解第一类算子方程的一种新的正则化方法

对算子与右端都为近似给定的第一类算子方程提出一种新的正则化方法，依据广义Ａｒｃａｎｇｅｌｉ方法选取正则参数，建立了正则解的收敛性。这种新的正则化方法与通常的Ｔｉｋｈｏｎｏｖ正则化方法相比较，提高了正则解的渐近阶估计。     

A Modified Tikhonov Regularization for Linear Operator Equations

We construct with the aid of regularizing filters a new class of improved regularization methods, called modified Tikhonov regularization (MTR), for solving ill-posed linear operator equations. Regularizing properties and asymptotic order of the regularized solutions are analyzed in the presence of noisy data and perturbation error in the operator. With some accurate estimates in the solution errors, optimal convergence order of the regularized solutions is obtained by a priori choice of the regularization parameter. Furthermore, numerical results are given for several ill-posed integral equations, which not only roughly coincide with the theoretical results but also show that MTR can be more accurate than ordinary Tikhonov regularization (OTR).     

求解病态问题的一种改进的Tikhonov正则化--(1)正则化方法的建立

对于带有右端扰动数据的第一类紧算子方程的病态问题 ,本文应用正则化子建立了一类新的正则化求解方法 ,称之为改进的Tikonov正则化 ;通过适当选取正则参数 ,证明了正则解具有最优的渐近收敛阶 .与通常的Tikhonov正则化相比 ,这种改进的正则化可使正则解取到足够高的最优渐近阶     

Asymptotic analysis of singularly perturbed integro-differential equations with zero operator in the differential part

We consider an integro-differential system with identically zero operator in the differential part. We construct a regularized asymptotics of the solution of this system for two cases, one in which the integral operator contains an exponentially varying factor and the other in which it does not. On the basis of the resulting asymptotic expansion, we study the passage to the limit in the system and present conditions under which this passage is uniform on the entire range of the independent variable (including the boundary layer region).     

The Schr?dinger equation on cylinders and the n-torus

In this paper, we study the solutions to the Schr?dinger equation on some conformally flat cylinders and on the n-torus. First, we apply an appropriate regularization procedure. Using the Clifford algebra calculus with an appropriate Witt basis, the solutions can be expressed as multiperiodic eigensolutions to the regularized parabolic-type Dirac operator. We study their fundamental properties, give representation formulas of all these solutions in terms of multiperiodic generalizations of the elliptic functions in the context of the regularized parabolic-type Dirac operator. Furthermore, we also develop some integral representation formulas. In particular, we set up a Green type integral formula for the solutions to the homogeneous regularized Schr?dinger equation on cylinders and n-tori. Then, we treat the inhomogeneous Schr?dinger equation with prescribed boundary conditions in Lipschitz domains on these manifolds. We present an L _p -decomposition where one of the components is the kernel of the first-order differential operator that factorizes the cylindrical (resp. toroidal) Schr?dinger operator. Finally, we study the behavior of our results in the limit case where the regularization parameter tends to zero.     

Degeneration and regularization of the operator in the Cauchy problem for the Schrödinger equation

The Cauchy problem for the Schrödinger equation whose operator degenerates on a half-line is studied. In order to approximate a solution to the problem with degeneracy by solutions to well-posed problems, the notion of regularization for an operator with degeneracy is introduced; an approximative solution to a problem with degeneracy is defined as the limit of a sequence of regularized problems. The dependence of the approximative solution on the choice of the class of admissible regularizations is studied. The weak compactness of sequences of states determined by sequences of solutions to regularized problems in the topologies determined by the space of all bounded linear operators and by subspaces of mutually commuting bounded linear operators is investigated.     

