A Mathematical Approach to the Rock Interface Problem

In the present paper a method is presented for the analysis of rock interface. The mechanical response of rock interface involves nonlinear unilateral phenomena as debonding and slip along the interface. Under the usual simplifying assumption of shear-normal stress decoupling, the problem has been studied in [1,2] by means of variational and hemivariational inequalities formulations. This assumption is here dropped and the problem is formulated in the form of a system of inequalities. Using Leray-Schauder degree theory together with regularization techniques based on mollifiers, we develop a mathematical theory applicable to the analysis of the aforementioned problem. This revised version was published online in August 2006 with corrections to the Cover Date.     

本刊英文版Vol.30(2014),No.7论文摘要(英文)

<正>Perturbations of Frames Dong Yang CHEN;;Lei LI;;Ben Tuo ZHENG Abstract In this paper,we give some sufficient conditions under which perturbations preserve Hilbert frames and near-Riesz bases.Similar results are also extended to frame sequences,Riesz sequences and Schauder frames.It is worth mentioning that some of our perturbation conditions are quite different from those used in the previous literatures on this topic.Perturbations of Moore-Penrose Metric Generalized Inverses of Linear Operators in Banach Spaces Hai Feng MA;;Shuang SUN;;Yu Wen WANG;;Wen Jing ZHENG     

Convergence analysis of the loping OS-EM iterative version of the circular radon transform

The loping OS-EM iteration is a numerically efficient regularization method for solving ill-posed problems. In this article we investigate the loping OS-EM iterative method in connection with the circular Radon transform. We show that the proposed method converges weakly for the noisy data. Numerical tests are presented for a linear problem related to photoacoustic tomography.     

HEISENBERG-GREINER p-退化椭圆算子的一类HARDY不等式及其应用

本文研究了p-退化椭圆Heisenberg-Greiner算子的一类Hardy不等式的证明问题.利用正则化及逼近的方法,获得了该算子的一类强Hardy不等式的结果,推广了已有文献中p的取值范围,并进一步弥补了去除原点的缺陷.作为应用,讨论了与该算子相关的一类非线性算子的正定性与下无界性,并给出了它的一个正解.     

A new development for the Tikhonov Theorem in nonlinear singular perturbation systems

This paper deals with the exponential stability of nonlinear perturbation systems under a new condition. A novel criterion of exponential stability of nonlinear systems is firstly given in a general form. In this criterion, a new kind of characteristic value is introduced, which makes the exponential stability measurable. Based on this criterion, a new development for the Tikhonov Theorem in nonlinear singular perturbation systems is presented.     

Sobolev regularization of a class of forward–backward parabolic equations

We address existence and asymptotic behaviour for large time of Young measure solutions   of the Dirichlet initial–boundary value problem for the equation _ut=∇⋅[φ(∇u)]$u_t=\mathrm{\nabla }\cdot [\phi (\mathrm{\nabla }u)]$, where the function φ need not satisfy monotonicity conditions. Under suitable growth conditions on φ  , these solutions are obtained by a “vanishing viscosity” method from solutions of the corresponding problem for the regularized equation _ut=∇⋅[φ(∇u)]+?Δ_ut$u_t=\mathrm{\nabla }\cdot [\phi (\mathrm{\nabla }u)]+?\mathrm{\Delta }{u}_t$. The asymptotic behaviour as t→∞$t\to \infty$ of Young measure solutions of the original problem is studied by ω-limit set techniques, relying on the tightness   of sequences of time translates of the limiting Young measure. When N=1$N=1$ this measure is characterized as a linear combination of Dirac measures with support on the branches of the graph of φ.     

Convergence rates in ℓ1-regularization when the basis is not smooth enough

Sparsity promoting regularization is an important technique for signal reconstruction and several other ill-posed problems. Theoretical investigation typically bases on the assumption that the unknown solution has a sparse representation with respect to a fixed basis. We drop this sparsity assumption and provide error estimates for nonsparse solutions. After discussing a result in this direction published earlier by one of the authors and co-authors, we prove a similar error estimate under weaker assumptions. Two examples illustrate that this set of weaker assumptions indeed covers additional situations which appear in applications.     

基于迭代正则化高斯-牛顿法的非线性Urysohn积分方程数值解

非线性Urysohn积分方程在许多领域中都有广泛的应用,但由于该方程具有不适定性的特点,数据的微小扰动可能导致解的巨大变化,给数值求解带来很大困难.为了获得稳定的、准确的数值解,本文利用迭代正则化高斯-牛顿法对此方程进行求解,给出了利用Sigmoid-型函数确定迭代正则化参数的方法.对一类重力测定问题进行了数值模拟,将得到的数值解和相应的精确解作比较.结果表明,本文提出的方法在求解非线性Urysohn积分方程时是可行的也是有效的.     

Deep learning for spatio‐temporal modeling: Dynamic traffic flows and high frequency trading

Deep learning applies hierarchical layers of hidden variables to construct nonlinear high dimensional predictors. Our goal is to develop and train deep learning architectures for spatio‐temporal modeling. Training a deep architecture is achieved by stochastic gradient descent and dropout for parameter regularization with a goal of minimizing out‐of‐sample predictive mean squared error. To illustrate our methodology, we first predict the sharp discontinuities in traffic flow data, and secondly, we develop a classification rule to predict short‐term futures market prices using order book depth. Finally, we conclude with directions for future research.     

Homogenization of contact problem with Coulomb's friction on periodic cracks

We consider the elasticity problem in a domain with contact on multiple periodic open cracks. The contact is described by the Signorini and Coulomb‐friction conditions. The problem is nonlinear, the dissipative functional depends on the unknown solution, and the existence of the solution for fixed period of the structure is usually proven by the fix‐point argument in the Sobolev spaces with a little higher regularity, H^1+α. We rescaled norms, trace, jump, and Korn inequalities in fractional Sobolev spaces with positive and negative exponents, using the unfolding technique, introduced by Griso, Cioranescu, and Damlamian. Then we proved the existence and uniqueness of the solution for friction and period fixed. Then we proved the continuous dependency of the solution to the problem with Coulomb's friction on the given friction and then estimated the solution using fixed‐point theorem. However, we were not able to pass to the strong limit in the frictional dissipative term. For this reason, we regularized the problem by adding a fourth‐order term, which increased the regularity of the solution and allowed the passing to the limit. This can be interpreted as micro‐polar elasticity.