Decomposition Of Matrices Research Articles

N‐BANDS: A new algorithm for estimating the extension of feasible bands in multivariate curve resolution of multicomponent systems in the presence of noise and rotational ambiguity

AbstractA new algorithm named N‐BANDS has been developed for the estimation of the combined effect of noise and rotational ambiguity in the bilinear decomposition of data matrices using the popular multivariate curve resolution–alternating least‐squares model. It is based on a nonlinear maximization and minimization of a component‐wise signal contribution function (SCF), with a single‐objective function and a separate module for applying a variety of constraints. The algorithm can be applied to multicomponent systems and efficiently estimates the extreme component profiles corresponding to maximum and minimum SCF in the presence of varying amounts of instrumental noise. Simulated systems mimicking multicomponent and multisample analytical calibration protocols have been studied, having uncalibrated interferents in the test samples. Different noise structures (independent and identically distributed, proportional and correlated) were added to the data matrices, with results that indicate an increase in the extension of feasible bands as the noise level increases.

Fast randomized matrix and tensor interpolative decomposition using CountSketch

We propose a new fast randomized algorithm for interpolative decomposition of matrices which utilizes CountSketch. We then extend this approach to the tensor interpolative decomposition problem introduced by Biagioni et al. (J. Comput. Phys. 281, pp. 116-134, 2015). Theoretical performance guarantees are provided for both the matrix and tensor settings. Numerical experiments on both synthetic and real data demonstrate that our algorithms maintain the accuracy of competing methods, while running in less time, achieving at least an order of magnitude speed-up on large matrices and tensors.

Efficient Reduced-Scaling Second-Order Møller-Plesset Perturbation Theory with Cholesky-Decomposed Densities and an Attenuated Coulomb Metric.

We present a novel, highly efficient method for the computation of second-order Møller-Plesset perturbation theory (MP2) correlation energies, which uses the resolution of the identity (RI) approximation and local molecular orbitals obtained from a Cholesky decomposition of pseudodensity matrices (CDD), as in the RI-CDD-MP2 method developed previously in our group [Maurer, S. A.; Clin, L.; Ochsenfeld, C. J. Chem. Phys. 2014, 140, 224112]. In addition, we introduce an attenuated Coulomb metric and subsequently redesign the RI-CDD-MP2 method in order to exploit the resulting sparsity in the three-center integrals. Coulomb and exchange energy contributions are computed separately using specialized algorithms. A simple, yet effective integral screening protocol based on Schwarz estimates is used for the MP2 exchange energy. The Coulomb energy computation and the preceding transformations of the three-center integrals are accelerated using a modified version of the natural blocking approach [Jung, Y.; Head-Gordon, M. Phys. Chem. Chem. Phys. 2006, 8, 2831-2840]. Effective subquadratic scaling for a wide range of molecule sizes is demonstrated in test calculations in conjunction with a low prefactor. The method is shown to enable cost-efficient MP2 calculations on large molecular systems with several thousand basis functions.

The inductive blockwise Alperin weight condition for PSp2n(q) and odd primes

In this paper, using a criterion given by J. Brough and B. Späth recently, we verify the inductive blockwise Alperin weight condition for the simple groups PSp2n(q) and any odd prime ℓ not dividing q under some assumptions concerning the decomposition matrices.

On improving the efficiency of mathematical modeling of the problem of stability of construction

