Seismic data reconstruction is an essential and yet fundamental step in seismic data processing workflow,which is of profound significance to improve migration imaging quality,multiple suppression effect,and seismic i...Seismic data reconstruction is an essential and yet fundamental step in seismic data processing workflow,which is of profound significance to improve migration imaging quality,multiple suppression effect,and seismic inversion accuracy.Regularization methods play a central role in solving the underdetermined inverse problem of seismic data reconstruction.In this paper,a novel regularization approach is proposed,the low dimensional manifold model(LDMM),for reconstructing the missing seismic data.Our work relies on the fact that seismic patches always occupy a low dimensional manifold.Specifically,we exploit the dimension of the seismic patches manifold as a regularization term in the reconstruction problem,and reconstruct the missing seismic data by enforcing low dimensionality on this manifold.The crucial procedure of the proposed method is to solve the dimension of the patches manifold.Toward this,we adopt an efficient dimensionality calculation method based on low-rank approximation,which provides a reliable safeguard to enforce the constraints in the reconstruction process.Numerical experiments performed on synthetic and field seismic data demonstrate that,compared with the curvelet-based sparsity-promoting L1-norm minimization method and the multichannel singular spectrum analysis method,the proposed method obtains state-of-the-art reconstruction results.展开更多
We present a new algorithm for manifold learning and nonlinear dimensionality reduction. Based on a set of unorganized data points sampled with noise from a parameterized manifold, the local geometry of the manifold i...We present a new algorithm for manifold learning and nonlinear dimensionality reduction. Based on a set of unorganized data points sampled with noise from a parameterized manifold, the local geometry of the manifold is learned by constructing an approximation for the tangent space at each point, and those tangent spaces are then aligned to give the global coordinates of the data points with respect to the underlying manifold. We also present an error analysis of our algorithm showing that reconstruction errors can be quite small in some cases. We illustrate our algorithm using curves and surfaces both in 2D/3D Euclidean spaces and higher dimensional Euclidean spaces. We also address several theoretical and algorithmic issues for further research and improvements.展开更多
Our aim in the present article is to introduce and study types of retraction of one dimensional manifold. New types of geodesics in one dimensional manifold are presented. The deformation retracts of one dimensional m...Our aim in the present article is to introduce and study types of retraction of one dimensional manifold. New types of geodesics in one dimensional manifold are presented. The deformation retracts of one dimensional manifold into itself and onto geodesics is deduced. Also, the isometric and topological folding in each case and the relation between the deformations retracts after and before folding has been obtained. New types of conditional folding are described.展开更多
The three-dimensional numerical manifold method（NMM） is studied on the basis of two-dimensional numerical manifold method. The three-dimensional cover displacement function is studied. The mechanical analysis and Ha...The three-dimensional numerical manifold method（NMM） is studied on the basis of two-dimensional numerical manifold method. The three-dimensional cover displacement function is studied. The mechanical analysis and Hammer integral method of three-dimensional numerical manifold method are put forward. The stiffness matrix of three-dimensional manifold element is derived and the dissection rules are given. The theoretical system and the numerical realizing method of three-dimensional numerical manifold method are systematically studied. As an example, the cantilever with load on the end is calculated, and the results show that the precision and efficiency are agreeable.展开更多
In this paper, we proposed a new semi-supervised multi-manifold learning method, called semi- supervised sparse multi-manifold embedding (S3MME), for dimensionality reduction of hyperspectral image data. S3MME exploit...In this paper, we proposed a new semi-supervised multi-manifold learning method, called semi- supervised sparse multi-manifold embedding (S3MME), for dimensionality reduction of hyperspectral image data. S3MME exploits both the labeled and unlabeled data to adaptively find neighbors of each sample from the same manifold by using an optimization program based on sparse representation, and naturally gives relative importance to the labeled ones through a graph-based methodology. Then it tries to extract discriminative features on each manifold such that the data points in the same manifold become closer. The effectiveness of the proposed multi-manifold learning algorithm is demonstrated and compared through experiments on a real hyperspectral images.展开更多
Based on locally compact perturbations of the identity map similar to the Fredholm structures on real Banach manifolds, complex manifolds with inverse mapping theorem as part of the defintion are proposed. Standard to...Based on locally compact perturbations of the identity map similar to the Fredholm structures on real Banach manifolds, complex manifolds with inverse mapping theorem as part of the defintion are proposed. Standard topics including holomorphic maps, morphisms, derivatives, tangent bundles, product manifolds and submanifolds are presented. Although this framework is elementary, it lays the necessary foundation for all subsequent developments.展开更多
