New regularization method and iteratively reweighted algorithm for sparse vector recovery

Motivated by the study of regularization for sparse problems,we propose a new regularization method for sparse vector recovery.We derive sufficient conditions on the well-posedness of the new regularization,and design an iterative algorithm,namely the iteratively reweighted algorithm(IR-algorithm),for efficiently computing the sparse solutions to the proposed regularization model.The convergence of the IR-algorithm and the setting of the regularization parameters are analyzed at length.Finally,we present numerical examples to illustrate the features of the new regularization and algorithm.

Two-level Bregmanized method for image interpolation with graph regularized sparse coding

A two-level Bregmanized method with graph regularized sparse coding （TBGSC） is presented for image interpolation. The outer-level Bregman iterative procedure enforces the observation data constraints, while the inner-level Bregmanized method devotes to dictionary updating and sparse represention of small overlapping image patches. The introduced constraint of graph regularized sparse coding can capture local image features effectively, and consequently enables accurate reconstruction from highly undersampled partial data. Furthermore, modified sparse coding and simple dictionary updating applied in the inner minimization make the proposed algorithm converge within a relatively small number of iterations. Experimental results demonstrate that the proposed algorithm can effectively reconstruct images and it outperforms the current state-of-the-art approaches in terms of visual comparisons and quantitative measures.

ITERATIVE REGULARIZATION METHODS FOR NONLINEAR ILL-POSED OPERATOR EQUATIONS WITH M-ACCRETIVE MAPPINGS IN BANACH SPACES

In this paper, a modified Newton type iterative method is considered for ap- proximately solving ill-posed nonlinear operator equations involving m-accretive mappings in Banach space. Convergence rate of the method is obtained based on an a priori choice of the regularization parameter. Our analysis is not based on the sequential continuity of the normalized duality mapping.

Downward continuation of airborne geomagnetic data based on two iterative regularization methods in the frequency domain

Downward continuation is a key step in processing airborne geomagnetic data. However,downward continuation is a typically ill-posed problem because its computation is unstable; thus, regularization methods are needed to realize effective continuation. According to the Poisson integral plane approximate relationship between observation and continuation data, the computation formulae combined with the fast Fourier transform（FFT）algorithm are transformed to a frequency domain for accelerating the computational speed. The iterative Tikhonov regularization method and the iterative Landweber regularization method are used in this paper to overcome instability and improve the precision of the results. The availability of these two iterative regularization methods in the frequency domain is validated by simulated geomagnetic data, and the continuation results show good precision.

Graph Regularized Sparse Coding Method for Highly Undersampled MRI Reconstruction

The imaging speed is a bottleneck for magnetic resonance imaging( MRI) since it appears. To alleviate this difficulty,a novel graph regularized sparse coding method for highly undersampled MRI reconstruction( GSCMRI) was proposed. The graph regularized sparse coding showed the potential in maintaining the geometrical information of the data. In this study, it was incorporated with two-level Bregman iterative procedure that updated the data term in outer-level and learned dictionary in innerlevel. Moreover,the graph regularized sparse coding and simple dictionary updating stages derived by the inner minimization made the proposed algorithm converge in few iterations, meanwhile achieving superior reconstruction performance. Extensive experimental results have demonstrated GSCMRI can consistently recover both real-valued MR images and complex-valued MR data efficiently,and outperform the current state-of-the-art approaches in terms of higher PSNR and lower HFEN values.

Solving Severely Ill⁃Posed Linear Systems with Time Discretization Based Iterative Regularization Methods

Recently,inverse problems have attracted more and more attention in computational mathematics and become increasingly important in engineering applications.After the discretization,many of inverse problems are reduced to linear systems.Due to the typical ill-posedness of inverse problems,the reduced linear systems are often illposed,especially when their scales are large.This brings great computational difficulty.Particularly,a small perturbation in the right side of an ill-posed linear system may cause a dramatical change in the solution.Therefore,regularization methods should be adopted for stable solutions.In this paper,a new class of accelerated iterative regularization methods is applied to solve this kind of large-scale ill-posed linear systems.An iterative scheme becomes a regularization method only when the iteration is early terminated.And a Morozov’s discrepancy principle is applied for the stop criterion.Compared with the conventional Landweber iteration,the new methods have acceleration effect,and can be compared to the well-known acceleratedν-method and Nesterov method.From the numerical results,it is observed that using appropriate discretization schemes,the proposed methods even have better behavior when comparing withν-method and Nesterov method.

