Smooth Subspace Research Articles

Isogeometric Residual Minimization Method (iGRM) with direction splitting preconditioner for stationary advection-dominated diffusion problems

In this paper, we introduce the isoGeometric Residual Minimization (iGRM) method. The method solves stationary advection-dominated diffusion problems.We stabilize the method via residual minimization. We discretize the problem using B-spline basis functions. We then seek to minimize the isogeometric residual over a spline space built on a tensor product mesh. We construct the solution over a smooth subspace of the residual. We can specify the solution subspace by reducing the polynomial order, by increasing the continuity, or by a combination of these. The Gramm matrix for the residual minimization method is approximated by a weighted H1 norm, which we can express as Kronecker products, due to the tensor-product structure of the approximations. We use the Gramm matrix as a preconditional which can be applied in a computational cost proportional to the number of degrees of freedom in 2D and 3D. Building on these approximations, we construct an iterative algorithm. We test the residual minimization method on several numerical examples, and we compare it to the Discontinuous Petrov–Galerkin (DPG) and the Streamline Upwind Petrov–Galerkin (SUPG) stabilization methods. The iGRM method delivers similar quality solutions as the DPG method, it uses smaller grids, it does not require breaking of the spaces, but it is limited to tensor-product meshes. The computational cost of the iGRM is higher than for SUPG, but it does not require the determination of problem specific parameters.

Transfer Dictionary Learning Method for Cross-Domain Multimode Process Monitoring and Fault Isolation

Data-driven methods have shown its great latent capacity in the field of industrial process monitoring. However, the existing methods usually achieve good results under the assumption that the offline learning data and the online monitoring data are drawn from the same distribution. Unfortunately, in the industrial system, the assumption is often violated due to the harsh operating environment. Especially, with the increasing complexity and scale of industrial production, the supervisory control and data acquisition (SCADA) data of the industrial production process often collected from different machines, seasons, or operating modes. In addition, due to the cost of manual data labeling and real-time requirement of process monitoring, the offline learning data, which was used to build the model, often have abundant source-domain data and insufficient target-domain data. Consequently, these methods have bad performance on the online monitoring data collected from the target domain. In order to make full use of the knowledge from the abundant source-domain data, a transfer dictionary learning method is proposed to address the cross-domain problem in this article. The proposed method can learn an initial dictionary from the abundant source-domain data, and then, the final dictionary is updated by incorporating the feature of insufficient target-domain data in a smooth subspace interpolation way. The effectiveness of the proposed method is evaluated through a numerical simulation case, a continuous stirred tank heater (CSTH) case, and a wind turbine system case, from which we can see the proposed method has a better performance compared with some state-of-the-art methods.

Fault Classification of Rotary Machinery Based on Smooth Local Subspace Projection Method and Permutation Entropy

Collected mechanical signals usually contain a number of noises, resulting in erroneous judgments of mechanical condition diagnosis. The mechanical signals, which are nonlinear or chaotic time series, have a high computational complexity and intrinsic broadband characteristic. This paper proposes a method of gear and bearing fault classification, based on the local subspace projection noise reduction and PE. A novel nonlinear projection noise reduction method, smooth orthogonal decomposition (SOD), is proposed to denoise the vibration signals of various operation conditions. SOD can decompose the reconstructed multiple strands to identify smooth local subspace. In the process of projection from a high dimension to a low dimension, a new weight matrix is put forward to achieve a better denoising effect. Afterwards, permutation entropy (PE) is applied in the detection of time sequence randomness and dynamic mutation behavior, which can effectively detect and amplify the variation of vibration signals. Hence PE can characterize the working conditions of gear and bearing under different conditions. The experimental results illustrate the effectiveness and superiority of the proposed approach. The theoretical derivations, numerical simulations and experimental studies, all confirm that the proposed approach based on the smooth local subspace projection method and PE, is promising in the field of the fault classification of rotary machinery.

Time series k-means: A new k-means type smooth subspace clustering for time series data

Existing clustering algorithms are weak in extracting smooth subspaces for clustering time series data. In this paper, we propose a new k-means type smooth subspace clustering algorithm named Time Series k-means (TSkmeans) for clustering time series data. The proposed TSkmeans algorithm can effectively exploit inherent subspace information of a time series data set to enhance clustering performance. More specifically, the smooth subspaces are represented by weighted time stamps which indicate the relative discriminative power of these time stamps for clustering objects. The main contributions of our work include the design of a new objective function to guide the clustering of time series data and the development of novel updating rules for iterative cluster searching with respect to smooth subspaces. Based on a synthetic data set and five real-life data sets, our experimental results confirm that the proposed TSkmeans algorithm outperforms other state-of-the-art time series clustering algorithms in terms of common performance metrics such as Accuracy, Fscore, RandIndex, and Normal Mutual Information.

