Orthonormalization Procedure Research Articles

Semi‐implicit method of high‐index saddle dynamics and application to construct solution landscape

AbstractWe analyze the semi‐implicit scheme of high‐index saddle dynamics, which provides a powerful numerical method for finding the any‐index saddle points and constructing the solution landscape. Compared with the explicit schemes of saddle dynamics, the semi‐implicit discretization relaxes the step size and accelerates the convergence, but the corresponding numerical analysis encounters new difficulties compared to the explicit scheme. Specifically, the orthonormal property of the eigenvectors at each time step could not be fully employed due to the semi‐implicit treatment, and computations of the eigenvectors are coupled with the orthonormalization procedure, which further complicates the numerical analysis. We address these issues to prove error estimates of the semi‐implicit scheme via, for example, technical splittings and multi‐variable circulating induction procedure. We further analyze the convergence rate of the generalized minimum residual solver for solving the semi‐implicit system. Extensive numerical experiments are carried out to substantiate the efficiency and accuracy of the semi‐implicit scheme in constructing solution landscapes of complex systems.

Preconditioned weighted full orothogonalization method for solving singular linear systems from PageRank problems

AbstractThe PageRank model, which was first proposed by Google for its web search engine application, has since become a popular computational tool in a wide range of scientific fields, including chemistry, bioinformatics, neuroscience, bibliometrics, social networks, and others. PageRank calculations necessitate the use of fast computational techniques with low algorithmic and memory complexity. In recent years, much attention has been paid to Krylov subspace algorithms for solving difficult PageRank linear systems, such as those with large damping parameters close to one. In this article, we examine the full orthogonalization method (FOM). We present a convergence study of the method that extends and clarifies part of the conclusions reached in Zhang et al. (J Comput Appl Math. 2016; 296:397–409.). Furthermore, we demonstrate that FOM is breakdown free when solving singular PageRank linear systems with index one and we investigate the effect of using weighted inner‐products instead of conventional inner‐products in the orthonormalization procedure on FOM convergence. Finally, we develop a shifted polynomial preconditioner that takes advantage of the special structure of the PageRank linear system and has a good ability to cluster most of the eigenvalues, making it a good choice for an iterative method like FOM or GMRES. Numerical experiments are presented to support the theoretical findings and to evaluate the performance of the new weighted preconditioned FOM PageRank solver in comparison to other established solvers for this class of problem, including conventional stationary methods, hybrid combinations of stationary and Krylov subspace methods, and multi‐step splitting strategies.

Observability Analysis and Restoration for Distribution Grids: A Robust Greedy Numerical Algorithm

This paper presents a robust and efficient algorithm for analyzing and restoring the observability of distribution grids. A linearized load flow model is used to characterize three-and multi-phase distribution grids under balanced and unbalanced conditions. Assuming that metering devices are installed to measure nodal magnitudes at some buses, the rank of the matrix associated with the non-metered magnitudes is determined by using a greedy strategy combined with a Gram-Schmidt orthonormalization procedure. Unlike other numerical-based techniques, the proposed algorithm avoids common errors caused by denominators close to zero. As an additional salient feature, a mathematically sound stopping criterion is provided to guarantee the robustness of the proposed algorithm. Furthermore, for unobservable grids, the algorithm identifies the linearly dependent rows in the aforementioned matrix, which correspond to the buses that should be metered to restore grid observability. The effective performance of the proposed algorithm is illustrated using several medium-and low-voltage benchmarks with up to 906 buses.