Optimisation Method for Determination of Crack Tip Position Based on Gauss-Newton Iterative Technique

In the digital image correlation research of fatigue crack growth rate,the accuracy of the crack tip position determines the accuracy of the calculation of the stress intensity factor,thereby affecting the life prediction.This paper proposes a Gauss-Newton iteration method for solving the crack tip position.The conventional linear fitting method provides an iterative initial solution for this method,and the preconditioned conjugate gradient method is used to solve the ill-conditioned matrix.A noise-added artificial displacement field is used to verify the feasibility of the method,which shows that all parameters can be solved with satisfactory results.The actual stress intensity factor solution case shows that the stress intensity factor value obtained by the method in this paper is very close to the finite element result,and the relative error between the two is only−0.621%;The Williams coefficient obtained by this method can also better define the contour of the plastic zone at the crack tip,and the maximum relative error with the test plastic zone area is−11.29%.The relative error between the contour of the plastic zone defined by the conventional method and the area of the experimental plastic zone reached a maximum of 26.05%.The crack tip coordinates,stress intensity factors,and plastic zone contour changes in the loading and unloading phases are explored.The results show that the crack tip change during the loading process is faster than the change during the unloading process;the stress intensity factor during the unloading process under the same load condition is larger than that during the loading process;under the same load,the theoretical plastic zone during the unloading process is higher than that during the loading process.

Two-Level Bregman Method for MRI Reconstruction with Graph Regularized Sparse Coding

In this paper, a two-level Bregman method is presented with graph regularized sparse coding for highly undersampled magnetic resonance image reconstruction. The graph regularized sparse coding is incorporated with the two-level Bregman iterative procedure which enforces the sampled data constraints in the outer level and updates dictionary and sparse representation in the inner level. Graph regularized sparse coding and simple dictionary updating applied in the inner minimization make the proposed algorithm converge with a relatively small number of iterations. Experimental results demonstrate that the proposed algorithm can consistently reconstruct both simulated MR images and real MR data efficiently, and outperforms the current state-of-the-art approaches in terms of visual comparisons and quantitative measures.

Multilevel Iteration Methods for Solving Linear Operator Equations of the First Kind

In this paper we develop two multilevel iteration methods for solving linear systems resulting from the Galerkin method and Tikhonov regularization for linear ill-posed problems. The two algorithms and their convergence analyses are presented in an abstract framework.

MULTILEVEL ITERATION METHODS FOR SOLVING LINEAR ILL-POSED PROBLEMS

In this paper we develop multilevel iteration methods for solving linear systems resulting from the Galerkin method and Tikhonov regularization for ill-posed problems. The algorithm and its convergence analysis are presented in an abstract framework.

Weighted Nuclear Norm Minimization-Based Regularization Method for Image Restoration

Regularization methods have been substantially applied in image restoration due to the ill-posedness of the image restoration problem.Different assumptions or priors on images are applied in the construction of image regularization methods.In recent years,matrix low-rank approximation has been successfully introduced in the image denoising problem and significant denoising effects have been achieved.Low-rank matrix minimization is an NP-hard problem and it is often replaced with the matrix’s weighted nuclear norm minimization(WNNM).The assumption that an image contains an extensive amount of self-similarity is the basis for the construction of the matrix low-rank approximation-based image denoising method.In this paper,we develop a model for image restoration using the sum of block matching matrices’weighted nuclear norm to be the regularization term in the cost function.An alternating iterative algorithm is designed to solve the proposed model and the convergence analyses of the algorithm are also presented.Numerical experiments show that the proposed method can recover the images much better than the existing regularization methods in terms of both recovered quantities and visual qualities.

大型离散不适定问题的广义G-K双对角正则化算法

不适定问题常常出现于科学和工程等诸多领域,求解此类问题的难点在于其解对扰动的高度敏感性。正则化方法由于用与原不适定问题相邻近的适定问题的解逼近原问题的解,成为求解不适定问题的一类有效算法。近来,用不同范数分别约束保真项和正则项的极小化模型求解不适定问题的正则化方法引起了广泛关注。本文针对大型离散不适定问题的不同范数约束优化模型,基于Majorization-Minimization优化算法和Golub-Kahan Lanczos双对角化过程,采用基于偏差原理的正则化参数选择策略,提出了一种求解大型离散不适定问题的广义Golub-Kahan双对角化正则化算法,并给出了所提算法的收敛性理论证明。本文对新算法进行了数值实验,并与已有算法进行了比较,数值结果表明所提算法与已有算法相比在计算效能等方面更具优势;新算法应用到图像恢复问题的算例验证了新算法在图像恢复应用中的实用性和有效性。新算法由于其更低迭代运算和更高计算效率而更具吸引力。

磁流体方程全局吸引子的正则性

本文考虑磁流体方程的长时间行为,研究其全局吸引子的正则性。首先,利用分数次空间的嵌入定理和全局吸引子的存在性定理分别得到该方程在空间H 1和H 2中存在全局吸引子;然后,利用迭代方法、线性算子半群的正则性理论和全局吸引子的存在性定理,证明该方程在任意Sobolev空间H^(k)(其中k≥0)中存在全局吸引子,并以H^(k)-范数吸引空间H^(k)中的任意有界集。

