The current manuscript makes use of the prominent iterative procedure, called the Adomian Decomposition Method (ADM), to tackle some important special differential equations. The equations of curiosity in this study a...The current manuscript makes use of the prominent iterative procedure, called the Adomian Decomposition Method (ADM), to tackle some important special differential equations. The equations of curiosity in this study are the singular equations that arise in many physical science applications. Thus, through the application of the ADM, a generalized recursive scheme was successfully derived and further utilized to obtain closed-form solutions for the models under consideration. The method is, indeed, fascinating as respective exact analytical solutions are accurately acquired with only a small number of iterations.展开更多
The low-field nuclear magnetic resonance(NMR)technique has been used to probe the pore size distribution and the fluid composition in geophysical prospecting and related fields.However,the speed and accuracy of the ex...The low-field nuclear magnetic resonance(NMR)technique has been used to probe the pore size distribution and the fluid composition in geophysical prospecting and related fields.However,the speed and accuracy of the existing numerical inversion methods are still challenging due to the ill-posed nature of the first kind Fredholm integral equation and the contamination of the noises.This paper proposes a novel inversion algorithmto accelerate the convergence and enhance the precision using empirical truncated singular value decompositions(TSVD)and the linearized Bregman iteration.The L1 penalty term is applied to construct the objective function,and then the linearized Bregman iteration is utilized to obtain fast convergence.To reduce the complexity of the computation,empirical TSVD is proposed to compress the kernel matrix and determine the appropriate truncated position.This novel inversion method is validated using numerical simulations.The results indicate that the proposed novel method is significantly efficient and can achieve quick and effective data solutions with low signal-to-noise ratios.展开更多
Higher-order singular value decomposition （HOSVD） is an efficient way for data reduction and also eliciting intrinsic structure of multi-dimensional array data. It has been used in many applications, and some of the...Higher-order singular value decomposition （HOSVD） is an efficient way for data reduction and also eliciting intrinsic structure of multi-dimensional array data. It has been used in many applications, and some of them involve incomplete data. To obtain HOSVD of the data with missing values, one can first impute the missing entries through a certain tensor completion method and then perform HOSVD to the reconstructed data. However, the two-step procedure can be inefficient and does not make reliable decomposition. In this paper, we formulate an incomplete HOSVD problem and combine the two steps into solving a single optimization problem, which simultaneously achieves imputation of missing values and also tensor decomposition. We also present one algorithm for solving the problem based on block coordinate update （BCU）. Global convergence of the algorithm is shown under mild assumptions and implies that of the popular higher-order orthogonality iteration （HOOI） method, and thus we, for the first time, give global convergence of HOOI. In addition, we compare the proposed method to state-of-the-art ones for solving incom- plete HOSVD and also low-rank tensor completion problems and demonstrate the superior performance of our method over other compared ones. Furthermore, we apply it to face recognition and MRI image reconstruction to show its practical performance.展开更多
针对大型矩阵奇异值分解(singular value decomposition,SVD)时使用经典算法时间复杂度较高,以及已有的量子SVD算法要求待分解的矩阵必须具有非稀疏低秩的性质,并且在计算过程中构造任意大小酉矩阵对目前的量子计算机来说实现起来并不...针对大型矩阵奇异值分解(singular value decomposition,SVD)时使用经典算法时间复杂度较高,以及已有的量子SVD算法要求待分解的矩阵必须具有非稀疏低秩的性质,并且在计算过程中构造任意大小酉矩阵对目前的量子计算机来说实现起来并不容易等问题,提出基于QR迭代的量子SVD。QR迭代使用的是Householder变换,通过量子矩阵乘法运算完成经典矩阵乘法运算过程。实验结果表明,该方法能够得到所求矩阵的奇异值及奇异矩阵,使大型矩阵的SVD具有可行性。展开更多
总体最小二乘估计能够同时顾及线性模型中系数矩阵A和观测向量L的误差,平差理论相对更为严密。如果系数矩阵A的部分元素没有误差,这种总体最小二乘模型为混合总体最小二乘模型。针对混合总体最小二乘(Least squares-total least squares...总体最小二乘估计能够同时顾及线性模型中系数矩阵A和观测向量L的误差,平差理论相对更为严密。如果系数矩阵A的部分元素没有误差,这种总体最小二乘模型为混合总体最小二乘模型。针对混合总体最小二乘(Least squares-total least squares,LS-TLS)解算问题,应用测量平差中的原理和方法,推导了混合总体最小二乘的迭代逼近解算公式,通过与奇异值分解法分析比较,分析了两种解算方法具有等价性,最后通过实验数据分析得出迭代算法的有效性和合理性。展开更多
利用目标辐射源空间分布的稀疏性,提出了一种基于稀疏表示的多快拍联合波达方向(direction of arrival,DOA)估计方法。该方法首先利用采样数据矩阵大奇异值对应的左奇异向量估计信号子空间,然后采用加权迭代最小方差方法对信号空间进行...利用目标辐射源空间分布的稀疏性,提出了一种基于稀疏表示的多快拍联合波达方向(direction of arrival,DOA)估计方法。该方法首先利用采样数据矩阵大奇异值对应的左奇异向量估计信号子空间,然后采用加权迭代最小方差方法对信号空间进行稀疏表示。与传统的角度高分辨估计方法不同,该方法没有利用样本的统计信息,因而对具有任意相关性的信号源能进行有效的波达方向估计,不需要进行去相关处理,且具有很高的分辨力及估计精度。实验表明在该方法能准确的对目标源方位进行估计,且极大地降低了稀疏表示的计算量。展开更多
文摘The current manuscript makes use of the prominent iterative procedure, called the Adomian Decomposition Method (ADM), to tackle some important special differential equations. The equations of curiosity in this study are the singular equations that arise in many physical science applications. Thus, through the application of the ADM, a generalized recursive scheme was successfully derived and further utilized to obtain closed-form solutions for the models under consideration. The method is, indeed, fascinating as respective exact analytical solutions are accurately acquired with only a small number of iterations.