A Matrix-Valued Inner Product for Matrix-Valued Signals and Matrix-Valued Lattices

A matrix-valued inner product was proposed before to construct orthonormal matrix-valued wavelets for matrix-valued signals. It introduces a weaker orthogonality for matrix-valued signals than the orthogonality of all components in a matrix that is commonly used in orthogonal multiwavelet constructions. With the weaker orthogonality, it is easier to construct orthonormal matrix-valued wavelets. In this paper, we re-study the matrix-valued inner product more from the inner product viewpoint that is more fundamental and propose a new but equivalent norm for matrix-valued signals. We show that although it is not scalar-valued, it maintains most of the scalarvalued inner product properties. We introduce a new linear independence concept for matrix-valued signals and present some related properties. We then present the Gram-Schmidt orthonormalization procedure for a set of linearly independent matrix-valued signals. Finally we define matrix-valued lattices, where the newly introduced Gram-Schmidt orthogonalization might be applied.

Enrichment of the nonconforming virtual element method with singular functions

We construct a nonconforming virtual element method (ncVEM) based on approximation spaces that are enriched with special singular functions. This enriched ncVEM is tailored for the approximation of solutions to elliptic problems, which have singularities due to the geometry of the domain. Differently from the traditional extended Galerkin method approach, based on the enrichment of local spaces with singular functions, no partition of unity is employed. Rather, the design of the method hinges upon the special structure of the nonconforming virtual element spaces. We discuss the theoretical analysis of the method and support it with several numerical experiments. We also present an orthonormalization procedure that drastically trims the ill-conditioning of the final system.

A Greedy Algorithm for Observability Analysis

This letter presents a numerical algorithm for observability analysis. The proposed algorithm is based on a greedy strategy combined with a Gram-Schmidt orthonormalization procedure of the rows of the measurement Jacobian matrix. The algorithm outputs the system is observable with the available measurements, and if not, it looks for the best candidate measurements to restore observability. Also, an efficient calculation of the null space of the Jacobian matrix can be obtained. The algorithm is an improvement with respect to other existing algorithms that calculate a factorization of the Jacobian matrix. Throughout the process it is not necessary to invert matrices, nor select pivots and the errors caused when dividing by values close to zero are avoided. Four examples are given to illustrate the effectiveness of the proposed method.

Parallelizable Algorithms for Optimization Problems with Orthogonality Constraints

To construct a parallel approach for solving optimization problems with orthogonality constraints is usually regarded as an extremely difficult mission, due to the low scalability of the orthonormalization procedure. However, such demand is particularly huge in some application areas such as materials computation. In this paper, we propose a proximal linearized augmented Lagrangian algorithm (PLAM) for solving optimization problems with orthogonality constraints. Unlike the classical augmented Lagrangian methods, in our algorithm, the prime variables are updated by minimizing a proximal linearized approximation of the augmented Lagrangian function, meanwhile the dual variables are updated by a closed-form expression which holds at any first-order stationary point. The orthonormalization procedure is only invoked once at the last step of the above mentioned algorithm if high-precision feasibility is needed. Consequently, the main parts of the proposed algorithm can be parallelized naturally. We establish global subsequence convergence, worst-case complexity and local convergence rate for PLAM under some mild assumptions. To reduce the sensitivity of the penalty parameter, we put forward a modification of PLAM, which is called parallelizable column-wise block minimization of PLAM (PCAL). Numerical experiments in serial illustrate that the novel updating rule for the Lagrangian multipliers significantly accelerates the convergence of PLAM and makes it comparable with the existent feasible solvers for optimization problems with orthogonality constraints, and the performance of PCAL does not highly rely on the choice of the penalty parameter. Numerical experiments under parallel environment demonstrate that PCAL attains good performance and high scalability in solving discretized Kohn-Sham total energy minimization problems.