Autofocus algorithm using blind homomorphic deconvolution for synthetic aperture radar imaging

In airborne case, synthetic aperture radar images generally suffer from the deterioration because of the unknown phase error caused by unstable platform and atmosphere perturbation. To obtain the phase error, a novel autofocus algorithm referred to as blind homomorphic deconvolution autofocus algorithm is proposed in this study. In this method, a wavelet scaling function is used to construct a smooth subspace that is orthogonal to noise subspace. Then, the phase error can be separated and reconstructed by the generated smooth subspace based on the differences of the smoothness properties between phase error and image reflectivity. Compared with the traditional autofocus methods, the proposed method does not require an iterative estimation. Thus, the computational complexity can be significantly reduced. Simulation and real data processing results validate the effectiveness of the proposed method.

SOME COMBINATORIAL REMARKS ON NORMAL FLATNESS IN ANALYTIC SPACES

In this article we present a combinatorial treatment of normal flatness in analytic spaces, using the idea of equimultiple standard bases. We will prove, using purely combinatorial methods, a characterization theorem for normal flatness. This will lead us to a new proof of a classical theorem on normal flatness, which can be stated by saying that normal flatness at a point along a smooth subspace is equivalent to the Hilbert function being locally constant. Though these topics belong to classical analytic geometry, we believe that this approach is valuable, since it replaces extremely general algebraic theorems by combinatorial objects, obtaining new results and striking the combinatorial nature of the classical (and basic) ideas in the resolution of singularities.

Deformations of Kähler manifolds with nonvanishing holomorphic vector fields

We study compact Kähler manifolds X admitting nonvanishing holomorphic vector fields, extending the classical birational classification of projective varieties with tangent vector fields to a classification modulo deformation in the Kähler case, and biholomorphic in the projective case. We introduce and analyze a new class of {tangential \ deformations} , and show that they form a smooth subspace in the Kuranishi space of deformations of the complex structure of X . We extend Calabi's theorem on the structure of compact Kähler manifolds X with c_1(X) =0 to compact Kähler manifolds with nonvanishing tangent fields, proving that any such manifold X admits an arbitrarily small tangential deformation which is a suspension over a torus; that is, a quotient of F\times \mathbb C^s fibering over a torus T=\mathbb C^s/\Lambda . We further show that either X is uniruled or, up to a finite Abelian covering, it is a small deformation of a product F\times T where F is a Kähler manifold without tangent vector fields and T is a torus. A complete classification when X is a projective manifold, in which case the deformations may be omitted, or when \dim X\leq s+2 is also given. As an application, it is shown that the study of the dynamics of holomorphic tangent fields on compact Kähler manifolds reduces to the case of rational varieties.

Ordinary Smooth Topological Spaces

In this paper, we introduce the concept of ordinary smooth topology on a set X by considering the gradation of openness of ordinary subsets of X. And we obtain the result [Corollary 2.13] : An ordinary smooth topology is fully determined its decomposition in classical topologies. Also we introduce the notion of ordinary smooth [resp. strong and weak] continuity and study some its properties. Also we introduce the concepts of a base and a subbase in an ordinary smooth topological space and study their properties. Finally, we investigate some properties of an ordinary smooth subspace.

Geometric properties on non-complete spaces

The purpose of this paper is to study certain geometrical properties for non-complete normed spaces. We show the existence of a non-rotund Banach space with a rotund dense maximal subspace. As a consequence, we prove that every separable Banach space can be renormed to be non-rotund and to contain a dense maximal rotund subspace. We then construct a non-smooth Banach space with a dense maximal smooth subspace. We also study the Krein-Milman property on non-complete normed spaces and provide a sufficient condition for an infinite dimensional Banach space to have an infinite dimensional, separable quotient.

Learning an Orthogonal and Smooth Subspace for Image Classification

The recent years have witnessed a surge of interests of learning a subspace for image classification, which has aroused considerable researches from the pattern recognition and signal processing fields. However, for image classification, the accuracies of previous methods are not so high since they neglect some particular characters of the image data. In this paper, we propose a new subspace learning method. It constrains that the transformation basis is orthonormal and the derived coefficients are spatially smooth. Classification is then performed in the image subspace. The proposed method can not only represent the intrinsic structure of the image data, but also avoid over-fitting. More importantly, it can be considered as a general framework, within which the performances of other subspace learning methods can be improved in the same way. Some related analyses of the proposed approach are presented. Promising experimental results on different kinds of real images demonstrate the effectiveness of our algorithm for image classification.

Composite Noise Reduction of ERPs Using Wavelet, Model-Based, and Principal Component Subspace Methods

This paper used three theoretically different algorithms for reducing noise in event-related potential (ERP) data. It examined the possibility that a hybrid of these methods could show gains in noise reduction beyond that obtained with any single method. The well-known ERP oddball paradigm was used to evaluate three denoising methods: statistical wavelet transform (wavelet-Z), a smooth subspace wavelet filter (wavelet-S), and subspace PCA. The six possible orders of serial application of these methods to the oddball waveforms were compared for efficacy in signal enhancement. It was found that the order was not commutative, with the best results obtained from applying the wavelet-Z first. Comparison of oddball and frequent trials in the grand average and in individual averages showed considerable enhancement of the differences. It was concluded that denoising to remove variance caused by rare sizeable artifacts is best done first, followed by state space PCA and a light-bias model-based wavelet denoising. The ability to detect and distinguish the effects of variables (such as task, drug effects, individual differences, etc.) on ERPs related to human cognition could be considerably advanced using the denoising methods described here.