Comparison of Iterative Feedback Tuning Search Techniques

Iterative feedback tuning is an attractive method for industry as it is a model free approach using experiments conducted on the plant to tune controller parameters. Classically Gauss-Newton iterative methods are used in IFT to update the controller parameters in the negative gradient direction of a specified design criterion function. Levenburg-Marquardt and Trust-Region strategies offer attractive advantages to Gauss-Newton in many applications,these alternative methods are given and results from simulation presented. A discussion on the differences between line search methods and Trust-Region methods is given showing the Trust-Region search direction is more flexible. Step size selection is often the limiting factor and it is found that with unknown step size values and initial controller parameters the Trust-Region is the best selection,where as if overshoot is a concern Levenburg-Marquardt is a good choice.Gauss-Newton method provides quick convergence and a fast response time however it shows more dependence on the step size.

A wavelet multiscale method for inversion of Maxwell equations

This paper is concerned with estimation of electrical conductivity in Maxwell equations. The primary difficulty lies in the presence of numerous local minima in the objective functional. A wavelet multiscale method is introduced and applied to the inversion of Maxwell equations. The inverse problem is decomposed into multiple scales with wavelet transform, and hence the original problem is reformulated to a set of sub-inverse problems corresponding to different scales, which can be solved successively according to the size of scale from the shortest to the longest. The stable and fast regularized Gauss-Newton method is applied to each scale. Numerical results show that the proposed method is effective, especially in terms of wide convergence, computational efficiency and precision.

基于微波热声层析成像定量重建生物组织电导率的改进方法

目的生物电磁学参数中的电导率与组织的功能性信息直接相关,精准重建生物组织电导率在医学成像技术和医学诊断领域中有着重要意义。本文改进定量微波热声层析成像(microwave-induced thermoacoustic tomography,MTAT)算法,使组织电导率的重建精度提高。方法本文在利用有限元离散法求解热声波动方程和亥姆霍兹方程的基础之上,提出了一种基于正则化牛顿迭代法(regularized Newton iteration method,RNIM)定量重建组织电导率的改进方法。结果通过数值模拟实验和含不同浓度NaCl溶液的仿体实验,验证了算法改进的有效性。组织仿体实验结果表明,目标在不同位置、不同大小、不同对比度情况下,相比于定量微波热声层析成像采用拟合(fitting)的方法,采用正则化牛顿法定量重建的仿体电导率相对误差明显降低,重建目标精度提高。在仿体实验中采用RNIM方法重建相同浓度的单目标在不同位置的电导率变化幅度更小,以及重建多目标电导率的相对比值与实际更接近,实验结果验证了改进方法的稳定性。结论研究结果表明优化算法能更加准确地定量重建组织仿体的电导率,这对于肿瘤的定位和分期的早期筛查及精准诊疗,预防疾病恶化具有重要意义。

径向热传导方程源项的分数阶Tikhonov-Landweber反演方法

本文研究了时间分数阶的径向热传导方程的源项反演问题.根据径向区域的终止时刻数据,利用分数阶Tikhonov-Landweber迭代正则化方法,求得在先验和后验的条件下的正则化解并得到相应误差.最后,数值实验表明了该正则化方法是有效的.

M-矩阵代数Riccati方程的一类简单迭代法

文章研究了M-矩阵代数Riccati方程的数值解法。当方程的系数矩阵为正则奇异M-矩阵时,现有的一些数值方法在计算中存在一定程度的困难。为此提出了一类简单迭代法以求解方程,该方法在每步迭代中只用到矩阵乘法,运算量小且易于实现。理论分析和数值实验表明该方法是可行的,而且在一定情况下有效。

一种迭代正则化方法求解一类同时带有两个扰动数据的反向问题

该文考虑了一类带有扰动扩散系数和扰动终值数据的空间分数阶扩散方程反向问题,从终值时刻的测量数据来反演初始时刻数据.该问题是严重不适定的,因此该文提出了一种迭代正则化方法来处理该反向问题,并利用先验正则化参数选取规则得到了正则化解和精确解之间的误差估计,最后进行了一些数值模拟,验证了方法的有效性.

时间分数阶扩散方程柯西问题的迭代正则化方法

本文研究一类时间分数阶扩散方程柯西问题,该问题是严重不适定的.基于傅里叶截断理论,构造了一种迭代方法来克服其不适定性,并且通过正则化参数的先验和后验选取规则获得了正则化方法的收敛性估计.最后,通过数值实验验证了该方法的有效性.数值结果表明,该方法求解时间分数阶扩散方程柯西问题是稳定可行的.