Algorithmic software for mathematical modeling of structural stability is considered, which is reduced to solving a partial generalized eigenvalues problem of sparse matrices, with automatic parallelization of calculations on modern parallel computers with graphics processors. Peculiarities of realization of parallel algorithms for different structures of sparse matrices are presented. The times of solving the problem of stability of composite materialsusing a three-dimensional model of "finite size fibers" on computers of different architectures are given. In mathematical modeling of physical and technical processes in many cases there is a need to solve problems of algebraic problem of eigenvalues (APVZ) with sparse matrices of large volumes. In particular, such problems arise in the analysis of the strength of structures in civil and industrial construction, aircraft construction, electric welding, etc. The solving to these problems is to determine the eigenvalues and eigenvectors of sparse matrices of different structure. The efficiency of solving these problems largely depends on the effectiveness of mathematical modeling of the problem as a whole. Continuous growth of task parameters, calculation of more complete models of objects and processes on computers require an increase in computer productivity. High-performance computing requirements are far ahead of traditional parallel computing, even with multicore processors. High-performance computing requirements are far ahead of traditional parallel computing, even with multicore processors. Today, this problem is solved by using powerful supercomputers of hybrid architecture, such as computers with multicore processors (CPUs) and graphics processors (GPUs), which combine MIMD and SIMD architectures. But the potential of high-performance computers can be used to the fullest only with algorithmic software that takes into account both the properties of the task and the features of the hybrid architecture. Complicating the architecture of modern high-performance supercomputers of hybrid architecture, which are actively used for mathematical modeling (increasing the number of computer processors and cores, different types of computer memory, different programming technologies, etc.) means a significant complication of efficient use of these resources in creating parallel algorithms and programs. here are problems with the creation of algorithmic software with automatic execution of stages of work, which are associated with the efficient use of computing resources, ways to store and process sparse matrices, analysis of the reliability of computer results. This makes it possible to significantly increase the efficiency of mathematical modeling of practical problems on modern high-performance computers, as well as free users from the problems of parallelization of complex problems. he developed algorithmic software automatically implements all stages of parallel computing and processing of sparse matrices on a hybrid computer. It was used at the Institute of Mechanics named after S.P. Tymoshenko NAS of Ukraine in modeling the strength problems of composite material. A significant improvement in the time characteristics of mathematical modeling was obtained. Problems of mathematical modeling of the properties of composite materials has an important role in designing the processes of deformation and destruction of products in various subject areas. Algorithmic software for mathematical modeling of structural stability is considered, which is reduced to solving a partial generalized problem of eigen values of sparse matrices of different structure of large orders, with automatic parallelization of calculations on modern parallel computers with graphics processors. The main methodological principles and features of implementation of parallel algorithms for different structures of sparse matrices are presented, which ensure effective implementation of multilevel parallelism of a hybrid system and reduce data exchange time during the computational process. As an example of these approaches, a hybrid algorithm of the iteration method in subspace for tape and block-diagonal matrices with a frame for computers of hybrid architecture is given. Peculiarities of data decomposition for matrices of profile structure at realization of parallel algorithms are considered. The proposed approach provides automatic determination of the required topology of the hybrid computer and the optimal amount of resources for the organization of an efficient computational process. The results of testing the developed algorithmic software for problems from the collection of the University of Florida, as well as the times of solving the problem of stability of composite materials using a three-dimensional model of "finite size fibers" on computers of different architectures. The results show a significant improvement in the time characteristics of solving problems.

Spatial–Temporal Variability of the Calculated Characteristics of the Ocean in the Arctic Zone of Russia by Using the NEMO Model with Altimetry Data Assimilation

The spatial–temporal variability of the calculated characteristics of the ocean in the Arctic zone of Russia is studied. In this study, the known hydrodynamic model of the ocean Nucleus for European Modelling of the Ocean (NEMO) is used with assimilation of observation data on the sea surface height taken from the Archiving, Validating and Interpolation Satellite Observation (AVISO) archive. We use the Generalized Kalman filter (GKF) method, developed earlier by the authors of this study, in conjunction with the method of decomposition of symmetric matrices into empirical orthogonal functions (EOF, Karhunen–Loeve decomposition). The investigations are focused mostly on the northern seas of Russia. The main characteristics of the ocean, such as the current velocity, sea surface height, and sea surface temperature are calculated with data assimilation (DA) and without DA (the control calculation). The calculation results are analyzed and their spatial–temporal variability over a time period of 14 days is studied. It is shown that the main spatial variability of characteristics after DA is in good agreement with the localization of currents in the North Atlantic and in the Arctic zone of Russia. The contribution of each of the eigenvectors and eigenvalues of the covariation matrix to the spatial–temporal variability of the calculated characteristics is shown by using the EOF analysis.