改进的Tikhonov 正则化及其正则解的最优渐近阶估计

对于算子与右端都有扰动的第一类算子方程建立了一类新的正则化方法(称为改进的 Ｔｉｋｈｏｎｏｖ正则化)．应用紧算子的奇异系统和广义 Ａｒｃａｎｇｅｌｉ方法后验选取正则参数,证明了正则解具有最优的渐近阶并给出了相应的算例分析．     

Substantiation of a Theorem on Limit Transition in Singularly Perturbed Integral System With Diagonal Degeneration of a Kernel

We study a problem of limit transition (as the small parameter tends to zero) in integral singularly perturbed system with diagonal degeneration of a kernel. In the proof of the corresponding theorem on the limit transition we essentially use the structure of the main term of asymptotic behavior, the construction of which is performed by use of algorithm of regularization method developed by S. A. Lomov for integro-differential equations. The spectrum of the operator responsible for the regularization is composed of purely imaginary points, therefore the passage to the limit in the classical sense (i.e., in a continuous metric) in general case is impossible. In work we allocate the class of right parts in which a uniform transition in the classical sense will take place.     

Spectral aspects of regularization of the Cauchy problem for a degenerate equation

We study the Cauchy problem for an equation whose generating operator is degenerate on some subset of the coordinate space. To approximate a solution of the degenerate problem by solutions of well-posed problems, we define a class of regularizations of the degenerate operator in terms of conditions on the spectral properties of approximating operators. We show that the behavior (convergence, compactness, and the set of partial limits in some topology) of the sequence of solutions of regularized problems is determined by the deficiency indices of the degenerate operator. We define an approximative solution of the degenerate problem as the limit of the sequence of solutions of regularized problems and analyze the dependence of the approximative solution on the choice of an admissible regularization.     

A METHOD TO APPROACH OPTIMAL RESTORATION IN IMAGE RESTORATION PROBLEMS WITHOUT NOISE ENERGY INFORMATION

This paper proposes a new image restoration technique, in which the resulting regularized image approximates the optimal solution steadily. The affect of the regular-ization operator and parameter on the lower band and upper band energy of the residue of the regularized image is theoretically analyzed by employing wavelet transform. This paper shows that regularization operator should generally be lowstop and highpass. So this paper chooses a lowstop and highpass operator as regularization operator, and construct an optimization model which minimizes the mean squares residue of regularized solution to determine regularization parameter. Although the model is random, on the condition of this paper, it can be solved and yields regularization parameter and regularized solu-tion. Otherwise, the technique has a mechanism to predict noise energy. So, without noisei nformation, it can also work and yield good restoration results.     

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.     

Regularized collocation method for Fredholm integral equations of the first kind

In this paper we consider a collocation method for solving Fredholm integral equations of the first kind, which is known to be an ill-posed problem. An “unregularized” use of this method can give reliable results in the case when the rate at which smallest singular values of the collocation matrices decrease is known a priori. In this case the number of collocation points plays the role of a regularization parameter. If the a priori information mentioned above is not available, then a combination of collocation with Tikhonov regularization can be the method of choice. We analyze such regularized collocation in a rather general setting, when a solution smoothness is given as a source condition with an operator monotone index function. This setting covers all types of smoothness studied so far in the theory of Tikhonov regularization. One more issue discussed in this paper is an a posteriori choice of the regularization parameter, which allows us to reach an optimal order of accuracy for deterministic noise model without any knowledge of solution smoothness.     

应用正则化子建立求解不适定问题的正则化方法的探讨

根据紧算子的奇异系统理论,提出一种新的正则化子进而建立了一类新的求解不适定问题的正则化方法。分别通过正则参数的先验选取和后验确定方法,证明了正则解的收敛性并得到了其最优的渐近收敛阶;验证了应用Newton迭代法计算最佳参数的可行性。最后建立了当算子与右端均有扰动时相应的正则化求解策略。文中所述方法完善了一般优化正则化策略的构造理论。     

Analysis of Wave Propagation in 1D Inhomogeneous Media

In this paper, we consider the one-dimensional inhomogeneous wave equation with particular focus on its spectral asymptotic properties and its numerical resolution. In the first part of the paper, we analyze the asymptotic nodal point distribution of high-frequency eigenfunctions, which, in turn, gives further information about the asymptotic behavior of eigenvalues and eigenfunctions. We then turn to the behavior of eigenfunctions in the high- and low-frequency limit. In the latter case, we derive a homogenization limit, whereas in the first we show that a sort of self-homogenization occurs at high frequencies. We also remark on the structure of the solution operator and its relation to desired properties of any numerical approximation. We subsequently shift our focus to the latter and present a Galerkin scheme based on a spectral integral representation of the propagator in combination with Gaussian quadrature in the spectral variable with a frequency-dependent measure. The proposed scheme yields accurate resolution of both high- and low-frequency components of the solution and as a result proves to be more accurate than available schemes at large time steps for both smooth and nonsmooth speeds of propagation.     

