SOLUTION OF BACKWARD HEAT PROBLEM BY MOROZOV DISCREPANCY PRINCIPLE AND CONDITIONAL STABILITY

Consider a 1-D backward heat conduction problem with Robin boundary condition. We recover u(x, 0) and u(x, t0) for to ∈(0, T) from the measured data u(x, T) respectively. The first problem is solved by the Morozov discrepancy principle for which a 3-order iteration procedure is applied to determine the regularizing parameter. For the second one, we combine the conditional stability with the Tikhonov regularization together to construct the regularizing solution for which the convergence rate is also established. Numerical results are given to show the validity of our inversion method     

A FAST CONVERGENT METHOD OF ITERATED REGULARIZATION

This article presents a fast convergent method of iterated regularization based on the idea of Landweber iterated regularization, and a method for a-posteriori choice by the Morozov discrepancy principle and the optimum asymptotic convergence order of the regularized solution is obtained. Numerical test shows that the method of iterated regularization can quicken the convergence speed and reduce the calculation burden efficiently.     

BLOCK-TRIANGULAR PRECONDITIONERS FOR SYSTEMS ARISING FROM EDGE-PRESERVING IMAGE RESTORATION

Signal and image restoration problems are often solved by minimizing a cost function consisting of an l2 data-fidelity term and a regularization term. We consider a class of convex and edge-preserving regularization functions. In specific, half-quadratic regularization as a fixed-point iteration method is usually employed to solve this problem. The main aim of this paper is to solve the above-described signal and image restoration problems with the half-quadratic regularization technique by making use of the Newton method. At each iteration of the Newton method, the Newton equation is a structured system of linear equations of a symmetric positive definite coefficient matrix, and may be efficiently solved by the preconditioned conjugate gradient method accelerated with the modified block SSOR preconditioner. Our experimental results show that the modified block-SSOR preconditioned conjugate gradient method is feasible and effective for further improving the numerical performance of the half-quadratic regularization approach.     

MULTI-PARAMETER TIKHONOV REGULARIZATION FOR LINEAR ILL-POSED OPERATOR EQUATIONS

We consider solving linear ill-posed operator equations. Based on a multi-scale decomposition for the solution space, we propose a multi-parameter regularization for solving the equations. We establish weak and strong convergence theorems for the multi-parameter regularization solution. In particular, based on the eigenfunction decomposition, we develop a posteriori choice strategy for multi-parameters which gives a regularization solution with the optimal error bound. Several practical choices of multi-parameters are proposed. We also present numerical experiments to demonstrate the outperformance of the multiparameter regularization over the single parameter regularization.     

Convergence and Optimality of Adaptive Regularization for Ill-posed Deconvolution Problems in Infinite Spaces

The adaptive regularization method is first proposed by Ryzhikov et al. in [6] for the deconvolution in elimination of multiples which appear frequently in geoscience and remote sensing. They have done experiments to show that this method is very effective. This method is better than the Tikhonov regularization in the sense that it is adaptive, i.e., it automatically eliminates the small eigenvalues of the operator when the operator is near singular. In this paper, we give theoretical analysis about the adaptive regularization. We introduce an a priori strategy and an a posteriori strategy for choosing the regularization parameter, and prove regularities of the adaptive regularization for both strategies. For the former, we show that the order of the convergence rate can approach O（｜｜n｜｜^4v/4v＋1） for some 0 〈 v 〈 1, while for the latter, the order of the convergence rate can be at most O（｜｜n｜｜^2v/2v＋1） for some 0 〈 v 〈 1.     

Regularity of Solutions to Elliptic Equations with VMO Coefficients

The aim of this paper is to study the regularity of solutions to the Dirichlet problems for general second-order elliptic equations in Lebesgue and Morrey spaces. We consider both nondivergence and divergence forms and the coefficients of principle terms are assumed to be in VMO.     

