A modified Gram–Schmidt algorithm with iterative orthogonalization and column pivoting

AbstractIterative orthogonalization is aimed to ensure small deviation from orthogonality in the Gram–Schmidt process. Former applications of this technique are restricted to classical Gram–Schmidt (CGS) and column-oriented modified Gram–Schmidt (MGS). The major aim of this paper is to explain how iterative orthogonalization is incorporated into row-oriented MGS. The interest that we have in a row-oriented iterative MGS comes from the observation that this method is capable of performing column pivoting. The use of column pivoting delays the deteriorating effects of rounding errors and helps to handle rank-deficient least-squares problems.A second modification proposed in this paper considers the use of Gram–Schmidt QR factorization for solving linear least-squares problems. The standard solution method is based on one orthogonalization of the r.h.s. vector b against the columns of Q. The outcome of this process is the residual vector, r∗, and the solution vector, x∗. The modified scheme is a natural extension of the standard solution method that allows it to apply iterative orthogonalization. This feature ensures accurate computation of small residuals and helps in cases when Q has some deviation from orthogonality

An overview of block Gram-Schmidt methods and their stability properties

Block Gram-Schmidt algorithms serve as essential kernels in many scientific computing applications, but for many commonly used variants, a rigorous treatment of their stability properties remains open. This survey provides a comprehensive categorization of block Gram-Schmidt algorithms, particularly those used in Krylov subspace methods to build orthonormal bases one block vector at a time. All known stability results are assembled, and new results are summarized or conjectured for important communication-reducing variants. Additionally, new block versions of low-synchronization variants are derived, and their efficacy and stability are demonstrated for a wide range of challenging examples. Low-synchronization variants appear remarkably stable for s-step-like matrices built with Newton polynomials, pointing towards a new stable and efficient backbone for Krylov subspace methods. Numerical examples are computed with a versatile MATLAB package hosted at https://github.com/katlund/BlockStab, and scripts for reproducing all results in the paper are provided. Block Gram-Schmidt implementations in popular software packages are discussed, along with a number of open problems. An appendix containing all algorithms type-set in a uniform fashion is provided.Comment: 42 pages, 5 tables, 17 figures, 20 algorithm

An overview of optimal and sub-optimal detection techniques for a non orthogonal spectrally efficient FDM

Spectrally Efficient non orthogonal Frequency Division Multiplexing (SEFDM) Systems occupy less bandwidth than equivalent orthogonal FDM (OFDM). However, enhanced spectral efficiency comes at the expense of an increased complexity in the signal detection. In this work, we present an overview of different detection techniques that trade the error performance optimality for the signal recovery computational effort. Linear detection methods like Zero Forcing (ZF) and Minimum Mean Squared Error (MMSE) offer fixed complexity but suffer from a significant degradation of the Bit Error Rate (BER). On the other hand optimal receivers like Sphere Decoders (SD) achieve the optimal solution in terms of error performance. Notwithstanding, their applicability is severely constrained by the SEFDM signal dimension, the frequency separation between the carriers as well as the noise level in the system

Signal detection for 3GPP LTE downlink: algorithm and implementation.

In this paper, we investigate an efficient signal detection algorithm, which combines lattice reduction (LR) and list decoding (LD) techniques for the 3rd generation long term evolution (LTE) downlink systems. The resulting detector, called LRLD based detector, is carried out within the framework of successive interference cancellation (SIC), which takes full advantages of the reliable LR detection. We then extend our studies to the implementation possibility of the LRLD based detector and provide reference for the possible real silicon implementation. Simulation results show that the proposed detector provides a near maximum likelihood (ML) performance with a significantly reduced complexity

Accurate and efficient evaluation of the a posteriori error estimator in the reduced basis method

The reduced basis method is a model reduction technique yielding substantial savings of computational time when a solution to a parametrized equation has to be computed for many values of the parameter. Certification of the approximation is possible by means of an a posteriori error bound. Under appropriate assumptions, this error bound is computed with an algorithm of complexity independent of the size of the full problem. In practice, the evaluation of the error bound can become very sensitive to round-off errors. We propose herein an explanation of this fact. A first remedy has been proposed in [F. Casenave, Accurate \textit{a posteriori} error evaluation in the reduced basis method. \textit{C. R. Math. Acad. Sci. Paris} \textbf{350} (2012) 539--542.]. Herein, we improve this remedy by proposing a new approximation of the error bound using the Empirical Interpolation Method (EIM). This method achieves higher levels of accuracy and requires potentially less precomputations than the usual formula. A version of the EIM stabilized with respect to round-off errors is also derived. The method is illustrated on a simple one-dimensional diffusion problem and a three-dimensional acoustic scattering problem solved by a boundary element method.Comment: 26 pages, 10 figures. ESAIM: Mathematical Modelling and Numerical Analysis, 201

A Novel Parallel Algorithm Based on the Gram-Schmidt Method for Tridiagonal Linear Systems of Equations

This paper introduces a new parallel algorithm based on the Gram-Schmidt orthogonalization method. This parallel algorithm can find almost exact solutions of tridiagonal linear systems of equations in an efficient way. The system of equations is partitioned proportional to number of processors, and each partition is solved by a processor with a minimum request from the other partitions' data. The considerable reduction in data communication between processors causes interesting speedup. The relationships between partitions approximately disappear if some columns are switched. Hence, the speed of computation increases, and the computational cost decreases. Consequently, obtained results show that the suggested algorithm is considerably scalable. In addition, this method of partitioning can significantly decrease the computational cost on a single processor and make it possible to solve greater systems of equations. To evaluate the performance of the parallel algorithm, speedup and efficiency are presented. The results reveal that the proposed algorithm is practical and efficient