Numerical differentiation from a viewpoint of regularization theory

In this paper, we discuss the classical ill-posed problem of numerical differentiation, assuming that the smoothness of the function to be differentiated is unknown. Using recent results on adaptive regularization of general ill-posed problems, we propose new rules for the choice of the stepsize in the finite-difference methods, and for the regularization parameter choice in numerical differentiation regularized by the iterated Tikhonov method. These methods are shown to be effective for the differentiation of noisy functions, and the order-optimal convergence results for them are proved.

     

求解参数识别反问题的同伦正则化方法

提出了求解参数识别反问题的同伦正则化方法,给出了相应的收敛性定理.数值结果表明该方法是一种快速的大范围收敛方法.     

Identification of nonstationary axisymmetric load distributed along a cylindrical shell

The paper outlines a procedure to identify the space-and time-dependent external nonstationary load acting on a closed circular cylindrical shell of medium thickness. Time-dependent deflections at several points of the shell are used as input data to solve the inverse problem. Examples of numerical identification of various nonstationary loads, including moving ones are presented. The relationship between the external load and the stress-strain state of the shell is described by the Volterra equation of the first kind. The identification problem is solved using Tikhonov's regularization method and Apartsin's h-regularization method __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 7, pp. 91–100, July 2008.     

广义病态方程的Tikhonov正则解

对于双方带扰动数据的病态方程(即所谓广义病态方程)，借助对Tikhonov正则化算法的改进，给出一种优良的正则化求解方法。     

Convergent Algorithm Based on Progressive Regularization for Solving Pseudomonotone Variational Inequalities

In this paper, we extend the Moreau-Yosida regularization of monotone variational inequalities to the case of weakly monotone and pseudomonotone operators. With these properties, the regularized operator satisfies the pseudo-Dunn property with respect to any solution of the variational inequality problem. As a consequence, the regularized version of the auxiliary problem algorithm converges. In this case, when the operator involved in the variational inequality problem is Lipschitz continuous (a property stronger than weak monotonicity) and pseudomonotone, we prove the convergence of the progressive regularization introduced in Refs. 1, 2.     

Characterization of Titania/Silica Gel by Means of Low-Pressure Nitrogen Adsorption

Adsorbents synthesized by grafting of titania onto mesoporous silica gel surfaces at different temperatures were studied by means of nitrogen adsorption–desorption and water desorption. The pore size distribution f(R_p) of titania/silica gel depends on the titania concentration (C_TiO2) and the temperature of titania synthesis. Nonuniformity of TiO₂ phase is maximal at a low C_TiO2 value (3.2 wt.% anatase deposited at 473 K), and two peaks of the fractal dimension distribution f(D) are observed at such a concentration of titania, but at larger C_TiO2 values, only one f(D) peak is seen. More ordered filling of pores and adsorption sites by nitrogen, reflecting in the shape of adsorption energy distributions f(E) at different pressures of adsorbate, is observed for adsorbent with titania (rutile+anatase) grafted on silica gel at a higher temperature (673 K).     

Parameter choice by discrepancy principles for ill-posed problems leading to optimal convergence rates

Schock (Ref. 1) considered a general a posteriori parameter choice strategy for the Tikhonov regularization of the ill-posed operator equationTx=y which provides nearly the optimal rate of convergence if the minimal-norm least-squares solution belongs to the range of the operator (T ^* T) ^v , o<v1. Recently, Nair (Ref. 2) improved the result of Schock and also provided the optimal rate ifv=1. In this note, we further improve the result and show in particular that the optimal rate can be achieved for 1/2v1.The final version of this work was written while M. T. Nair was a Visiting Fellow at the Centre for Mathematics and Its Applications, Australian National University, Canberra, Australia. The work of S. George was supported by a Senior Research Fellowship from CSIR, India.     

Prox-regular functions in variational analysis

The class of prox-regular functions covers all l.s.c., proper, convex functions, lower- functions and strongly amenable functions, hence a large core of functions of interest in variational analysis and optimization. The subgradient mappings associated with prox-regular functions have unusually rich properties, which are brought to light here through the study of the associated Moreau envelope functions and proximal mappings. Connections are made between second-order epi-derivatives of the functions and proto-derivatives of their subdifferentials. Conditions are identified under which the Moreau envelope functions are convex or strongly convex, even if the given functions are not.

     

2D Burgers equation with large Reynolds number using POD/DEIM and calibration

Model order reduction of the two‐dimensional Burgers equation is investigated. The mathematical formulation of POD/discrete empirical interpolation method (DEIM)‐reduced order model (ROM) is derived based on the Galerkin projection and DEIM from the existing high fidelity‐implicit finite‐difference full model. For validation, we numerically compared the POD ROM, POD/DEIM, and the full model in two cases of Re = 100 and Re = 1000, respectively. We found that the POD/DEIM ROM leads to a speed‐up of CPU time by a factor of O(10). The computational stability of POD/DEIM ROM is maintained by means of a careful selection of POD modes and the DEIM interpolation points. The solution of POD/DEIM in the case of Re = 1000 has an accuracy with error O(10^?3) versus O(10^?4) in the case of Re = 100 when compared with the high fidelity model. For this turbulent flow, a closure model consisting of a Tikhonov regularization is carried out in order to recover the missing information and is developed to account for the small‐scale dissipation effect of the truncated POD modes. It is shown that the computational results of this calibrated ROM exhibit considerable agreement with the high fidelity model, which implies the efficiency of the closure model used. Copyright © 2016 John Wiley & Sons, Ltd.