СНИЖЕНИЕ РАЗМЕРНОСТИ ОБУЧАЮЩИХ ВЫБОРОК ПРИ РАСПОЗНАВАНИИ ОБРАЗОВ НА КОСМИЧЕСКИХ ИЗОБРАЖЕНИЯХ С ПОМОЩЬЮ МЕТОДА ГЛАВНЫХ КОМПОНЕНТ

The essence of principal components analysis and the problem of dimension reduction are described. A method of principal components calculation is presented, which is based on the covariance matrix eigenvalues determination. Practical implementations of principal components analysis are described, which are based on QR-algorithm. Application of principal components analysis in space images classification for the reduction of training samples dimension is discussed.Описываются сущность метода главных компонент и задача снижения размерности в про-цессе статистической обработки. Приводится способ вычисления главных компонент на основе оп-ределения собственных значений ковариационной матрицы. Описываются алгоритмы практической реализации метода главных компонент на основе QR-алгоритма. Проводится анализ возможности использования метода главных компонент при классификации космических изображений с целью снижения размерности обучающих выборок

Efficient Numerical Algorithms for Balanced Stochastic Truncation

We propose an efficient numerical algorithm for relative error model reduction based on balanced stochastic truncation. The method uses full-rank factors of the Gramians to be balanced versus each other and exploits the fact that for large-scale systems these Gramians are often of low numerical rank. We use the easy-to-parallelize sign function method as the major computational tool in determining these full-rank factors and demonstrate the numerical performance of the suggested implementation of balanced stochastic truncation model reduction

Communication and Matrix Computations on Large Message Passing Systems

This paper is concerned with the consequences for matrix computations of having a rather large number of general purpose processors, say ten or twenty thousand, connected in a network in such a way that a processor can communicate only with its immediate neighbors. Certain communication tasks associated with most matrix algorithms are defined and formulas developed for the time required to perform them under several communication regimes. The results are compared with the times for a nominal $n^3$ floating point operations. The results suggest that it is possible to use a large number of processors to solve matrix problems at a relatively fine granularity, provided fine grain communication is available. Additional figures are available at ftp thales.cs.umd.edu in the directory pub/reports (Also cross-referenced as UMIACS-TR-88-81

A Novel Parallel QR Algorithm For Hybrid Distributed Memory HPC Systems

A novel variant of the parallel QR algorithm for solving dense nonsymmetric eigenvalue problems on hybrid distributed high performance computing systems is presented. For this purpose, we introduce the concept of multiwindow bulge chain chasing and parallelize aggressive early deflation. The multiwindow approach ensures that most computations when chasing chains of bulges are performed in level 3 BLAS operations, while the aim of aggressive early deflation is to speed up the convergence of the QR algorithm. Mixed MPI-OpenMP coding techniques are utilized for porting the codes to distributed memory platforms with multithreaded nodes, such as multicore processors. Numerous numerical experiments confirm the superior performance of our parallel QR algorithm in comparison with the existing ScaLAPACK code, leading to an implementation that is one to two orders of magnitude faster for sufficiently large problems, including a number of examples from applications

Linear structure of nonlinear dynamic systems via Koopman decomposition

Doctor of PhilosophyDepartment of Mechanical and Nuclear EngineeringMingjun WeiLinear structure and invariant subspaces of nonlinear dynamics are revealed, extending the superposition principle and invariant subspaces from linear dynamics. They are achieved by considering dynamics in its dual space and the local spectral Koopman theory. The Koopman eigenfunctions constitute invariant subspaces under the given dynamic system, providing convenient bases for the linear structure. On the other hand, the locality and infinite dimensionality are identified as two unique properties of nonlinear dynamics, where the former refers to the spectral problem is locally defined, and the latter refers to Koopman spectrums are recursively proliferated by nonlinear interaction. Koopman spectral theory is studied. For a linear time-invariant (LTI) system, its linear spectrum is a subset of Koopman spectrums. High order Koopman spectrum can be obtained for nonlinear observables using the proliferation rule. For a linear time-variant system (LTV), Koopman decomposition is obtained by the eigenvalue problem of its fundamental matrix. Besides the general LTV, the periodic LTV system is studied using the Floquet theory. The Floquet spectrums are found to be Koopman spectrums. For a nonlinear system, a local Koopman spectrum problem is defined for a parameterized semigroup Koopman operator, and the simple local spectra are found to be conditionally continuous from the operator perturbation theory. The proliferation is found to recursively applicable to nonlinear dynamics. Moreover, the hierarchy structure of the Koopman decomposition of nonlinear systems is discovered, by decomposing dynamics into base and perturbation on top of it. The numerical algorithm, dynamic mode decomposition (DMD), is examined for its applicability to capture the spectra and modes for a variety of dynamic systems. A more robust and efficient framework based on generalized eigenvalue problem (GEV) is proposed, which is then solved by a least-square solution (LS) or a total least square solution (TLS). Therefore, two algorithms, DMD-LS and DMD-TLS algorithm, are developed. DMD-LS algorithm is mathematically equivalent to the standard DMD algorithm first proposed by Schmid (2010) but more robust. DMD-TLS is more accurate for noise data. A residue-based criterion is developed to choose dynamically important or true DMD modes from trivial or spurious modes that often appear in DMD computations. Linear structure via Koopman decomposition is first applied to a linear dynamic system and an asymptotic nonlinear system, for example. Then flow past fixed cylinder of a Hopf bifurcation process is numerically studied via DMD technique. The equivalence of Koopman decomposition to the GSA is verified at the primary instability stage. The Fourier modes, the least stable Floquet modes, and their high-order derived Koopman modes are found to be the superposition of countable infinite Koopman modes when the flow reaches periodic by considering continuity of Koopman spectrum and the invariance of Koopman modes to the nonlinear transition process. The nonlinear modulation effects, namely, the modulation of the mean flow and the resonance phenomena is explained similarly. The coherent structures are also found to be related to the decomposition. A DMD based model order reduction method is implemented based on Galerkin projection. The model reduction approach is applied to both the transitional and the periodic stages of flow passing a fixed cylinder. Accurate dynamics and frequencies are rebuilt

A Parallel Implementation of the QR Algorithm

In this paper a parallel implementation of the QR algorithm for the eigenvalues of a non-Hermitian matrix is proposed. The algorithm is designed to run efficiently on a linear array of processors that communicate by accessing their neighbors' memory. A module for building such arrays, the Maryland crab, is also described. 1 Introduction This paper has two purposes. The first is to describe a parallel version of the QR algorithm for the eigenvalues of a nonsymmetric matrix. The second is to show how the algorithm may be implemented on a network of processors that communicate through locally shared memory. We shall be especially concerned to insure that our algorithm runs well at medium granularity. To see what this means, consider the problem of performing Gaussian elimination on a matrix A of order n with a linear array of p processors. Typically the matrix might be partitioned by columns in the form A = (A 1 ; A 2 ; : : : ; A p ); (1:1) where each A k has roughly n=p columns. Thes..