The Singular Value Decomposition over Completed Idempotent Semifields

In this paper, we provide a basic technique for Lattice Computing: an analogue of the Singular Value Decomposition for rectangular matrices over complete idempotent semifields (i-SVD). These algebras are already complete lattices and many of their instances—the complete schedule algebra or completed max-plus semifield, the tropical algebra, and the max-times algebra—are useful in a range of applications, e.g., morphological processing. We further the task of eliciting the relation between i-SVD and the extension of Formal Concept Analysis to complete idempotent semifields (K-FCA) started in a prior work. We find out that for a matrix with entries considered in a complete idempotent semifield, the Galois connection at the heart of K-FCA provides two basis of left- and right-singular vectors to choose from, for reconstructing the matrix. These are join-dense or meet-dense sets of object or attribute concepts of the concept lattice created by the connection, and they are almost surely not pairwise orthogonal. We conclude with an attempt analogue of the fundamental theorem of linear algebra that gathers all results and discuss it in the wider setting of matrix factorization.

Boolean decomposition of binary matrices using a post-nonlinear mixture approach

We introduce a novel binary matrix factorization (BMF) approach based on a post-nonlinear mixture model. Unlike the existing BMF methods, which are based on the classical matrix product, the proposed mixture model is equivalent to the Boolean matrix factorization model when the entries of the factor matrices are exactly binary. Consequently, our approach yields interpretable results in the case of overlapping sources and more accurate low-rank binary matrix approximations compared to the state-of-the-art. We propose a simple yet efficient algorithm for solving the proposed BMF problem based on multiplicative update rules. In addition, we provide for the first time in the binary data literature, a necessary and sufficient condition for the uniqueness of the Boolean matrix factorization, as well as several other uniqueness results. The interest of this new approach is illustrated in numerical simulation and on real datasets.

Convex Optimization for the Densest Subgraph and Densest Submatrix Problems

We consider the densest k-subgraph problem, which seeks to identify the k-node subgraph of a given input graph with maximum number of edges. This problem is well-known to be NP-hard, by reduction to the maximum clique problem. We propose a new convex relaxation for the densest k-subgraph problem, based on a nuclear norm relaxation of a low-rank plus sparse decomposition of the adjacency matrices of k-node subgraphs to partially address this intractability. We establish that the densest k-subgraph can be recovered with high probability from the optimal solution of this convex relaxation if the input graph is randomly sampled from a distribution of random graphs constructed to contain an especially dense k-node subgraph with high probability. Specifically, the relaxation is exact when the edges of the input graph are added independently at random, with edges within a particular k-node subgraph added with higher probability than other edges in the graph. We provide a sufficient condition on the size of this subgraph k and the expected density under which the optimal solution of the proposed relaxation recovers this k-node subgraph with high probability. Further, we propose a first-order method for solving this relaxation based on the alternating direction method of multipliers, and empirically confirm our predicted recovery thresholds using simulations involving randomly generated graphs, as well as graphs drawn from social and collaborative networks.

Simplified methods of the 3D-SVPWM for four-wire three-leg inverter

This paper introduces a novel technique for simplification of a three-dimension space vector pulse width modulation (3D-SVPWM). Conventional SVPWM calculations are based on trigonometric functions or several decomposition matrices. Thus, a very fast embedded controller is needed either for two-dimension space vector pulse width modulation (2D-SVPWM) or 3D-SVPWM. The proposed simplified 3D-SVPWM is dedicated to four-wire three-leg inverters. This technique is based on first-order equations of the curve fitting technique to reduce calculation time. Therefore, this method can be implemented with low memory storage and Central Processing Unit (CPU) capability. In addition, high accuracy can be achieved using higher-order equations. Many approaches are introduced for simplifications. The applications of the proposed simplified 3D-SVPWM method are also provided. Simulation results of the proposed simplified 3D-SVPWM and the conventional 3D-SVPWM are discussed. Finally, the results are verified using a real-time simulator based on Typhoon device (402) as Hardware-In-the-Loop. In addition, the controller of the simplified method is implemented and operated on the C2000 LAUNCHXL-F28379D as a low-cost microcontroller kit.

Partial orders based on the CS decomposition

UDC 512.5 A new decomposition for square matrices is given by using two known matrix decompositions, a new characterization of the core-EP order is obtained by using this new matrix decomposition. Also, we will use a matrix decomposition to investigate the minus, star, sharp and core partial orders in the setting of complex matrices.

