We study boundary value problems for fractional integro-differential equations involving Caputo derivative of order α∈ （n-1, n） in Banach spaces. Existence and uniqueness results of solutions are established by vi...We study boundary value problems for fractional integro-differential equations involving Caputo derivative of order α∈ （n-1, n） in Banach spaces. Existence and uniqueness results of solutions are established by virtue of the Holder＇s inequality, a suitable singular Cronwall＇s inequality and fixed point theorem via a priori estimate method. At last, examples are given to illustrate the results.展开更多
A necessary and sufficient condition for the existence of simultaneous (M,N)singular value decomposition of matrices is given.Some properties about the weighted partial ordering are discussed with the help of the deco...A necessary and sufficient condition for the existence of simultaneous (M,N)singular value decomposition of matrices is given.Some properties about the weighted partial ordering are discussed with the help of the decomposition.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
Tensor analysis approaches are of great importance in various fields such as computa-tion vision and signal processing.Thereinto,the definitions of tensor-tensor product(t-product)and tensor singular value decompositi...Tensor analysis approaches are of great importance in various fields such as computa-tion vision and signal processing.Thereinto,the definitions of tensor-tensor product(t-product)and tensor singular value decomposition(t-SVD)are significant in practice.This work presents new t-product and t-SVD definitions based on the discrete simplified fractional Fourier transform(DSFRFT).The proposed definitions can effectively deal with special complex tenors,which fur-ther motivates the transform based tensor analysis approaches.Then,we define a new tensor nucle-ar norm induced by the DSFRFT based t-SVD.In addition,we analyze the computational complex-ity of the proposed t-SVD,which indicates that the proposed t-SVD can improve the computation-al efficiency.展开更多
A type of high-order integral observers for matrix second-order linear systems is proposed on the basis of generalized eigenstructure assignment via unified parametric approaches. Through establishing two general para...A type of high-order integral observers for matrix second-order linear systems is proposed on the basis of generalized eigenstructure assignment via unified parametric approaches. Through establishing two general parametric solutions to this type of generalized matrix second-order Sylvester matrix equations, two unified complete parametric methods for the proposed observer design problem are presented. Both methods give simple complete parametric expressions for the observer gain matrices. The first one mainly depends on a series of singular value decompositions, and is thus numerically simple and reliable; the second one utilizes the fight factorization of the system, and allows eigenvalues of the error system to be set undetermined and sought via certain optimization procedures. A spring-mass-dashpot system is utilized to illustrate the design procedure and show the effect of the proposed approach.展开更多
In practical applications, we often have to deal with high-order data, for example, a grayscale image and a video clip are intrinsically a 2nd-order tensor and a 3rd-order tensor, respectively. In order to satisty the...In practical applications, we often have to deal with high-order data, for example, a grayscale image and a video clip are intrinsically a 2nd-order tensor and a 3rd-order tensor, respectively. In order to satisty these high-order data, it is conventional to vectorize these data in advance, which often destroys the intrinsic structures of the data and includes the curse of dimensionality. For this reason, we consider the problem of high-order data representation and classification, and propose a tensor based fisher discriminant analysis （FDA）, which is a generalized version of FDA, named as GFDA. Experimental results show our GFDA outperforms the existing methods, such as the 2-directional 2-dimensional principal component analysis （（2D）2pCA）, 2-directional 2-dimensional linear discriminant analysis （（2D）2LDA）, and multilinear discriminant analysis （MDA）, in high-order data classification under a lower compression ratio.展开更多
Nowadays,the modeling of systems may be quite large,even up to tens of thousands orders.In spite of the increasing computational powers,direct simulation of these large-scale systems may be impractical.Thus,to industr...Nowadays,the modeling of systems may be quite large,even up to tens of thousands orders.In spite of the increasing computational powers,direct simulation of these large-scale systems may be impractical.Thus,to industry requirements,analytically tractable and computationally cheap models must be designed.This is the essence task of Model Order Reduction(MOR).This article describes the basics of MOR optimization,various way of designing MOR,and gives the conclusion about existing methods.In addition,it proposed some heuristic footpath.展开更多
基金supported by Grant In Aid research fund of Virginia Military Instittue, USA
文摘We study boundary value problems for fractional integro-differential equations involving Caputo derivative of order α∈ （n-1, n） in Banach spaces. Existence and uniqueness results of solutions are established by virtue of the Holder＇s inequality, a suitable singular Cronwall＇s inequality and fixed point theorem via a priori estimate method. At last, examples are given to illustrate the results.
基金The Guangxi Science Foundation(0575032,06400161)the support program for 100 Young and Middle-aged Disciplinary Leaders in Guangxi Higher Education Institutions
文摘A necessary and sufficient condition for the existence of simultaneous (M,N)singular value decomposition of matrices is given.Some properties about the weighted partial ordering are discussed with the help of the decomposition.
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.
基金supported by the National Key R&D Program of China(No.2018YFC2000600).
文摘Tensor analysis approaches are of great importance in various fields such as computa-tion vision and signal processing.Thereinto,the definitions of tensor-tensor product(t-product)and tensor singular value decomposition(t-SVD)are significant in practice.This work presents new t-product and t-SVD definitions based on the discrete simplified fractional Fourier transform(DSFRFT).The proposed definitions can effectively deal with special complex tenors,which fur-ther motivates the transform based tensor analysis approaches.Then,we define a new tensor nucle-ar norm induced by the DSFRFT based t-SVD.In addition,we analyze the computational complex-ity of the proposed t-SVD,which indicates that the proposed t-SVD can improve the computation-al efficiency.
基金This work was supported by the Chinese National Natural Science Foundation ( No. 69925308).
文摘A type of high-order integral observers for matrix second-order linear systems is proposed on the basis of generalized eigenstructure assignment via unified parametric approaches. Through establishing two general parametric solutions to this type of generalized matrix second-order Sylvester matrix equations, two unified complete parametric methods for the proposed observer design problem are presented. Both methods give simple complete parametric expressions for the observer gain matrices. The first one mainly depends on a series of singular value decompositions, and is thus numerically simple and reliable; the second one utilizes the fight factorization of the system, and allows eigenvalues of the error system to be set undetermined and sought via certain optimization procedures. A spring-mass-dashpot system is utilized to illustrate the design procedure and show the effect of the proposed approach.
文摘In practical applications, we often have to deal with high-order data, for example, a grayscale image and a video clip are intrinsically a 2nd-order tensor and a 3rd-order tensor, respectively. In order to satisty these high-order data, it is conventional to vectorize these data in advance, which often destroys the intrinsic structures of the data and includes the curse of dimensionality. For this reason, we consider the problem of high-order data representation and classification, and propose a tensor based fisher discriminant analysis （FDA）, which is a generalized version of FDA, named as GFDA. Experimental results show our GFDA outperforms the existing methods, such as the 2-directional 2-dimensional principal component analysis （（2D）2pCA）, 2-directional 2-dimensional linear discriminant analysis （（2D）2LDA）, and multilinear discriminant analysis （MDA）, in high-order data classification under a lower compression ratio.
文摘Nowadays,the modeling of systems may be quite large,even up to tens of thousands orders.In spite of the increasing computational powers,direct simulation of these large-scale systems may be impractical.Thus,to industry requirements,analytically tractable and computationally cheap models must be designed.This is the essence task of Model Order Reduction(MOR).This article describes the basics of MOR optimization,various way of designing MOR,and gives the conclusion about existing methods.In addition,it proposed some heuristic footpath.