In dealing with high-dimensional data, such as the global climate model, facial data analysis, human gene distribution and so on, the problem of dimensionality reduction is often encountered, that is, to find the low ...In dealing with high-dimensional data, such as the global climate model, facial data analysis, human gene distribution and so on, the problem of dimensionality reduction is often encountered, that is, to find the low dimensional structure hidden in high-dimensional data. Nonlinear dimensionality reduction facilitates the discovery of the intrinsic structure and relevance of the data and can make the high-dimensional data visible in the low dimension. The isometric mapping algorithm (Isomap) is an important algorithm for nonlinear dimensionality reduction, which originates from the traditional dimensionality reduction algorithm MDS. The MDS algorithm is based on maintaining the distance between the samples in the original space and the distance between the samples in the lower dimensional space;the distance used here is Euclidean distance, and the Isomap algorithm discards the Euclidean distance, and calculates the shortest path between samples by Floyd algorithm to approximate the geodesic distance along the manifold surface. Compared with the previous nonlinear dimensionality reduction algorithm, the Isomap algorithm can effectively compute a global optimal solution, and it can ensure that the data manifold converges to the real structure asymptotically.展开更多
In this paper foe Liapunov functionals has been constructed.the decay property of the high dimensional modes of the J-J equations in the Josephson junctions is obtained,and thus the approxtmate inertial manifolds are...In this paper foe Liapunov functionals has been constructed.the decay property of the high dimensional modes of the J-J equations in the Josephson junctions is obtained,and thus the approxtmate inertial manifolds are given.展开更多
A short description of structural and virtual Kirichenko tensors that form a complete system of first-order differential-geometrical invariants of an arbitrary almost Hermitian structure is given.A characterization of...A short description of structural and virtual Kirichenko tensors that form a complete system of first-order differential-geometrical invariants of an arbitrary almost Hermitian structure is given.A characterization of nearly-Khlerian structures in terms of Kirichenko tensors is also given.展开更多
As an effective way in finding the underlying parameters of a high-dimension space, manifold learning is popular in nonlinear dimensionality reduction which makes high-dimensional data easily to be observed and analyz...As an effective way in finding the underlying parameters of a high-dimension space, manifold learning is popular in nonlinear dimensionality reduction which makes high-dimensional data easily to be observed and analyzed. In this paper, Isomap, one of the most famous manifold learning algorithms, is applied to process closing prices of stocks of CSI 300 index from September 2009 to October 2011. Results indicate that Isomap algorithm not only reduces dimensionality of stock data successfully, but also classifies most stocks according to their trends efficiently.展开更多
In this paper, the boundary layer equations (abbreviation BLE) for exterior flow around an obstacle are established using semi-geodesic coordinate system (S-coordinate) based on the curved two dimensional surface of t...In this paper, the boundary layer equations (abbreviation BLE) for exterior flow around an obstacle are established using semi-geodesic coordinate system (S-coordinate) based on the curved two dimensional surface of the obstacle. BLE are nonlinear partial differential equations on unknown normal viscous stress tensor and pressure on the obstacle and the existence of solution of BLE is proved. In addition a dimensional split method for dimensional three Navier-Stokes equations is established by applying several 2D-3C partial differential equations on two dimensional manifolds to approach 3D Navier-Stokes equations. The examples for the exterior flow around spheroid and ellipsoid are presents here.展开更多
The existence of approximate inertial manifold Using wavelet to Burgers' equation, and numerical solution under multiresolution analysis with the low modes were studied. It is shown that the Burgers' equation ...The existence of approximate inertial manifold Using wavelet to Burgers' equation, and numerical solution under multiresolution analysis with the low modes were studied. It is shown that the Burgers' equation has a good localization property of the numerical solution distinguishably.展开更多
基金supported by National Natural Science Foundation of China(Grant No.41874146 and No.42030103)Postgraduate Innovation Project of China University of Petroleum(East China)(No.YCX2021012)
文摘Seismic data reconstruction is an essential and yet fundamental step in seismic data processing workflow,which is of profound significance to improve migration imaging quality,multiple suppression effect,and seismic inversion accuracy.Regularization methods play a central role in solving the underdetermined inverse problem of seismic data reconstruction.In this paper,a novel regularization approach is proposed,the low dimensional manifold model(LDMM),for reconstructing the missing seismic data.Our work relies on the fact that seismic patches always occupy a low dimensional manifold.Specifically,we exploit the dimension of the seismic patches manifold as a regularization term in the reconstruction problem,and reconstruct the missing seismic data by enforcing low dimensionality on this manifold.The crucial procedure of the proposed method is to solve the dimension of the patches manifold.Toward this,we adopt an efficient dimensionality calculation method based on low-rank approximation,which provides a reliable safeguard to enforce the constraints in the reconstruction process.Numerical experiments performed on synthetic and field seismic data demonstrate that,compared with the curvelet-based sparsity-promoting L1-norm minimization method and the multichannel singular spectrum analysis method,the proposed method obtains state-of-the-art reconstruction results.
