Complete Orthogonal Decomposition Research Articles

An efficient algorithm for the minimal least squares solution of linear systems with indefinite symmetric matrices

In this work, a new algorithm for solving symmetric indefinite systems of linear equations is presented. It factorizes the matrix into the form LDLt using Jacobi rotations in order to increase the pivot´s absolute value. Furthermore, Rook´s pivoting strategy is also adapted and implemented. In determinate compatible systems, the computational cost of the algorithm was similar to the cost of the Bunch-Kaufman method, but the error was approximately 50 % smaller for intermediate and large matrices, regardless of the condition number of the coefficient matrix. Furthermore, unlike Bunch-Kaufman, the new algorithm calculates with little additional cost the fundamental basis of the null space, and obtains the minimal least squares and minimum norm solutions. In minimal least squares with minimum norm problems, the new algorithm was compared with the LAPACK Complete Orthogonal Decomposition algorithm, among others. The obtained error with both algorithms was similar but the computational cost was at least 20 % smaller with the new algorithm, even though the Complete Orthogonal Decomposition is implemented in a blocked form.

Short-, Medium-, and Long-Term Prediction of Carbon Dioxide Emissions using Wavelet-Enhanced Extreme Learning Machine

Carbon dioxide (CO2) is the main greenhouse gas responsible for global warming. Early prediction of CO2 is critical for developing strategies to mitigate the effects of climate change. A sophisticated version of the extreme learning machine (ELM), the wavelet enhanced extreme learning machine (W-EELM), is used to predict CO2 on different time scales (weekly, monthly, and yearly). Data were collected from the Mauna Loa Observatory station in Hawaii, which is ideal for global air sampling. Instead of the traditional method (singular value decomposition), a complete orthogonal decomposition (COD) was used to accurately calculate the weights of the ELM output layers. Another contribution of this study is the removal of noise from the input signal using the wavelet transform technique. The results of the W-EELM model are compared with the results of the classical ELM. Various statistical metrics are used to evaluate the models, and the comparative figures confirm the superiority of the applied models over the ELM model. The proposed W-EELM model proves to be a robust and applicable computer-based technology for modeling CO2concentrations, which contributes to the fundamental knowledge of the environmental engineering perspective. Doi: 10.28991/CEJ-2023-09-04-04 Full Text: PDF

EpiReSIM: A Resampling Method of Epistatic Model without Marginal Effects Using Under-Determined System of Equations.

Simulation experiments are essential to evaluate epistasis detection methods, which is the main way to prove their effectiveness and move toward practical applications. However, due to the lack of effective simulators, especially for simulating models without marginal effects (eNME models), epistasis detection methods can hardly verify their effectiveness through simulation experiments. In this study, we propose a resampling simulation method (EpiReSIM) for generating the eNME model. First, EpiReSIM provides two strategies for solving eNME models. One is to calculate eNME models using prevalence constraints, and another is by joint constraints of prevalence and heritability. We transform the computation of the model into the problem of solving the under-determined system of equations. Introducing the complete orthogonal decomposition method and Newton's method, EpiReSIM calculates the solution of the underdetermined system of equations to obtain the eNME model, especially the solution of the high-order model, which is the highlight of EpiReSIM. Second, based on the computed eNME model, EpiReSIM generates simulation data by a resampling method. Experimental results show that EpiReSIM has advantages in preserving the biological properties of minor allele frequencies and calculating high-order models, and it is a convenient and effective alternative method for current simulation software.

Study of a fiber spectrometer based on offset fusion.

A near-infrared spectrometer based on offset fused multimode fiber (MMF) is investigated in this study. The light spectrum is recovered by analyzing the speckle images when light is passing through the MMF. In order to generate adequate speckles, a polarization maintaining fiber (PMF) and a 30 cm long MMF are fused with a vertical offset. Seven different offset displacements are implemented in the fiber fusion. The follow-up experiments show that the fiber offset fusion has a significant influence on the spectral correlation and the resolution. Larger offset fusion can excite more high-order modes in the MMF, and it greatly improves the spectrometer's performance. The simulation results also show that more modes are excited in MMF, and the increase of mode number leads to lower correlation coefficients of the neighboring spectral channels. However, large offset fusion increases the fusion and the insertion loss of the whole system, which may bring difficulties in the low-light cases. In addition, an image denoising algorithm based on dynamic threshold filtering and a spectral reconstruction algorithm originated from complete orthogonal decomposition were used to remove the speckle pattern noise and recover the spectrum. The final speckle-based spectrometer has a spectral resolution of 0.6∼0.016nm depending on the different offset fusions.

