Modified Gram-Schmidt Research Articles

Parallel QR Factorization of Block Low-rank Matrices

We present two new algorithms for Householder QR factorization of Block Low-Rank (BLR) matrices: one that performs block-column-wise QR and another that is based on tiled QR. We show how the block-column-wise algorithm exploits BLR structure to achieve arithmetic complexity of 𝒪( mn ), while the tiled BLR-QR exhibits 𝒪( mn 1.5 complexity. However, the tiled BLR-QR has finer task granularity that allows parallel task-based execution on shared memory systems. We compare the block-column-wise BLR-QR using fork-join parallelism with tiled BLR-QR using task-based parallelism. We also compare these two implementations of Householder BLR-QR with a block-column-wise Modified Gram–Schmidt (MGS) BLR-QR using fork-join parallelism and a state-of-the-art vendor-optimized dense Householder QR in Intel MKL. For a matrix of size 131k × 65k, all BLR methods are more than an order of magnitude faster than the dense QR in MKL. Our methods are also robust to ill conditioning and produce better orthogonal factors than the existing MGS-based method. On a CPU with 64 cores, our parallel tiled Householder and block-column-wise Householder algorithms show a speedup of 50 and 37 times, respectively.

FPGA Implementation of Sparsity Independent Regularized Pursuit for Fast CS Reconstruction

Sparse recovery algorithms are integral to compressed sensing (CS) as they facilitate the reconstruction of higher dimensional signals from sub-Nyquist measurements. Although Orthogonal Matching Pursuit (OMP) has been ubiquitously adopted in hardware implementations to curb the computational complexity of CS recovery, increasing sparsity levels incur higher reconstruction costs. Sparsity independent regularized pursuit (SIRP) is a recent algorithm that employs parallel index selection and regularization to curtail the number of iterations required to reconstruct the signal and thereby enhance the reconstruction speed. This paper proposes a novel reformulation of the SIRP algorithm from the hardware perspective, incorporating a cheaper regularization strategy and a modified Gram-Schmidt (MGS) based incremental QR decomposition (QRD) approach. The proposed design incorporates an iterative QRD architecture with feedback circuitry to exploit the parallelism of the triangularization step in MGS. Additionally, a fast inverse square block circumvents the need for parallel divider blocks giving considerable hardware and latency savings. The design reuses the iterative QRD block to implement the interdependent computations of the algorithm by sophisticated scheduling techniques. The proposed implementation on a Xilinx Virtex Ultrascale FPGA can recover 1024-dimensional signals from 25% measurements within <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$77~\mu \text{s}$ </tex-math></inline-formula> , which translates to a 37% reduction in processing cycles from state-of-art.

CHAOTIC COMPRESSIVE SENSING OF TV –UHF BAND IN IRAQ USING CHEBYSHEV GRAM SCHMIDT SENSING MATRIX

Cognitive radio (CR) is a promising technology for solving spectrum sacristy problem. Spectrum sensing is the main step of CR. Sensing the wideband spectrum produces more challenges. Compressive sensing (CS) is a technology used as spectrum sening in CR to solve these challenges. CS consists of three stages: sparse representation, encoding and decoding. In encoding stage sensing matrix are required, and it plays an important role for performance of CS. The design of efficient sensing matrix requires achieving low mutual coherence . In decoding stage the recovery algorithm is applied to reconstruct a sparse signal. İn this paper a new chaotic matrix is proposed based on Chebyshev map and modified gram Schmidt (MGS). The CS based proposed matrix is applied for sensing real TV signal as a PU. The proposed system is tested under two types of recovery algorithms. The performance of CS based proposed matrix is measured using recovery error (Re), mean square error (MSE), and probability of detection (Pd) and evaluated by comparing it with Gaussian, Bernoulli and chaotic matrix in the literature. The simulation results show that the proposed system has low Re and high Pd under low SNR values and has low MSE with high compression.