文摘We present a new algorithm for manifold learning and nonlinear dimensionality reduction. Based on a set of unorganized data points sampled with noise from a parameterized manifold, the local geometry of the manifold is learned by constructing an approximation for the tangent space at each point, and those tangent spaces are then aligned to give the global coordinates of the data points with respect to the underlying manifold. We also present an error analysis of our algorithm showing that reconstruction errors can be quite small in some cases. We illustrate our algorithm using curves and surfaces both in 2D/3D Euclidean spaces and higher dimensional Euclidean spaces. We also address several theoretical and algorithmic issues for further research and improvements.
文摘Our aim in the present article is to introduce and study types of retraction of one dimensional manifold. New types of geodesics in one dimensional manifold are presented. The deformation retracts of one dimensional manifold into itself and onto geodesics is deduced. Also, the isometric and topological folding in each case and the relation between the deformations retracts after and before folding has been obtained. New types of conditional folding are described.
文摘The three-dimensional numerical manifold method（NMM） is studied on the basis of two-dimensional numerical manifold method. The three-dimensional cover displacement function is studied. The mechanical analysis and Hammer integral method of three-dimensional numerical manifold method are put forward. The stiffness matrix of three-dimensional manifold element is derived and the dissection rules are given. The theoretical system and the numerical realizing method of three-dimensional numerical manifold method are systematically studied. As an example, the cantilever with load on the end is calculated, and the results show that the precision and efficiency are agreeable.
文摘In this paper, we proposed a new semi-supervised multi-manifold learning method, called semi- supervised sparse multi-manifold embedding (S3MME), for dimensionality reduction of hyperspectral image data. S3MME exploits both the labeled and unlabeled data to adaptively find neighbors of each sample from the same manifold by using an optimization program based on sparse representation, and naturally gives relative importance to the labeled ones through a graph-based methodology. Then it tries to extract discriminative features on each manifold such that the data points in the same manifold become closer. The effectiveness of the proposed multi-manifold learning algorithm is demonstrated and compared through experiments on a real hyperspectral images.
基金This paper is a talk on the held in Nanjing, P. R. China, July, 2004.
文摘Based on locally compact perturbations of the identity map similar to the Fredholm structures on real Banach manifolds, complex manifolds with inverse mapping theorem as part of the defintion are proposed. Standard topics including holomorphic maps, morphisms, derivatives, tangent bundles, product manifolds and submanifolds are presented. Although this framework is elementary, it lays the necessary foundation for all subsequent developments.
文摘In dealing with high-dimensional data, such as the global climate model, facial data analysis, human gene distribution and so on, the problem of dimensionality reduction is often encountered, that is, to find the low dimensional structure hidden in high-dimensional data. Nonlinear dimensionality reduction facilitates the discovery of the intrinsic structure and relevance of the data and can make the high-dimensional data visible in the low dimension. The isometric mapping algorithm (Isomap) is an important algorithm for nonlinear dimensionality reduction, which originates from the traditional dimensionality reduction algorithm MDS. The MDS algorithm is based on maintaining the distance between the samples in the original space and the distance between the samples in the lower dimensional space;the distance used here is Euclidean distance, and the Isomap algorithm discards the Euclidean distance, and calculates the shortest path between samples by Floyd algorithm to approximate the geodesic distance along the manifold surface. Compared with the previous nonlinear dimensionality reduction algorithm, the Isomap algorithm can effectively compute a global optimal solution, and it can ensure that the data manifold converges to the real structure asymptotically.
文摘In this paper foe Liapunov functionals has been constructed.the decay property of the high dimensional modes of the J-J equations in the Josephson junctions is obtained,and thus the approxtmate inertial manifolds are given.
文摘A short description of structural and virtual Kirichenko tensors that form a complete system of first-order differential-geometrical invariants of an arbitrary almost Hermitian structure is given.A characterization of nearly-Khlerian structures in terms of Kirichenko tensors is also given.
文摘As an effective way in finding the underlying parameters of a high-dimension space, manifold learning is popular in nonlinear dimensionality reduction which makes high-dimensional data easily to be observed and analyzed. In this paper, Isomap, one of the most famous manifold learning algorithms, is applied to process closing prices of stocks of CSI 300 index from September 2009 to October 2011. Results indicate that Isomap algorithm not only reduces dimensionality of stock data successfully, but also classifies most stocks according to their trends efficiently.
文摘In this paper, the boundary layer equations (abbreviation BLE) for exterior flow around an obstacle are established using semi-geodesic coordinate system (S-coordinate) based on the curved two dimensional surface of the obstacle. BLE are nonlinear partial differential equations on unknown normal viscous stress tensor and pressure on the obstacle and the existence of solution of BLE is proved. In addition a dimensional split method for dimensional three Navier-Stokes equations is established by applying several 2D-3C partial differential equations on two dimensional manifolds to approach 3D Navier-Stokes equations. The examples for the exterior flow around spheroid and ellipsoid are presents here.
文摘The existence of approximate inertial manifold Using wavelet to Burgers' equation, and numerical solution under multiresolution analysis with the low modes were studied. It is shown that the Burgers' equation has a good localization property of the numerical solution distinguishably.