Unitriangular shape of decomposition matrices of unipotent blocks

We show that the decomposition matrix of unipotent $\ell$-blocks of a finite reductive group $\mathbf{G}(\mathbb{F}_q)$ has a unitriangular shape, assuming $q$ is a power of a good prime and $\ell$ is very good for $\mathbf{G}$. This was conjectured by Geck in 1990 as part of his PhD thesis. We establish this result by constructing projective modules using a modification of generalised Gelfand--Graev characters introduced by Kawanaka. We prove that each such character has at most one unipotent constituent which occurs with multiplicity one. This establishes a 30 year old conjecture of Kawanaka.

A Unified Scalable Equivalent Formulation for Schatten Quasi-Norms

The Schatten quasi-norm is an approximation of the rank, which is tighter than the nuclear norm. However, most Schatten quasi-norm minimization (SQNM) algorithms suffer from high computational cost to compute the singular value decomposition (SVD) of large matrices at each iteration. In this paper, we prove that for any p, p1, p2>0 satisfying 1/p=1/p1+1/p2, the Schatten p-(quasi-)norm of any matrix is equivalent to minimizing the product of the Schatten p1-(quasi-)norm and Schatten p2-(quasi-)norm of its two much smaller factor matrices. Then, we present and prove the equivalence between the product and its weighted sum formulations for two cases: p1=p2 and p1≠p2. In particular, when p>1/2, there is an equivalence between the Schatten p-quasi-norm of any matrix and the Schatten 2p-norms of its two factor matrices. We further extend the theoretical results of two factor matrices to the cases of three and more factor matrices, from which we can see that for any 0<p<1, the Schatten p-quasi-norm of any matrix is the minimization of the mean of the Schatten (⌊1/p⌋+1)p-norms of ⌊1/p⌋+1 factor matrices, where ⌊1/p⌋ denotes the largest integer not exceeding 1/p.

Simulation of structural timber properties based on geographical data and stand-level forest inventory data

ABSTRACT A method for simulation of distributions of structural timber properties based on geographical data and stand-level forest inventory data is presented. The method is based on Monte Carlo simulation of randomly selected sites from volume distributions of standing timber, in our case from the Norwegian National Forest Inventory. Simulation models of density, modulus of elasticity and bending strength were estimated from data from 2369 boards of Norway spruce from 27 sites in Norway. The models were estimated as mixed models including covariate models with fixed effects of latitude, altitude, site index and stand age, random site effects and residual variances, representing the within-site variances. Correlations between the simulated properties were obtained by transformation of the random effects in the simulations. The transformations were based on Cholesky decomposition of the correlation matrices of the random site-effects and the residuals of the simulation models. The simulations provide distributions for timber from each selected site and from all selected sites within defined areas, with different limitations on geographical data and stand-level forest inventory data. Simulations of Norway spruce from Eastern Norway are presented for demonstration, comparing timber from maturity classes, altitudes and site indices.

Construction of many-body-localized models where all the eigenstates are matrix-product-states

The inverse problem of ‘eigenstates-to-Hamiltonian’ is considered for an open chain of N quantum spins in the context of many-body-localization. We first construct the simplest basis of the Hilbert space made of 2N orthonormal matrix-product-states (MPS), that will thus automatically satisfy the entanglement area-law. We then analyze the corresponding N local integrals of motions (LIOMs) that can be considered as the local building blocks of these 2N MPS, in order to construct the parent Hamiltonians that have these 2N MPS as eigenstates. Finally we study the matrix-product-operator form of the diagonal ensemble density matrix that allows to compute long-time-averaged observables of the unitary dynamics. Explicit results are given for the memory of local observables and for the entanglement properties in operator-space, via the generalized notion of Schmidt decomposition for density matrices describing mixed states.

Efficient calculation of interior scattering from cavities with partial IBC wall

The magnetic field integral equation based method of moments (MoM) with the Kirchhoff approximation can predict the interior scattering from cavities with impedance boundary condition (IBC) walls. However, if the IBC wall is a small part of the whole wall and is modified multiple times, the conventional MoM may be time-consuming due to the repeated and expensive lower–upper (LU) decompositions of impedance matrices. In this Letter, the Sherman–Morrison–Woodbury (SMW) formula is used to mitigate the problem. The authors only require to perform the LU decomposition of the cavity without IBC walls, and to efficiently obtain the solutions of the cavity with different IBC walls by reusing the LU decomposition and the SMW formula.