基金support by the National Nature Science Foundation of China(42174142)CNPC Innovation Found(2021DQ02-0402)National Key Foundation for Exploring Scientific Instrument of China(2013YQ170463).
文摘The low-field nuclear magnetic resonance(NMR)technique has been used to probe the pore size distribution and the fluid composition in geophysical prospecting and related fields.However,the speed and accuracy of the existing numerical inversion methods are still challenging due to the ill-posed nature of the first kind Fredholm integral equation and the contamination of the noises.This paper proposes a novel inversion algorithmto accelerate the convergence and enhance the precision using empirical truncated singular value decompositions(TSVD)and the linearized Bregman iteration.The L1 penalty term is applied to construct the objective function,and then the linearized Bregman iteration is utilized to obtain fast convergence.To reduce the complexity of the computation,empirical TSVD is proposed to compress the kernel matrix and determine the appropriate truncated position.This novel inversion method is validated using numerical simulations.The results indicate that the proposed novel method is significantly efficient and can achieve quick and effective data solutions with low signal-to-noise ratios.
文摘Higher-order singular value decomposition （HOSVD） is an efficient way for data reduction and also eliciting intrinsic structure of multi-dimensional array data. It has been used in many applications, and some of them involve incomplete data. To obtain HOSVD of the data with missing values, one can first impute the missing entries through a certain tensor completion method and then perform HOSVD to the reconstructed data. However, the two-step procedure can be inefficient and does not make reliable decomposition. In this paper, we formulate an incomplete HOSVD problem and combine the two steps into solving a single optimization problem, which simultaneously achieves imputation of missing values and also tensor decomposition. We also present one algorithm for solving the problem based on block coordinate update （BCU）. Global convergence of the algorithm is shown under mild assumptions and implies that of the popular higher-order orthogonality iteration （HOOI） method, and thus we, for the first time, give global convergence of HOOI. In addition, we compare the proposed method to state-of-the-art ones for solving incom- plete HOSVD and also low-rank tensor completion problems and demonstrate the superior performance of our method over other compared ones. Furthermore, we apply it to face recognition and MRI image reconstruction to show its practical performance.
文摘针对大型矩阵奇异值分解(singular value decomposition,SVD)时使用经典算法时间复杂度较高,以及已有的量子SVD算法要求待分解的矩阵必须具有非稀疏低秩的性质,并且在计算过程中构造任意大小酉矩阵对目前的量子计算机来说实现起来并不容易等问题,提出基于QR迭代的量子SVD。QR迭代使用的是Householder变换,通过量子矩阵乘法运算完成经典矩阵乘法运算过程。实验结果表明,该方法能够得到所求矩阵的奇异值及奇异矩阵,使大型矩阵的SVD具有可行性。
文摘总体最小二乘估计能够同时顾及线性模型中系数矩阵A和观测向量L的误差,平差理论相对更为严密。如果系数矩阵A的部分元素没有误差,这种总体最小二乘模型为混合总体最小二乘模型。针对混合总体最小二乘(Least squares-total least squares,LS-TLS)解算问题,应用测量平差中的原理和方法,推导了混合总体最小二乘的迭代逼近解算公式,通过与奇异值分解法分析比较,分析了两种解算方法具有等价性,最后通过实验数据分析得出迭代算法的有效性和合理性。
文摘利用目标辐射源空间分布的稀疏性,提出了一种基于稀疏表示的多快拍联合波达方向(direction of arrival,DOA)估计方法。该方法首先利用采样数据矩阵大奇异值对应的左奇异向量估计信号子空间,然后采用加权迭代最小方差方法对信号空间进行稀疏表示。与传统的角度高分辨估计方法不同,该方法没有利用样本的统计信息,因而对具有任意相关性的信号源能进行有效的波达方向估计,不需要进行去相关处理,且具有很高的分辨力及估计精度。实验表明在该方法能准确的对目标源方位进行估计,且极大地降低了稀疏表示的计算量。
