Small Subspace Research Articles

High Order Neural Networks for Simultaneous Diagnosis of Multiple Faults in Rotating Machines

To overcome the limitations of the standard feedforward neural networks, high-order neural networks (i.e. ellipsoidal unit networks), which are very useful for fault diagnosis applications due to their bounded generalisation and extrapolation, are proposed. This paper describes the theory and structure of such networks. A method for initialising hyperellipsoids and a training algorithm based on the standard backpropagation algorithm are proposed. Utilising the properties of bounded generalisation and extrapolation inherent in such networks, a Hierarchical Diagnostic Artificial Neural Network (HDANN) based on the ellipsoidal unit network is put forward with respect to simultaneous diagnosis of multiple faults on rotating machines, which consists of several subnetworks and aims at dividing a large pattern space into several smaller subspaces. The subnetworks can be trained in subspaces respectively and the whole network is capable of simultaneous diagnosis of multiple faults. Finally, typical fault data from rotating machines are tested in the network. The research results show that HDANN can obtain more accurate and efficient diagnostic results and is useful for real-time condition monitoring and on-line diagnosis of rotating machines.

Using implicitly filtered RKS for generalised eigenvalue problems

The rational Krylov sequence (RKS) method can be seen as a generalisation of Arnoldi's method. It projects a matrix pencil onto a smaller subspace; this projection results in a small upper Hessenberg pencil. As for the Arnoldi method, RKS can be restarted implicitly, using the QR decomposition of a Hessenberg matrix. This restart comes with a projection of the subspace using a rational function. In this paper, it is shown how the restart can be worked out in practice. In a second part, it is shown when the filtering of the subspace basis can fail and how this failure can be handled by deflating a converged eigenvector from the subspace, using a Schur-decomposition.

An alternative classification method for northern Wisconsin lakes

Wisconsin has nearly 15 000 lakes with great variation in limnology, morphometry, and origin. Classification of lakes into groups is a continuing goal. This study examines two alternative approaches to lake classification, one common and the other somewhat novel. Both approaches used lake morphometry and limnological variables and were compared for ability to form groups and assign lakes to groups with a high probability of correct classification. The first approach used nonhierarchical cluster analysis to form lake groups and discriminant analysis to put lakes into these groups. The second approach formed lake groups by iterative dichotomous splitting of the sampling space into smaller and smaller subspaces. Each binary split was done using nonhierarchical cluster analyses on a subset of the original variables. This iterative splitting resulted in a hierarchical classification tree with reduced dimensionality in comparison with the original data set. At each branch, multiple logistic regression was used to place lakes into nodes of the tree. Validation of both approaches was performed with a resubstitution analysis of the model building data set as well as a separate validation data set. The decision tree method yielded significantly lower rates of misclassification and was more easily interpreted than the discriminant analysis approach.

Orbits in a Stäckel Approximation

Because of their analytical simplicity and regularity, Stäckel potentials are attractive tools for modelling galaxies. The third integral I3 is explicitly known in a Stäckel potential, and can be used as an approximation to the effective third integral, in order to construct three-integral models (cf. Dejonghe, et al., 1996, A&A 306, 363).Moreover, Stäckel potentials turn out to yield good global descriptions for either axisymmetric or triaxial systems without central mass concentration (de Zeeuw 1985, MNRAS 216, 273, de Zeeuw & Lynden-Bell 1985, MNRAS 215, 713), and even for some systems with a black hole included (Sridhar & Touma 1997, MNRAS 292, 657).One long-standing concern though, is that Stäckel potentials form only a very small subspace in the family of all potentials. The main orbit families found by numerical integration in general triaxial potentials are present in a Stäckel potential (Schwarzschild 1981, ApJ 232, 236, de Zeeuw 1985, MNRAS 216, 273), but there is obviously no place in an integrable potential for smaller orbital families or stochastic orbits. However, since regular orbits are the rule rather than the exception, a potential which yields a good representation of those orbits is certainly a good basis for building models.