Modeling and optimization of ionospheric model coefficients based on adjusted spherical harmonics function

The range error caused by ionospheric delay in Global Positioning System (GPS) signals is currently the main factor that affected the positioning accuracy and navigation determination. Ionospheric modeling and optimization of its range error is a practical approach to improve the GPS positioning accuracy. The global ionospheric models are mostly unable to predict ionospheric delay corrections over the India region due to the Equatorial Ionization Anomaly (EIA)/fountain effect and lack of equatorial/low latitude GPS stations data. This paper proposes a method to facilitates regional ionospheric delay corrections based on the Adjusted Spherical Harmonic Function (ASHF) model by considering fewer coefficients into the algorithm. The ionospheric correction is driven with low-resolution harmonics coefficients with order ≤2 (≤9 coefficients), and coefficients are estimated with the Modified Gram-Schmidt (MGS) approach. The performance of the proposed model (ASHF) is compared with the Spherical Harmonic Function and Klobuchar models. Preliminary results reveal that the low-resolution ASHF model improved the ionospheric time delay corrections by 12.98% relative to the Klobuchar model. The results could be useful in the context of ionospheric time delay modeling for regional navigation satellite systems.

Direct reconstruction method for discontinuous Galerkin methods on higher-order mixed-curved meshes II. Surface integration

As the second extension of the direct reconstruction method (DRM), DRM for the surface integration of the discontinuous Galerkin (DG) weak formulation is presented on multi-dimensional high-order mixed meshes. From the previous study, DRM was successfully extended to the volume integration of the DG method, and produced a significant reduction in both the computational cost and memory overhead, particularly for high-order solution approximations on 3-D high-order meshes. DRM has the potential to reduce the computational cost and memory overhead further by treating the surface integration term consistently. To realize this, we design a linear dependency detector and formulate an adaptive orthonormalization process from the modified Gram-Schmidt (MGS) process. DRM is then applied to the restricted function space of the surface integration in an adaptive and optimal manner. The resulting methods, namely the DRM surface integration with the brute force points (BFP) and shape function points (SFP) methods, perform very well on high-order mixed-curved meshes. In addition, the adaptive MGS process is robust and accurate in extracting linear faces/edges from a given set of mixed-curved meshes. Various benchmark tests from the compressible Euler and Navier-Stokes equations verify the impact of DRM for 3-D large-scale computations on mixed-curved meshes. As an example, the DRM framework applied to the 3-D surface and volume integrations with DG-P3 approximation on P3-mesh may yield O(10)× speed-up, compared to conventional quadrature-based methods.

Novel Implementation Approach with Enhanced Memory Access Performance of MGS Algorithm for VLIW Architecture

Modified Gram–Schmidt (MGS) algorithm is one of the most-known forms of QR decomposition (QRD) algorithms. It has been used in many signal and image processing applications to solve least square problem and linear equations or to invert matrices. However, QRD is well-thought-out as a computationally expensive technique, and its sequential implementation fails to meet the requirements of many real-time applications. In this paper, we suggest a new parallel version of MGS algorithm that uses VLIW (Very Long Instruction Word) resources in an efficient way to get more performance. The presented parallel MGS is based on compact VLIW kernels that have been designed for each algorithm step taking into account architectural and algorithmic constraints. Based on instruction scheduling and software pipelining techniques, the proposed kernels exploit efficiently data, instruction and loop levels parallelism. Additionally, cache memory properties were used efficiently to enhance parallel memory access and to avoid cache misses. The robustness, accuracy and rapidity of the introduced parallel MGS implementation on VLIW enhance significantly the performance of systems under severe rea-time and low power constraints. Experimental results show great improvements over the optimized vendor QRD implementation and the state of art.

A Class of Derivative-Free CG Projection Methods for Nonsmooth Equations with an Application to the LASSO Problem

In this paper, based on a modified Gram–Schmidt (MGS) process, we propose a class of derivative-free conjugate gradient (CG) projection methods for nonsmooth equations with convex constraints. Two attractive features of the new class of methods are (1) its generated direction contains a free vector, which can be set as any vector such that the denominator of the direction does not equal to zero; (2) it adopts a new line search which can reduce its computing cost. The new class of methods includes many efficient iterative methods for the studied problem as its special cases. When the underlying mapping is monotone, we establish its global convergence and convergence rate. Finally, preliminary numerical results about the LASSO problem show that the new class of methods is promising compared to some existing ones.