Sufficient dimension reduction in the presence of controlling variables

We are concerned with partial dimension reduction for the conditional mean function in the presence of controlling variables. We suggest a profile least squares approach to perform partial dimension reduction for a general class of semi-parametric models. The asymptotic properties of the resulting estimates for the central partial mean subspace and the mean function are provided. In addition, a Wald-type test is proposed to evaluate a linear hypothesis of the central partial mean subspace, and a...     

A variational formulation for physical noised image segmentation

Image segmentation is a hot topic in image science. In this paper we present a new variational segmentation model based on the theory of Mumford-Shah model. The aim of our model is to divide noised image, according to a certain criterion, into homogeneous and smooth regions that should correspond to structural units in the scene or objects of interest. The proposed region-based model uses total variation as a regularization term, and different fidelity term can be used for image segmentation in the cases of physical noise, such as Gaussian, Poisson and multiplicative speckle noise. Our model consists of five weighted terms, two of them are responsible for image denoising based on fidelity term and total variation term, the others assure that the three conditions of adherence to the data, smoothing, and discontinuity detection are met at once. We also develop a primal-dual hybrid gradient algorithm for our model. Numerical results on various synthetic and real images are provided to compare our method with others,these results show that our proposed model and algorithms are effective.     

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.     

Dunkl Multiplier Operators on a Class of Reproducing Kernel Hilbert Spaces

We study some class of Dunkl multiplier operators; and we establish for them the Heisenberg-Pauli-Weyl uncertainty principle and the Donoho-Stark''s uncertainty principle. For these operators we give also an application of the theory of reproducing kernels to the Tikhonov regularization on the Sobolev-Dunkl spaces.     

ON SMOOTH LU DECOMPOSITIONS WITH APPLICATIONS TO SOLUTIONS OF NONLINEAR EIGENVALUE PROBLEMS

We study the smooth LU decomposition of a given analytic functional A-matrix A（A） and its block-analogue. Sufficient conditions for the existence of such matrix decompositions are given, some differentiability about certain elements arising from them are proved, and several explicit expressions for derivatives of the specified elements are provided. By using these smooth LU decompositions, we propose two numerical methods for computing multiple nonlinear eigenvalues of A（A）, and establish their locally quadratic convergence properties. Several numerical examples are provided to show the feasibility and effectiveness of these new methods.     

DURATION OF NEGATIVE SURPLUS FOR A TWO STATE MARKOV-MODULATED RISK MODEL

We consider a continuous time risk model based on a two state Markov process, in which after an exponentially distributed time, the claim frequency changes to a different level and can change back again in the same way. We derive the Laplace transform for the first passage time to surplus zero from a given negative surplus and for the duration of negative surplus. Closed-form expressions are given in the case of exponential individual claim. Finally, numerical results are provided to show how to estimate the moments of duration of negative surplus.     

DISSIPATIVE NUMERICAL METHODS FOR THE HUNTER-SAXTON EQUATION

In this paper, we present further development of the local discontinuous Galerkin （LDG） method designed in [21] and a new dissipative discontinuous Galerkin （DG） method for the HuntermSaxton equation. The numerical fluxes for the LDG and DG methods in this paper are based on the upwinding principle. The resulting schemes provide additional energy dissipation and better control of numerical oscillations near derivative singularities. Stability and convergence of the schemes are proved theoretically, and numerical simulation results are provided to compare with the scheme in [21].     

Generalized Maximum Principles and Stochastic Completeness for Pseudo-Hermitian Manifolds*

In this paper, the authors establish a generalized maximum principle for pseudoHermitian manifolds. As corollaries, Omori-Yau type maximum principles for pseudoHermitian manifolds are deduced. Moreover, they prove that the stochastic completeness for the heat semigroup generated by the sub-Laplacian is equivalent to the validity of a weak form of the generalized maximum principles. Finally, they give some applications of these generalized maximum principles.     