Restrictions on implicit filtering techniques for orthogonal projection methods

We consider the class of the Orthogonal Projection Methods (OPM) to solve iteratively large eigenvalue problems. An OPM is a method that projects a large eigenvalue problem on a smaller subspace. In this subspace, an approximation of the eigenvalue spectrum can be computed from a small eigenvalue problem using a direct method. Examples of OPMs are the Arnoldi and the Davidson method. We show how an OPM can be restarted — implicitly and explicitly. This restart can be used to remove a specific subset of vectors from the approximation subspace. This is called explicit filtering. An implicit restart can also be combined with an implicit filtering step, i.e. the application of a polynomial or rational function on the subspace, even if inaccurate arithmetic is assumed. However, the condition for the implicit application of a filter is that the rank of the residual matrix must be small.

Exact analytical partition function and energy levels for a Heisenberg ring of N=6 spin 1/2 sites

Using a method which combines reduction due to translational symmetry and elimination of degenerate states from smaller subspaces, the 64×64 Hamiltonian matrix of a Heisenberg ring of N=6 spin 1/2 atoms can be reduced to submatrices that are no larger than 3×3. This reduction makes possible the calculation of the exact eigenenergies E( k) of the system and subsequently the derivation of the partition function in an exact analytical form. The results are in perfect agreement with those obtained numerically by other authors.

Subspace Approach to Inversion by Genetic Algorithms Involving Multiple Frequencies

Based on waveguide physics, a subspace inversion approach is proposed. It is observed that the ability to estimate a given parameter depends on its sensitivity to the acoustic wavefield, and this sensitivity depends on frequency. At low frequencies it is mainly the bottom parameters that are most sensitive and at high frequencies the geometric parameters are the most sensitive. Thus, the parameter vector to be determined is split into two subspaces, and only part of the data that is most influenced by the parameters in each subspace is used. The data sets from the Geoacoustic Inversion Workshop (June 1997) are inverted to demonstrate the approach. In each subspace Genetic Algorithms are used for the optimization — it provides flexibility to search over a wide range of parameters and also helps in selecting data sets to be used in the inversion. During optimization, the responses from many environmental parameter sets are computed in order to estimate the a posteriori probabilities of the model parameters. Thus the uniqueness and uncertainty of the model parameters are assessed. Using data from several frequencies to estimate a smaller subspace of parameters iteratively provides stability and greater accuracy in the estimated parameters.

Subspace Orthogonalization for Substructuring Preconditioners for Non-self-adjoint Elliptic Problems

For non-self-adjoint elliptic boundary value problems which are preconditioned by a substructuring method, i.e., nonoverlapping domain decomposition, we introduce and study the concept of subspace orthogonalization. In subspace orthogonalization variants of Krylov methods, the computation of inner products and vector updates, and the storage of basis elements is restricted to a (presumably small) subspace, in this case the edge and vertex unknowns with respect to the partitioning into subdomains. We discuss the convergence properties of these iteration schemes and compare them with Krylov methods applied to the full preconditioned system.

Selective feature extraction via signal decomposition

In this article, a mathematical framework that jointly optimizes the parameters of classifier and feature extractor is presented. In this approach, feature extraction is formulated as a process of projecting the signals onto a smaller subspace in which the statistical properties of the signal can be efficiently modeled. An algorithm, called statistical matching pursuit (SMP), is proposed to learn from the training data the optimal projection dimensions and the extent of signal reduction. The algorithm is designed to achieve unconditional convergence and can be seamlessly incorporated into the expectation-maximization (EM) algorithm employed to train the classifier. Finally, we report some experimental results on speech recognition and elaborate the potential of the proposed method.

Nested Krylov methods based on GCR

Recently the GMRESR method for the solution of linear systems of equations has been introduced by Vuik and Van der Vorst (1991). Similar methods have been proposed by Axelsson and Vassilevski (1991) and Saad (1993) (FGMRES 1 1 Since FGMRES and GMRESR are very similar, the ideas presented will be relevant for FGMRES as well. ). GMRESR involves an outer and an inner method. The outer method is GCR, which is used to compute the optimal approximation over a given set of search vectors in the sense that the residual is minimized. The inner method is GMRES, which computes a new search vector by approximately solving the residual equation. This search vector is then used by the outer algorithm to compute a new approximation. However, the optimality of the approximation over the space of search vectors is ignored in the inner GMRES algorithm. This leads to suboptimal corrections to the solution in the outer algorithm. Therefore, we propose to preserve the orthogonality relations of GCR in the inner GMRES algorithm. This gives optimal corrections to the solution and also leads to solving the residual equation in a smaller subspace and with an “improved” operator, which should also lead to faster convergence. However, this involves using Krylov methods with a singular, nonsymmetric operator. We will discuss some important properties of this. We will show by experiments that in terms of matrix-vector products, this modification (almost) always leads to better convergence. Because we do more orthogonalizations, it does not always give an improved performance in time. This depends on the costs of the matrix-vector products relative to the costs of the orthogonalizations. Of course, we can also use methods other than GMRES as the inner method. Methods with short recurrences like BiCGSTAB seem especially interesting. The experimental results indicate that, especially for such methods, it is advantageous to preserve the orthogonality in the inner method.