Application of the Non-Hermitian Singular Spectrum Analysis to the Exponential Retrieval Problem

Introduction. In practical signal processing and its many applications, researchers and engineers try to find a number of harmonics and their frequencies in a time signal contaminated by noise. In this manuscript we propose a new approach to this problem. Aim. The main goal of this work is to embed the original time series into a set of multi-dimensional information vectors and then use shift-invariance properties of the exponentials. The information vectors are cast into a new basis where the exponentials could be separated from each other. Materials and methods. We derive a stable technique based on the singular value decomposition (SVD) of lagcovariance and cross-covariance matrices consisting of covariance coefficients computed for index translated copies of an original time series. For these matrices a generalized eigenvalue problem is solved. Results. The original time series is mapped into the basis of the generalized eigenvectors and then separated into components. The phase portrait of each component is analyzed by a pattern recognition technique to distinguish between the phase portraits related to exponentials constituting the signal and the noise. A component related to the exponential has a regular structure, its phase portrait resembles a unitary circle/arc. Any commonly used method could be then used to evaluate the frequency associated with the exponential. Conclusion. Efficiency of the proposed and existing methods is compared on the set of examples, including the white Gaussian and auto-regressive model noise. One of the significant benefits of the proposed approach is a way to distinguish false and true frequency estimates by the pattern recognition. Some automatization of the pattern recognition is completed by discarding noise-related components, associated with the eigenvectors that have a modulus less than a certain threshold.

It[formula omitted]-Taylor-based square-root unscented Kalman filtering methods for state estimation in nonlinear continuous-discrete stochastic systems

This paper addresses the problem of square-rooting in the Unscented Kalman Filtering (UKF) methods rooted in the Ito^-Taylor approximation of the strong order 1.5. Since its discovery the UKF has become one of the most powerful state estimation means because of its outstanding performance in numerous stochastic systems of practical value, including continuous-discrete ones. Besides, the main shortcoming of this technique is the need for the Cholesky decomposition of covariance matrices derived in its time and measurement update steps. Such a factorization is time-consuming and highly sensitive to round-off and other errors committed in the course of computation, which can result in losing the covariance’s positivity and, hence, failing the Cholesky decomposition. The latter problem is usually overcome by means of square-root filter implementations, which propagate not the covariance itself but its square root (Cholesky factor), only. Unfortunately, negative weights arising in applications of the UKF to high-dimensional stochastic systems preclude from designing conventional square-root UKF methods. We resolve it with low-rank Cholesky factor update procedures or with hyperbolic QR transforms used for yielding J-orthogonal square roots. Our novel square-root filters are justified theoretically and examined and compared numerically to the existing UKF in a flight control scenario.

A fast method for solving a block tridiagonal quasi-Toeplitz linear system

This paper addresses the problem of solving block tridiagonal quasi-Toeplitz linear systems. Inspired by [10], we propose a more general algorithm for such systems. The algorithm is based on a block decomposition for block tridiagonal quasi-Toeplitz matrices and the Sherman–Morrison–Woodbury inversion formula. We also compare the proposed approach to the standard block LU decomposition method and the Gauss algorithm. A theoretical error analysis is also presented. All algorithms have been implemented in Matlab. Numerical experiments performed on a wide variety of test problems show the effectiveness of our algorithm in terms of efficiency, stability and robustness.

Parallel solution of saddle point systems with nested iterative solvers based on the Golub‐Kahan Bidiagonalization

SummaryThe Golub‐Kahan bidiagonalization is widely used in the singular value decomposition of rectangular matrices and has been generalized to an iterative solver for symmetric indefinite linear systems with a two‐by‐two block structure. In this work, we present a scalability study of this generalized solver as implemented in a recent release of the parallel numerical library PETSc (Portable, Extensible Toolkit for Scientific Computation). We present an improved solver performance for the two‐dimensional (2D) Stokes equations as compared to previous work. Furthermore, we investigate the performance of different parallel inner solvers in the outer Golub‐Kahan iteration for a three‐dimensional Stokes problem. The study includes parallel sparse direct solvers and multigrid methods. When increasing the number of cores for a fixed total problem size, the solver exhibits good speedups of up to 50% at the 1024 core count. For the tests in which the total problem size grows while the workload in each core stays constant, the parallel performance of the solver scales almost linearly with the increase in the core counts. In particular, the computation time increases only by about 15% when the number of cores increases from 80 to 1024 for a 2D test case.