ON A REGULARIZATION OF INDEX 2 DIFFERENTIAL-ALGEBRAIC EQUATIONS WITH PROPERLY STATED LEADING TERM

In this article, linear regular index 2 DAEs A(t)[D(t)x(t)]′+B(t)x(t)=q(t) are considered. Using a decoupling technique, initial condition and boundary condition are properly formulated. Regular index 1 DAEs are obtained by a regularization method. We study the behavior of the solution of the regularization system via asymptotic expansions. The error analysis between the solutions of the DAEs and its regularization system is given.     

Construction of periodic wavelet frames with dilation matrix

An important tool for the construction of periodic wavelet frame with the help of extension principles was presented in the Fourier domain by Zhang and Saito [Appl. Comput. Harmon. Anal., 2008, 125： 68-186]. We extend their results to the dilation matrix cases in two aspects. We first show that the periodization of any wavelet frame constructed by the unitary extension principle formulated by Ron and Shen is still a periodic wavelet frame under weaker conditions than that given by Zhang and Saito, and then prove that the periodization of those generated by the mixed extension principle is also a periodic wavelet frame if the scaling functions have compact supports.     

A sharp nonasymptotic bound and phase diagram of L 1/2 regularization

We derive a sharp nonasymptotic bound of parameter estimation of the L1/2 regularization.The bound shows that the solutions of the L1/2 regularization can achieve a loss within logarithmic factor of an ideal mean squared error and therefore underlies the feasibility and effectiveness of the L1/2regularization.Interestingly,when applied to compressive sensing,the L1/2 regularization scheme has exhibited a very promising capability of completed recovery from a much less sampling information.As compared with the Lp(0 p 1) penalty,it is appeared that the L1/2 penalty can always yield the most sparse solution among all the Lp penalty when 1/2 ≤ p 1,and when 0 p 1/2,the Lp penalty exhibits the similar properties as the L1/2 penalty.This suggests that the L1/2 regularization scheme can be accepted as the best and therefore the representative of all the Lp(0 p 1) regularization schemes.     

THE PERMUTATION FORMULA OF SINGULAR INTEGRALS WITH BOCHNER-MARTINELLI KERNEL ON STEIN MANIFOLDS

Using the method of localization, the authors obtain the permutation formula of singular integrals with Bochner-Martinelli kernel for a relative compact domain with C(1) smooth boundary on a Stein manifold. As an application the authors discuss the regularization problem for linear singular integral equations with Bochner-Martinelli kernel and variable coefficients; using permutation formula, the singular integral equation can be reduced to a fredholm equation.     

Analysis of a Class of Symmetric Equilibrium Configurations for a Territorial Model

<正>Motivated by an animal territoriality model,we consider a centroidal Voronoi tessellation algorithm from a dynamical systems perspective.In doing so,we discuss the stability of an aligned equilibrium configuration for a rectangular domain that exhibits interesting symmetry properties.We also demonstrate the procedure for performing a center manifold reduction on the system to extract a set of coordinates which capture the long term dynamics when the system is close to a bifurcation.Bifurcations of the system restricted to the center manifold are then classified and compared to numerical results.Although we analyze a specific set-up,these methods can in principle be applied to any bifurcation point of any equilibrium for any domain.     

EXPONENTIAL CONVERGENCE OF SAMPLE AVERAGE APPROXIMATION METHODS FOR A CLASS OF STOCHASTIC MATHEMATICAL PROGRAMS WITH COMPLEMENTARITY CONSTRAINTS

In this paper,we propose a Sample Average Approximation(SAA)method for a class ofStochastic Mathematical Programs with Complementarity Constraints(SMPCC)recentlyconsidered by Birbil,Gürkan and Liste[3].We study the statistical properties of obtainedSAA estimators.In particular we show that under moderate conditions a sequence of weakstationary points of SAA programs converge to a weak stationary point of the true problemwith probability approaching one at exponential rate as the sample size tends to infinity.To implement the SAA method more efficiently,we incorporate the method with sometechniques such as Scholtes' regularization method and the well known smoothing NCPmethod.Some preliminary numerical results are reported.