Singularly Perturbed Nonlinear Integrodifferential Systems with Fast Varying Kernels

We consider nonlinear singularly perturbed integro-differential equations with fast varying kernels. It is assumed that the spectrum of the limiting operator lies in the closed left half-plane Re0. We derive an algorithm for obtaining regularized (in the sense of Lomov) asymptotic solutions in both the nonresonance and resonance cases. In deriving the algorithm, we essentially use the regularization apparatus for integral operators with fast varying kernels, developed earlier by the authors for linear integral and integro-differential systems. The algorithm is justified and the existence of a solution of the original nonlinear problem is proved by means of the Newton method for operator equations.     

The Geometry of the Phase Diffusion Equation

$ \tau(|{{\vec k}}|) \mbox{\bf $\Theta$}_T = -\nabla\cdot (B(|{{\vec k}}|)\cdot {{\vec k}}), \,\, {{\vec k}} = \nabla \mbox{\bf $\Theta$},$ and its regularization describes natural patterns and defects far from onset in large aspect ratio systems with rotational symmetry. In this paper we construct explicit solutions of the unregularized equation and suggest candidates for its weak solutions. We confirm these ideas by examining a fourth-order regularized equation in the limit of infinite aspect ratio. The stationary solutions of this equation include the minimizers of a free energy, and we show these minimizers are remarkably well-approximated by a second-order ``self-dual' equation. Moreover, the self-dual solutions give upper bounds for the free energy which imply the existence of weak limits for the asymptotic minimizers. In certain cases, some recent results of Jin and Kohn [28] combined with these upper bounds enable us to demonstrate that the energy of the asymptotic minimizers converges to that of the self-dual solutions in a viscosity limit. Received on October 30, 1998; final revision received July 6, 1999     

解第一类算子方程的一种迭代正则化方法

提出了一种求解第一类算子方程的新的迭代正则化方法,并依据广义Arcangeli方法选取正则参数,建立了正则解的收敛性.与通常的Tikhonov正则化方法相比较,提高了正则解的渐近阶估计.     

Stable numerical differentiation for the second order derivatives

Consider a numerical differential problem, which aims to compute the second order derivative of a function stably from its given noisy data. For this ill-posed problem, we introduce the Lavrent′ev regularization scheme by reformulating this differentiation problem as an integral equation of the first kind. The advantage of this proposed scheme is that we can give the regularizing solution by an explicit integral expression, therefore it is easy to be implemented. The a-priori and a-posterior choice strategies for the regularization parameter are considered, with convergence analysis and error estimate of the regularizing solution for noisy data based on the integral operator decomposition. The validity of the proposed scheme is shown by several numerical examples.     

Mathematical programs with vanishing constraints: a new regularization approach with strong convergence properties

Motivated by a recent method introduced by Kanzow and Schwartz [C. Kanzow and A. Schwartz, A new regularization method for mathematical programs with complementarity constraints with strong convergence properties, Preprint 296, Institute of Mathematics, University of Würzburg, Würzburg, 2010] for mathematical programs with complementarity constraints (MPCCs), we present a related regularization scheme for the solution of mathematical programs with vanishing constraints (MPVCs). This new regularization method has stronger convergence properties than the existing ones. In particular, it is shown that every limit point is at least M-stationary under a linear independence-type constraint qualification. If, in addition, an asymptotic weak nondegeneracy assumption holds, the limit point is shown to be S-stationary. Second-order conditions are not needed to obtain these results. Furthermore, some results are given which state that the regularized subproblems satisfy suitable standard constraint qualifications such that the existing software can be applied to these regularized problems.