An efficient LDU algorithm for the minimal least squares solution of linear systems

Abstract The minimal least squares solutions is a topic of interest due to the broad range of applications of this problem. Although it can be obtained from other algorithms, such as the Singular Value Decomposition (SVD) or the Complete Orthogonal Decomposition (COD), the use of LDU factorizations has its advantages, namely the computational cost and the low fill-in that can be obtained using this method. If the right and left null-subspaces (which can also be named as Null and Image subspaces, respectively) are to be obtained, the use of these factorizations leads to fundamental subspaces, which are sparse by definition. Here an algorithm that takes advantage of both the Peters–Wilkinson method and Sautter method is presented. This combination allows for a good performance in all cases. The method also optimizes memory use by storing the right null-subspace and the left null-subspace in the factored matrix.

Fault diagnosis of output‐related processes with multi‐block MOPLS

For fault diagnosis of output‐related processes, a relatively high false alarm rate (FAR) of output‐irrelevant faults exists because the output‐irrelevant variables are not removed completely by conventional approaches. A relatively large number of computational loads is thus required. Therefore, in this paper, a new fault diagnosis approach based on multiblock modified orthogonal projections to latent structures is proposed to complete fault diagnosis for complex chemical processes, particularly for the penicillin fermentation process. The main contributions are as follows: (1) Multiblock orthogonal projections to latent structures are applied to remove the noncorrelated variables of input blocks, which requires a relatively low computational load. In addition, block scores are obtained by block weights rather than super weights to better describe the character of each subblock. (2) Complete orthogonal decomposition between input block variables and output is explored to completely separate output‐related and output‐irrelevant variables to improve the accuracy of diagnosis. (3) A hierarchical diagnosis scheme, which is composed of block monitoring statistics and subblock monitoring statistics, is proposed to monitor and localize faults. Penicillin fermentation process is considered in this study, and the penicillin concentration‐relevant fault in each block and subblock is analyzed. The results of this study show that the proposed method has steadier performance on output‐related fault diagnoses and diagnoses faults more accurately.

Boundary-aware hodge decompositions for piecewise constant vector fields

We provide a theoretical framework for discrete Hodge-type decomposition theorems of piecewise constant vector fields on simplicial surfaces with boundary that is structurally consistent with decomposition results for differential forms on smooth manifolds with boundary. In particular, we obtain a discrete Hodge–Morrey–Friedrichs decomposition with subspaces of discrete harmonic Neumann fields Hh,N and Dirichlet fields Hh,D, which are representatives of absolute and relative cohomology and therefore directly linked to the underlying topology of the surface. In addition, we discretize a recent result that provides a further refinement of the spaces Hh,N and Hh,D, and answer the question in which case one can hope for a complete orthogonal decomposition involving both spaces at the same time.As applications, we present a simple strategy based on iterated L2-projections to compute refined Hodge-type decompositions of vector fields on surfaces according to our results, which give a more detailed insight than previous decompositions. As a proof of concept, we explicitly compute harmonic basis fields for the various significant subspaces and provide exemplary decompositions for two synthetic vector fields.

An integrated chaotic time series prediction model based on efficient extreme learning machine and differential evolution

In this paper, an integrated model based on efficient extreme learning machine (EELM) and differential evolution (DE) is proposed to predict chaotic time series. In the proposed model, a novel learning algorithm called EELM is presented and used to model the chaotic time series. The EELM inherits the basic idea of extreme learning machine (ELM) in training single hidden layer feedforward networks, but replaces the commonly used singular value decomposition with a reduced complete orthogonal decomposition to calculate the output weights, which can achieve a much faster learning speed than ELM. Moreover, in order to obtain a more accurate and more stable prediction performance for chaotic time series prediction, this model abandons the traditional two-stage modeling approach and adopts an integrated parameter selection strategy which employs a modified DE algorithm to optimize the phase space reconstruction parameters of chaotic time series and the model parameter of EELM simultaneously based on a hybrid validation criterion. Experimental results show that the proposed integrated prediction model can not only provide stable prediction performances with high efficiency but also achieve much more accurate prediction results than its counterparts for chaotic time series prediction.

Algorithm 930

The MATLAB™ backslash (x=A\b) is an elegant and powerful interface to a suite of high-performance factorization methods for the direct solution of the linear system Ax = b and the least-squares problem min x ‖ b - Ax ‖. It is a meta-algorithm that selects the best factorization method for a particular matrix, whether sparse or dense. However, the simplicity and elegance of its single-character interface prohibits the reuse of its factorization for subsequent systems. Requiring MATLAB users to find the best factorization method on their own can lead to suboptimal choices; even MATLAB experts can make the wrong choice. Furthermore, naive MATLAB users have a tendency to translate mathematical expressions from linear algebra directly into MATLAB, so that x = A -1 b becomes the inferior yet all-to-prevalent x=inv(A)*b. To address these issues, an object-oriented FACTORIZE method is presented. Via simple-to-use operator overloading, solving two linear systems can be written as F=factorize(A); x=F\b; y=F\c, where A is factorized only once. The selection of the best factorization method (LU, Cholesky, LDL T , QR, or a complete orthogonal decomposition for rank-deficient matrices) is hidden from the user. The mathematical expression x = A -1 b directly translates into the MATLAB expression x=inverse(A)*b, which does not compute the inverse at all, but does the right thing by factorizing A and solving the corresponding triangular systems.