An Efficient and High-Speed Implementation of QRD-MGS Algorithm for STAP Application Based on Floating Point FPGAs

Space-Time Adaptive Processing (STAP) can harness the efficacy of interference and clutter significantly. Calculations of the STAP weights involve solving linear equations which require very intensive computations. In this paper, the QR decomposition (QRD) using the modified gram-schmidt (MGS) algorithm is parameterized with vector size to create a trade-off between the hardware resources utilization and computation time. To achieve an efficient floating point structure, the proposed architecture of QRD-MGS algorithm is simulated and implemented in two modes: single-vector and multi-vector. Results show that the multi-vector method can lead to a high-performance design with higher operating frequency, lower power consumption, and less resource utilization than the single-vector method. For example, Modelism simulations show that the decomposition of a [Formula: see text] matrix with vector size of 17 takes 7.86[Formula: see text][Formula: see text]s with the maximum clock frequency of 282[Formula: see text]MHz, for implementation on the Arria10 FPGA. In real STAP applications, the matrix sizes are too large to be fit on FPGAs and the update rate of the weights are high. Therefore, this method can fit any matrix in the contemporary FPGAs with an acceptable update rate.

Efficient implementations of the modified Gram–Schmidt orthogonalization with a non-standard inner product

The modified Gram–Schmidt (MGS) orthogonalization is one of the most well-used algorithms for computing the thin QR factorization. MGS can be straightforwardly extended to a non-standard inner product with respect to a symmetric positive definite matrix A. For the thin QR factorization of an $$m \times n$$ matrix with the non-standard inner product, a naive implementation of MGS requires 2n matrix-vector multiplications (MV) with respect to A. In this paper, we propose n-MV implementations: a high accuracy (HA) type and a high performance type, of MGS. We also provide error bounds of the HA-type implementation. Numerical experiments and analysis indicate that the proposed implementations have competitive advantages over the naive implementation in terms of both computational cost and accuracy.

Comparison of practical methods for an efficient FPGA implementation of STAP

ABSTRACTThis study investigated and compared the practical methods used for the efficient Field- Programmable Gate Array (FPGA) implementation of space-time adaptive processing (STAP). The most important part of calculating the STAP weights is the QR decomposition (QRD), which can be implemented using the modified Gram-Schmidt (MGS) algorithm. The results show that the method that uses QRD with less computational burden leads to a more effective implementation. Its structure was parameterised with the vector size to create a trade-off between the hardware and performance factors. For this purpose, QRD-MGS algorithm was first modified to increase the speed, and then the STAP weight vector was calculated. The implementation results show that decreasing the vector size decreases the resource utilisation, computational burden and the consumption power. While the computation time increases slightly, the updated rate of the STAP weights is maintained. For example, the STAP weights in a system with 6 antenna arrays, 10 received pulses and 200 range samples computed in 262 µs using a vector size of 17 on the Arria10 FPGA that has a maximum of 155 µs correlates to the QRD-MGS algorithm and 107 µs correlates to the other parts. Therefore, QRD-MGS algorithm is the most important component of the calculation of the STAP weight vector, and its simplification leads to efficient implementation.

Modified Gram-Schmidt Method-Based Variable Projection Algorithm for Separable Nonlinear Models.

Separable nonlinear models are very common in various research fields, such as machine learning and system identification. The variable projection (VP) approach is efficient for the optimization of such models. In this paper, we study various VP algorithms based on different matrix decompositions. Compared with the previous method, we use the analytical expression of the Jacobian matrix instead of finite differences. This improves the efficiency of the VP algorithms. In particular, based on the modified Gram-Schmidt (MGS) method, a more robust implementation of the VP algorithm is introduced for separable nonlinear least-squares problems. In numerical experiments, we compare the performance of five different implementations of the VP algorithm. Numerical results show the efficiency and robustness of the proposed MGS method-based VP algorithm.