Uncertainty quantification for nonlinear difference equations with dependent random inputs via a stochastic Galerkin projection technique

Discrete stochastic systems model discrete response data on some phenomenon with inherent uncertainty. The main goal of uncertainty quantification is to derive the probabilistic features of the stochastic system. This paper deals with theoretical and computational aspects of uncertainty quantification for nonlinear difference equations with dependent random inputs. When the random inputs are independent random variables, a generalized Polynomial Chaos (gPC) approach has been usually used to computationally quantify the uncertainty of stochastic systems. In the gPC technique, the stochastic Galerkin projections are done onto linear spans of orthogonal polynomials from the Askey–Wiener scheme or from Gram–Schmidt orthonormalization procedures. In this regard, recent results have established the algebraic or exponential convergence of these Galerkin projections to the solution process. In this paper, as the random inputs of the difference equation may be dependent, we perform Galerkin projections directly onto linear spans of canonical polynomials. The main contribution of this paper is to study the spectral convergence of these Galerkin projections for the solution process of general random difference equations. Spectral convergence is important to derive the main statistics of the response process at a cheap computational expense. In this regard, the numerical experiments bring to light the theoretical discussion of the paper.

Symmetrized density matrix renormalization group algorithm for low-lying excited states of conjugated carbon systems: Application to 1,12-benzoperylene and polychrysene

The symmetry adapted density matrix renormalization group (SDMRG) technique has been an efficient method for studying low-lying eigenstates in one- and quasi-one-dimensional electronic systems. However, the SDMRG method had bottlenecks involving the construction of linearly independent symmetry adapted basis states as the symmetry matrices in the DMRG basis were not sparse. We have developed a modified algorithm to overcome this bottleneck. The new method incorporates end-to-end interchange symmetry $({C}_{2})$, electron-hole symmetry $(J)$, and parity or spin-flip symmetry $(P)$ in these calculations. The one-to-one correspondence between direct-product basis states in the DMRG Hilbert space for these symmetry operations renders the symmetry matrices in the new basis with maximum sparseness, just one nonzero matrix element per row. Using methods similar to those employed in the exact diagonalization technique for Pariser-Parr-Pople (PPP) models, developed in the 1980s, it is possible to construct orthogonal SDMRG basis states while bypassing the slow step of the Gram-Schmidt orthonormalization procedure. The method together with the PPP model which incorporates long-range electronic correlations is employed to study the correlated excited-state spectra of 1,12-benzoperylene and a narrow mixed graphene nanoribbon with a chrysene molecule as the building unit, comprising both zigzag and cove-edge structures.

A novel method of constructing compactly supported orthogonal scaling functions from splines

A novel construction of compactly supported orthogonal scaling functions and wavelets with spline functions is presented in this paper. Let M_{n} be the center B-spline of order n, except for the case of order one, we know M_{n} is not orthogonal. But by the formula of orthonormalization procedure, we can construct an orthogonal scaling function corresponding to M_{n}. However, unlike M_{n} itself, this scaling function no longer has compact support. To induce the orthogonality while keeping the compact support of M_{n}, we put forward a simple, yet efficient construction method that uses the formula of orthonormalization procedure and the weighted average method to construct the two-scale symbol of some compactly supported orthogonal scaling functions.

Downlink MU-MIMO With THP Combined With Pre- and Post-processing and Selection of the Processing Vectors for Maximization of Per-Stream SNR

In this paper, we consider a downlink multiuser multiple-input–multiple-output (MU-MIMO) system with multiple antennas at the transmitter and multiple antennas at each user, where the transmitter can send one or more data streams to each user. We propose a non-iterative method by combining Tomlinson–Harashima precoding (THP) with pre- and post-processing and selecting processing vectors based on the maximization of instantaneous signal-to-noise ratio (SNR) of the data stream at the input of the detector of each user. The postprocessing vectors for all users are found to be eigenvectors corresponding to the maximum eigenvalue of a certain matrix involving the channel matrix of the user. The transmitter computes the vectors of the linear processing matrix through an orthonormalization procedure in a single step. The feedback matrix at the transmitter is then obtained from the effective channel matrix and the linear processing matrix. We express the instantaneous SNR of all data streams in terms of eigenvalues of a Wishart matrix, obtained from the channel matrix of the user, and find the diversity order for each data stream. Considering multiple scenarios, we find the outage probability of the instantaneous SNR for all data streams and the cumulative distribution function (cdf) of the sum-rate capacity for all users using the proposed method and compare the results with those of recently proposed methods that provide a closed-form solution and with that of block diagonalization, channel inversion, and THP in scenarios where they are applicable.