Robust extreme learning machine

The output weights computing of extreme learning machine (ELM) encounters two problems, the computational and outlier robustness problems. The computational problem occurs when the hidden layer output matrix is a not full column rank matrix or an ill-conditioned matrix because of randomly generated input weights and biases. An existing solution to this problem is Singular Value Decomposition (SVD) method. However, the training speed is still affected by the large complexity of SVD when computing the Moore–Penrose (MP) pseudo inverse. The outlier robustness problem may occur when the training data set contaminated with outliers then the accuracy rate of ELM is extremely affected. This paper proposes the Extended Complete Orthogonal Decomposition (ECOD) method to solve the computational problem in ELM weights computing via ECODLS algorithm. And the paper also proposes the other three algorithms, i.e. the iteratively reweighted least squares (IRWLS-ELM), ELM based on the multivariate least-trimmed squares (MLTS-ELM), and ELM based on the one-step reweighted MLTS (RMLTS-ELM) to solve the outlier robustness problem. However, they also encounter the computational problem. Therefore, the ECOD via ECODLS algorithm is also used successfully in the three proposed algorithms. The experiments of regression problems were conducted on both toy and real-world data sets. The outlier types are one-sided and two-sided outliers. Each experiment was randomly contaminated with outliers, of one type only, with 10%, 20%, 30%, 40%, and 50% of the total training data size. Meta-metrics evaluation was used to measure the outlier robustness of the proposed algorithms compared to the existing algorithms, i.e. the minimax probability machine regression (MPMR) and the ordinary ELM. The experimental results showed that ECOD can effectively replace SVD. The ECOD is robust to the not full column rank or the ill-conditional problem. The speed of the ELM training using ECOD is also faster than the ordinary training algorithm. Moreover, the meta-metrics measure showed that the proposed algorithms are less affected by the increasing number of outliers than the existing algorithms.

Hodge decomposition for symmetric matrix fields and the elasticity complex in Lipschitz domains

In 1999 M. Eastwood has used the general construction known as the Bernstein-Gelfand-Gelfand (BGG) resolution to prove, at least in smooth situation, the equivalence of the linear elasticity complex and of the de Rham complex in $\mathbf{R}^{3}$. The main objective of this paper is to study the linear elasticity complex for general Lipschitz domains in $\mathbf{R}^{3}$ and deduce a complete Hodge orthogonal decomposition for symmetric matrix fields in $L^{2}$, counterpart of the Hodge decomposition for vector fields. As a byproduct one obtains that the finite dimensional terms of this Hodge decomposition can be interpreted in homological terms as the corresponding terms for the de Rham complex if one takes the homology with value in $RIG\cong \mathbf{R}^{6}$ as in the (BGG) resolution.

Solving Rank-Deficient and Ill-posed Problems Using UTV and QR Factorizations

The algorithm of Mathias and Stewart [Linear Algebra Appl., 182 (1993), pp. 91--100] is examined as a tool for constructing regularized solutions to rank-deficient and ill-posed linear equations. The algorithm is based on a sequence of QR factorizations. If it is stopped after the first step, it produces the same solution as the complete orthogonal decomposition used in LAPACK's xGELSY. However, we show that for low-rank problems a careful implementation can lead to an order of magnitude improvement in speed over xGELSY as implemented in LAPACK. We prove, under assumptions similar to assumptions used by others, that if the numerical rank is chosen at a gap in the singular value spectrum and if the initial factorization is rank-revealing, then, even if the algorithm is stopped after the first step, approximately half the time its solutions are closer to the desired solution than are the singular value decomposition (SVD) solutions. Conversely, the SVD will be closer approximately half the time, and in this case overall the two algorithms are very similar in accuracy. We confirm this with numerical experiments. Although the algorithm works best for problems with a gap in the singular value spectrum, numerical experiments suggest that it may work well for problems with no gap.

Complete orthogonal decomposition homomorphisms between matrix ordered Hilbert spaces

The purpose of this paper is to show that a complete order homomorphism and a complete orthogonal decomposition homomorphism between the non-commutative L 2 L^{2} -spaces induce respectively an isomorphism and a ∗ * -isomorphism between the associated reduced von Neumann algebras.