Flat connections and nonlocal conserved quantities in irrational conformal field theory

Irrational conformal field theory (ICFT) includes rational conformal field theory as a small subspace, and the affine-Virasoro Ward identities describe the biconformal correlators of ICFT. The Ward identities are reformulated as an equivalent linear partial differential system with flat connections and new nonlocal conserved quantities. As examples of the formulation, the system of flat connections is solved for the coset correlators, the correlators of the affine-Sugawara nests, and the high-level n-point correlators of ICFT.

Vapnik-Chervonenkis dimension and (pseudo-)hyperplane arrangements

An arrangement of oriented pseudohyperplanes in affined-space defines on its setX of pseudohyperplanes a set system (or range space) (X, ?), ? ? 2x of VC-dimensiond in a natural way: to every cellc in the arrangement assign the subset of pseudohyperplanes havingc on their positive side, and let ? be the collection of all these subsets. We investigate and characterize the range spaces corresponding tosimple arrangements of pseudohyperplanes in this way; such range spaces are calledpseudogeometric, and they have the property that the cardinality of ? is maximum for the given VC-dimension. In general, such range spaces are calledmaximum, and we show that the number of rangesR?? for whichX - R?? also, determines whether a maximum range space is pseudogeometric. Two other characterizations go via a simple duality concept and small subspaces. The correspondence to arrangements is obtained indirectly via a new characterization of uniforom oriented matroids: a range space (X, ?) naturally corresponds to a uniform oriented matroid of rank |X|--d if and only if its VC-dimension isd,R?? impliesX - R??, and |?| is maximum under these conditions.

Some Cardinal Inequalities in Lindelöf Spaces

ABSTRACT. Some bounds on the cardinality of Lindelöf T1‐spaces such that small subspaces verify a meta‐Lindelöf property are proved. A procedure to extend these results to a larger class of spaces is shown.

THE LIE h-INVARIANT CONFORMAL FIELD THEORIES AND THE LIE h-INVARIANT GRAPHS

We use the Virasoro master equation to study the space of Lie h-invariant conformal field theories, which includes the standard rational conformal field theories as a small subspace. In a detailed example, we apply the general theory to characterize and study the Lie h-invariant graphs, which classify the Lie h-invariant conformal field theories of the diagonal ansatz on SO(n). The Lie characterization of these graphs is another aspect of the recently observed Lie group-theoretic structure of graph theory.

Analysis of Some Krylov Subspace Approximations to the Matrix Exponential Operator

In this note a theoretical analysis of some Krylov subspace approximations to the matrix exponential operation $\exp (A)v$ is presented, and a priori and a posteriors error estimates are established. Several such approximations are considered. The main idea of these techniquesis to approximately project the exponential operator onto a small Krylov subspace and to carry out the resulting small exponential matrix computation accurately. This general approach, which has been used with success in several applications, provides a systematic way of defining high-order explicit-type schemes for solving systems of ordinary differential equations or time-dependent partial differential equations.

The use of reduced finite element models in system identification

AbstractAn a priori, analytical model in system identification of vibrating structures is improved by input and output measurements with least square fitting. The structure is modelled by the finite element method. Finite element models usually need a large number of degrees of freedom to simulate a small number of lower spectrum eigenfrequencies with accuracy. The large finite element model is reduced to a subspace of significant eigenfrequencies. The size of the subspace is chosen with regard to the frequency content of the measured data and the accuracy of the large analytical model. An identification method is formulated for the large analytical model. This procedure improves system parameters in the matrices of the large model by measured input and output data with a least square functional. The objective function is consistently reduced, so that the whole identification procedure can be performed in the small subspace. The proposed reduction method permits very large and accurate analytical models to be used, and it decreases the computational cost of the identification procedure significantly. The computational efficiency is demonstrated on an in situ experiment of a radar tower.