Euclidean Distance Between Haar Orthogonal and Gaussian Matrices

In this work we study a version of the general question of how well a Haar distributed orthogonal matrix can be approximated by a random gaussian matrix. Here, we consider a gaussian random matrix $Y_n$ of order $n$ and apply to it the Gram-Schmidt orthonormalization procedure by columns to obtain a Haar distributed orthogonal matrix $U_n$. If $F_i^m$ denotes the vector formed by the first $m$-coordinates of the $i$th row of $Y_n-\sqrt{n}U_n$ and $\alpha=\frac{m}{n}$, our main result shows that the euclidean norm of $F_i^m$ converges exponentially fast to $\sqrt{ \left(2-\frac{4}{3} \frac{(1-(1 -\alpha)^{3/2})}{\alpha}\right)m}$, up to negligible terms. To show the extent of this result, we use it to study the convergence of the supremum norm $\epsilon_n(m)=\sup_{1\leq i \leq n, 1\leq j \leq m} |y_{i,j}- \sqrt{n}u_{i,j}|$ and we find a coupling that improves by a factor $\sqrt{2}$ the recently proved best known upper bound of $\epsilon_n(m)$. Applications of our results to Quantum Information Theory are also explained.

Error analysis of analytical coarse alignment formulations for stationary SINS

This paper presents a comprehensive error analysis of coarse alignment formulations for stationary strapdown inertial navigation systems (SINS). Analytical expressions for the residual normality, orthogonality, alignment, and Euler angle errors are systematically derived, allowing us to establish comparisons in terms of rapidity, accuracy, and autonomy requirements. For the purpose of the error analysis, geographic latitude and local gravity acceleration uncertainties are considered, in addition to the inertial sensor uncertainties. As the main contribution of this paper, an improved coarse alignment method is proposed, which is based on orthonormality constraints existing between the Euler angles and the attitude matrix. In contrast to the traditional methods, the proposed formulation does not imply normality and orthogonality errors, besides producing time-independent alignment and Euler angle errors. The latter is seen to be particularly interesting for situations wherein, due to time constraints, posterior fine alignment and orthonormalization procedures cannot be implemented. The superiority of the proposed formulation is validated through simulated and experimental tests, regardless of the SINS initial orientation. The accuracy degradation produced by each alignment formulation in the navigation stage is used as the performance index.

A New Class of Vector Padé Approximants in the Asymptotic Numerical Method: Application in Nonlinear 2D Elasticity

The Asymptotic Numerical Method (ANM) is a family of algorithms for path following problems, where each step is based on the computation of truncated vector series [1]. The Vector Padé approximants were introduced in the ANM to improve the domain of validity of vector series and to reduce the number of steps needed to obtain the entire solution path [1,2]. In this paper and in the framework of the ANM, we define and build a new type of Vector Padé approximant from a truncated vector series by extending the definition of the Padé approximant of a scalar series without any orthonormalization procedure. By this way, we define a new class of Vector Padé approximants which can be used to extend the domain of validity in the ANM algorithms. There is a connection between this type of Vector Padé approximant and Vector Padé type approximant introduced in [3, 4]. We show also that the Vector Padé approximants introduced in the previous works [1,2], are special cases of this class. Applications in 2D nonlinear elasticity are presented.