Computation of general inner-outer and spectral factorizations

In this paper, we solve two problems in linear systems theory: the computation of the inner-outer and spectral factorizations of a continuous-time system considered in the most general setting. We show that these factorization problems rely essentially on solving for the stabilizing solution a standard algebraic Riccati equation of order usually much smaller than the McMillan degree of the transfer function matrix of the system. The proposed procedures are completely general, being applicable for a polynomial/proper/improper system whose transfer function matrix could be rank deficient and could have poles/zeros on the imaginary axis or at infinity. As an application we discuss the extension to the case of rational matrices of the complete orthogonal decomposition of a constant matrix. Numerical refinements and examples illustrating the proposed approach, are discussed in detail.

Computing with Differential-difference Operators

Computer algebra can be used to prove identities in the algebra of operators on polynomials which is generated by multiplication by coordinate functions, and the group translations and Dunkl operators associated with a reflection group. This technique is illustrated by a conceptual proof of a complete orthogonal decomposition of the harmonic polynomials associated with the abelian reflection groups.

Ordinary linear model estimation on a massively parallel SIMD computer

Efficient algorithms for estimating the coefficient parameters of the ordinary linear model on a massively parallel SIMD computer are presented. The numerical stability of the algorithms is ensured by using orthogonal transformations in the form of Householder reflections and Givens plane rotations to compute the complete orthogonal decomposition of the coefficient matrix. Algorithms for reconstructing the orthogonal matrices involved in the decompositions are also designed, implemented and analyzed. The implementation of all algorithms on the targeted SIMD array processor is considered in detail. Timing models for predicting the execution time of the implementations are derived using regression modelling methods. The timing models also provide an insight into how the algorithms interact with the parallel computer. The predetermined factors used in the regression fits are derived from the number of memory layers involved in the factorization process of the matrices. Experimental results show the high accuracy and predictive power of the timing models. Copyright © 1999 John Wiley & Sons, Ltd.

Efficient Solution Of The Rank-Deficient Linear Least Squares Problem

In this paper we present several new fast and reliable algorithms for solving rank-deficient linear least squares problems by means of the complete orthogonal decomposition. Experimental comparison of our algorithms with the existing implementations in LAPACK on a wide range of platforms shows considerable performance improvements. Moreover, for full-rank matrices the performance of the new algorithm is very close to that ofthe fast method based on QR factorization, thus providing a valuable general tool for full-rank matrices, rank-deficient matrices, and those matrices with unknown rank.

Complete Orthogonal Decomposition for Weighted Least Squares

This paper proposes a complete orthogonal decomposition (COD) algorithm for solving weighted least-squares problems. In applications, the weight matrix can be highly ill conditioned, and this can cause standard methods like QR factorization to return inaccurate answers in floating-point arithmetic. Stewart and Todd independently established a norm bound for the weighted least-squares problem that is independent of the weight matrix. Vavasis proposed a definition of a "stable" solution of weighted least squares based on this norm bound: The solution computed by a stable algorithm must satisfy an accuracy bound that is not affected by ill conditioning in the weight matrix. A forward error analysis shows that the COD algorithm is stable in this sense, but it is simpler and more efficient than the algorithm proposed by Vavasis. Our forward error bound is contrasted to the backward error analysis of other previous works on weighted least squares.

Phase space, wavelet transform and Toeplitz-Hankel type operators

Let IG(n) be the Euclidean group with dilations. It has a maximal compact subgroup SO(n−1). The homogeneous space can be realized as the phase space IG(n)/SO(n−1)≅R n ×R n . The square-integrable representation gives the admissible wavelets AW and wavelet transforms onL 2(R n ). With Laguerre polynomials and surface spherical harmonics an orthogonal decomposition of AW is given; it turns to give a complete orthogonal decomposition of theL 2-space on the phase spaceL 2(R n ×R n ,dxdy/|y| n+1) of the form ⊕ k=0 ∞ ⊕ l=0 ∞ ⊕ j=0 al A l,j k . The Schatten-von Neumann properties of the Toeplitz-Hankel type operators between these decomposition components are established.

Fast Polar Decomposition of an Arbitrary Matrix

The polar decomposition of an $m \times n$ matrix A of full rank, where $m \geqq n$, can be computed using a quadratically convergent algorithm of Higham [SIAM J. Sci. Statist. Comput., 7(1986), pp. 1160–1174]. The algorithm is based on a Newton iteration involving a matrix inverse. It is shown how, with the use of a preliminary complete orthogonal decomposition, the algorithm can be extended to arbitrary A. The use of the algorithm to compute the positive semidefinite square root of a Hermitian positive semidefinite matrix is also described. A hybrid algorithm that adaptively switches from the matrix inversion based iteration to a matrix multiplication based iteration due to Kovarik, and to Björck and Bowie, is formulated. The decision when to switch is made using a condition estimator. This “matrix multiplication rich” algorithm is shown to be more efficient on machines for which matrix multiplication can be executed 1.5 times faster than matrix inversion.