Three‐dimensional electromagnetic analysis by the subspace synthesis method

AbstractThe required computer memory capacity and the calculation time become enormous when analyzing problems for three‐dimensional electromagnetic fields by finite‐element methods. In many cases calculation is difficult even by a super‐computer. To save the required computer memory capacity and computer time, we propose a subspace synthesis method, an outline of which is as follows. First, the object space is divided into several subspaces. In each subspace we apply the finite‐element method. We formulate dispersed equations for each subspace. From these equations we derive reduced equations which relate only the unknown coefficients on connecting regions among sub‐spaces. By combining reduced equations from all subspaces, we obtain the reduced equations for the total object space. By solving them we can obtain the electromagnetic fields. In the proposed method, we execute calculations for a small subspace one at a time. Therefore the required computer memory capacity is reduced greatly from that for analyzing the object space by the direct application of the finite‐element method. The corresponding computer calculation time is also reduced. This paper shows the formulation of the subspace synthesis method, and applies it to several examples. The calculation results by the present method were compared with rigorous solutions or experimental data. The effectiveness of the proposed method was shown from these comparisons.

Multilevel adaptive technique for quantum reactive scattering

Discretization (with localized basis functions or grid points) of the coupled integral equations for molecular collisions leads to a very large system of linear algebraic equations. New methods, which are well adapted to vector supercomputers and parallel architectures, are developed for solving this large system. The multilevel adaptive technique (MLAT) is combined with recursive and iterative techniques. First, a multichannel solution is obtained on a low level grid. The basis is then adapted to this solution and the coarse solution is projected or interpolated onto the adapted basis. The scattering amplitudes (K-matrix elements) on the high level are then developed through use of either the recursion method (for single amplitudes, or a small batch of them) or the iterative technique (for all transitions from a specified initial state). In both of these methods, the original large system of algebraic equations is projected into a much smaller subspace (an orthonormalized Krylov space) spanned by a few basis vectors. Applications to very large systems are possible because it is not necessary to store or invert a large matrix. Computational results on a model chemical reaction are presented.

Exact calculation of level densities for many non-interacting fermions in large model spaces

A simple recursive method to calculate exactly level densities with fixed spinJ and parity is presented for an independent particle model of nuclei. This method can be used to calculate fixed particle-hole level densities as well. The results of these exact calculations are compared to those of commonly used approximation methods, like the saddle-point approximation and Gram-Charlier expansion. The former appears to be quite accurate, while the accuracy of Gram-Charlier expansion is strongly dependent on decomposition of then-fermion space into smaller subspaces. Comparison of exactly calculated with experimental densities at the neutron evaporation threshold shows unambiguously that some residual interaction must be introduced to describe nuclei at these excitation energies.

Universal spectra of quasirandom objects produced by off-equilibrium space divisions.

A simple model of quasirandomness is presented. The model is a self-similar space division: A d-dimensional space is divided successively into smaller subspaces with randomness. Positions of subspaces are fixed. Then centers of subspaces distribute quasirandomly. Two typical cases of the model are discussed. In one case (repulsive case) the power spectrum of centers of subspaces exhibits a ${k}^{\mathrm{\ensuremath{-}}\ensuremath{\alpha}}$ (\ensuremath{\alpha}\ensuremath{\approxeq}d) singularity in a broad range of a randomness parameter, where k is the wave number. This spectrum corresponds to a so-called 1/f noise spectrum for a one-dimensional time sequence. The ${k}^{\mathrm{\ensuremath{-}}d}$ singularity will be observed in the scattering intensity from a phase-separating system if droplets cannot move freely due to the strong correlation. This singularity does not seem to depend on details of the model, and is universal. We can show the same singularity analytically for a modified model. In the other case (attractive case) the power spectrum exhibits a ${k}^{\mathrm{\ensuremath{-}}\ensuremath{\alpha}}$ (\ensuremath{\alpha}\ensuremath{\approxeq}0.4d) singularity in a broad range of the randomness parameter. This singularity is equivalent to the typical singularity observed in the universe. The physical reason for this agreement is discussed.