A hybrid variant of the BiCOR method for a nonsymmetric linear system with a complex spectrum

A biconjugate A -orthogonal residual method based on the biconjugate A -orthonormalization procedure has been proposed and denoted as BiCOR by Jing, Huang, et al. [Y.-F. Jing, T.-Z. Huang, Y. Zhang, L. Li, G.-H. Cheng, Z.-G. Ren, Y. Duan, T. Sogabe, and B. Carpentieri, Lanczos-type variants of the COCR method for complex nonsymmetric linear systems, J. Comput. Phys. 228 (2009) 6376–6394]. Here, a hybrid variant of BiCOR is presented to avoid generating the transpose of the system matrix and to optimize the convergence of a nonsymmetric linear system with a complex spectrum.

AERIAL DEGREES OF FREEDOM OF PARASITIC ARRAYS FOR SINGLE RF FRONT-END MIMO TRANSCEIVERS

The beamspace domain of parasitic antenna arrays is explored in this paper, providing the aerial degrees of freedom available for use in Multiple Input-Multiple Output (MIMO) systems. The beamspace representation allows for the design of an alternative MIMO architecture based on single radio-frequency (RF) chains, and facilitates the inclusion of MIMO transceivers in devices with strict size limitations. A three dimensional orthogonal expansion is performed on the beamspace domain providing the basis patterns used for mapping of the transmitted symbols and for sampling at the receiver. The expansion is based on the Gram-Schmidt orthonormalization procedure and can be generalized for any parasitic antenna array. The multiplexing capability of ESPAR antennas is presented as a means for supporting future performance demanding communication systems. Performance evaluation results are illustrated in detail.

Numerical Gram–Schmidt orthonormalization

A numerical Gram–Schmidt orthonormalization procedure is presented for constructing an orthonormal basis function set from a non-orthonormal set, when the number of basis functions is large. This method will provide a pedagogical illustration of the Gram–Schmidt procedure and can be presented in classes on numerical methods or computational physics.

Multiresolution Analysis Adapted to Irregularly Spaced Data

This paper investigates the mathematical background of multiresolution analysis in the specific context where the signal is represented by irregularly sampled data at known locations. The study is related to the construction of nested piecewise polynomial multiresolution spaces represented by their corresponding orthonormal bases. Using simple spline basis orthonormalization procedures involves the construction of a large family of orthonormal spline scaling bases defined on consecutive bounded intervals. However, if no more additional conditions than those coming from multiresolution are imposed on each bounded interval, the orthonormal basis is represented by a set of discontinuous scaling functions. The spline wavelet basis also has the same problem. Moreover, the dimension of the corresponding wavelet basis increases with the spline degree. An appropriate orthonormalization procedure of the basic spline space basis, whatever the degree of the spline, allows us to (i) provide continuous scaling and wavelet functions, (ii) reduce the number of wavelets to only one, and (iii) reduce the complexity of the filter bank. Examples of the multiresolution implementations illustrate that the main important features of the traditional multiresolution are also satisfied.

EXPERIMENTS WITH LANCZOS BICONJUGATE A-ORTHONORMALIZATION METHODS FOR MOM DISCRETIZATIONS OF MAXWELL'S EQUATIONS

In this paper we consider a novel class of Krylov projection methods computed from the Lanczos biconjugate A- Orthonormalization procedure for the solution of dense complex non-Hermitian linear systems arising from the Method of Moments discretization of Maxwell's equations. We report on experiments on a set of model problems representative of realistic radar-cross section calculations to show their competitiveness with other popular Krylov solvers, especially when memory is a concern. The results presented in this study will contribute to assess the potential of iterative Krylov methods for solving electromagnetic scattering problems from large structures enriching the database of this technology.

An invariant analytic orthonormalization procedure with applications

We apply the orthonormalization procedure previously introduced by Bagarello and Triolo [J. Math. Phys. 48, 043505 (2007)] and adopted in connection with coherent states to Gabor frames and other examples. For instance, for Gabor frames, we show how to construct g(x)∊L2(R) in such a way the functions gṉ(x)=eian1xg(x+an2), ṉ∊Z2 and a some positive real number, are mutually orthogonal. We discuss in some detail the role of the lattice naturally associated with the procedure in this analysis.