribution of phases is affected by the choice of the parameters h and σ. Copyright 2002 John Wiley & Sons, Ltd.     

Orientational deconvolution of two-dimensional static disorder by a Tikhonov regularized method for 2H solid state NMR of nano-tubular-oriented structures

A general trend of supramolecular chemistry is the building of mesoscopic-oriented nanotubes. When the typical radius of the channel almost fits the mean supramolecular cross-sectional radius, the guest supramolecule is ultraconfined in the channel. In that case, only rotational disorder around the channel axis is allowed. For sufficiently low temperatures, we expect this 2D disorder to be static on the NMR timescale, and it should reflect the local symmetry of the channels. In this article, we show that experiments performed with deuterium single crystal solid-state NMR as a function of rotation around the channel axis may lead to important information concerning the static orientational disorder of the confined supramolecules. Using an inversion method with Tikhonov regularization and a positivity constraint, that also takes into account of the nonideal pulse sequence response, the orientational probability density can be obtained from 1D experiments performed at different angles, even for significant disorder. As a first step, the method is validated on four different theoretical distributions. It is then applied to ²H NMR single crystal experiments performed on an archetype of parallel channels intergrowth compounds: selectively deuterated 1,10-decanedicarboxylic diacids in hydrogenated urea channels. The diacids form one-dimensional infinite hydrogen-bonded chains ultraconfined in urea linear channels.     

Anomaly Problem in the N=1 Supersymmetric Electrodynamics as a Consequence of the Inconsistency of the Dimensional Reduction Method

We compare the calculation of the two-loop -function in the N=1 supersymmetric electrodynamics regularized via higher derivatives and via dimensional reduction. We show that the renormalized effective action is the same for both regularizations. But in the method of higher derivatives, unlike in the dimensional reduction, the -function defined as the derivative of the renormalized coupling constant with respect to log turns out to be purely one-loop. The anomaly problem therefore does not occur in this regularization, because in the method of higher derivatives, the diagrams with counterterm insertions make a nonzero contribution, which is evaluated exactly in all orders of the perturbation theory. When dimensional reduction is used, this contribution is zero. We argue that this result is a consequence of the mathematical inconsistency of the dimensional reduction method and that just this inconsistency leads to the anomaly problem.     

A New Technique for Ill-Conditioned Linear Systems

The technique we propose for solving ill-conditioned linear systems consists of two steps. First we compute the regularized solution on some values of the regularization parameter . Then we use these solutions either to extrapolate at =0 or to estimate the regularized solution with determined by the generalized cross validation or by the L-curve method.     

Kernel polynomials for the solution of indefinite and ill-posed problems

We introduce a new family of semiiterative schemes for the solution of ill-posed linear equations with selfadjoint and indefinite operators. These schemes avoid the normal equation system and thus benefit directly from the structure of the problem. As input our method requires an enclosing interval of the spectrum of the indefinite operator, based on some a priori knowledge. In particular, for positive operators the schemes are mathematically equivalent to the so-called -methods of Brakhage. In a way, they can therefore be seen as appropriate extensions of the -methods to the indefinite case. This extension is achieved by substituting the orthogonal polynomials employed by Brakhage in the definition of the -methods by appropriate kernel polynomials. We determine the rate of convergence of the new methods and establish their regularizing properties.     

A regularization of nonlinear diffusion equations in a multiresolution framework

We develop a regularization technique for Perona–Malik diffusion equations that relies on multiresolution techniques. The main result of this paper is to show that the chosen discretization overcomes the ill posedness of the nonlinear Perona–Malik model. The resulting algorithm is tested and the results are compared with pixel‐based methods. Copyright © 2007 John Wiley & Sons, Ltd.     

非正则线性增广拉格朗日乘子法的收敛速率研究CSCD

Recently,an indefinite linearized augmented Lagrangian method(IL-ALM)was proposed for the convex programming problems with linear constraints.The IL-ALM differs from the linearized augmented Lagrangian method in that the augmented Lagrangian is linearized by adding an indefinite quadratic proximal term.But,it preserves the algorithmic feature of the linearized ALM and usually has the advantage to improve the performance.The IL-ALM is proved to be convergent from contraction perspective,but its convergence rate is still missing.This is mainly because that the indefinite setting destroys the structures when we directly employ the contraction frameworks.In this paper,we derive the convergence rate for this algorithm by using a different analysis.We prove that a worst-case O(1/t)convergence rate is still hold for this algorithm,where t is the number of iterations.Additionally we show that the customized proximal point algorithm can employ larger step sizes by proving its equivalence to the linearized ALM.     

Extended Gauss–Markov Theorem for Nonparametric Mixed-Effects Models

The Gauss–Markov theorem provides a golden standard for constructing the best linear unbiased estimation for linear models. The main purpose of this article is to extend the Gauss–Markov theorem to include nonparametric mixed-effects models. The extended Gauss–Markov estimation (or prediction) is shown to be equivalent to a regularization method and its minimaxity is addressed. The resulting Gauss–Markov estimation serves as an oracle to guide the exploration for effective nonlinear estimators adaptively. Various examples are discussed. Particularly, the wavelet nonparametric regression example and its connection with a Sobolev regularization is presented.     

A new zero-finder for Tikhonov regularization

Tikhonov regularization with the regularization parameter determined by the discrepancy principle requires the computation of a zero of a rational function. We describe a cubically convergent zero-finder for this purpose. AMS subject classification (2000)  65F22, 65H05, 65R32