A Novel Architecture to Eliminate Bottlenecks in a Parallel Tiled QRD Algorithm for Future MIMO Systems

QR decomposition (QRD) is one of the performance bottlenecks in a lot of high performance wireless communication algorithms and should have the flexibility property for future multiple-input multiple-output systems. However, the existing QRD architectures only focus on several fixed dimension matrices. The parallel tiled QRD algorithm is a perfect choice to implement QRD for its flexibility and modularity property. The size of the tile is set to 2 $\times$ 2 instead of the traditional 200 $\times$ 200 or more to support flexible antenna configurations in this brief. Using a look-ahead technique and the property of unitary matrix, a novel algorithm based on a modified Gram–Schmidt (MGS) algorithm is proposed for the bottleneck operations (GEQRT and TTQRT) of the parallel tiled QRD algorithm. A corresponding hardware architecture is also designed with the proposed algorithm. The implementation results show that the hardware architecture based on the proposed algorithm achieves a 2.7× reduction in normalized processing latency, compared with the one based on the traditional MGS algorithm.

QR decomposition architecture using the iteration look‐ahead modified Gram–Schmidt algorithm

QR decomposition is extensively adopted in multiple-input–multiple-output orthogonal frequency-division multiplexing wireless communication systems, and is one of the performance bottlenecks in lots of high-performance wireless communication algorithms. To implement low processing latency QR decomposition with hardware, the authors propose a novel iterative look-ahead modified Gram–Schmidt (ILMGS) algorithm based on the traditional modified Gram–Schmidt (MGS) algorithm. They also design the corresponding triangular systolic array (TSA) architecture with the proposed ILMGS algorithm, which only needs n time slots for a n × n real matrix. For reducing the hardware overhead, they modify the TSA architecture into an iterative architecture. They also design a modified iterative architecture to further reduce the hardware overhead. The implementation results show that the normalised processing latency of the modified iterative architecture based on the proposed ILMGS algorithm is 1.36 times lower than the one based on the MGS algorithm. To the best of the authors’ knowledge, the designed architecture achieves the superior latency performance than the existing works.

Image Fusion and Evaluation of Geological Disaster Based on Remote Sensing

The investigation of geological disaster in our article locates in southern Australia, which is characterized by wide range, high relief, inaccessibility and other unfavorable factors. Multi-spectral ETM+ and SPOT 5 pan images were selected as the remote sensing data source, and Brovey transform (BT), intensity-hue-saturation (IHS), principal component analysis (PCA), high-pass filtering (HPF) and modified Gram-Schmidt (MGS) methods were used for image fusion. A comparison has been conducted between the resultant fusion images to assess the image quality both in subjective and objective evaluation. The results show that, the MGS method is the optimal image fusion method for geological disaster interpretation, and can provide abundant textural and spectral information in interpreting such geological disasters as landslide, rock fall and debris flow.

Iterative &lt;formula formulatype="inline"&gt;&lt;tex Notation="TeX"&gt;$QR$&lt;/tex&gt; &lt;/formula&gt; Decomposition Architecture Using the Modified Gram&amp;#x2013;Schmidt Algorithm for MIMO Systems

Implementation of an iterative QR decomposition (QRD) (IQRD) architecture based on the modified Gram-Schmidt (MGS) algorithm is proposed in this paper. A QRD is extensively adopted by the detection of multiple-input-multiple-output systems. In order to achieve computational efficiency with robust numerical stability, a triangular systolic array (TSA) for QRD of large-size matrices is presented. In addition, the TSA architecture can be modified into an iterative architecture that is called IQRD for reducing hardware cost. The IQRD hardware is constructed by the diagonal and the triangular process with fewer gate counts and lower power consumption than TSAQRD. For a 4 t 4 matrix, the hardware area of the proposed IQRD can reduce about 41% of the gate counts in TSAQRD. For a generic square matrix of order m IQRD, the latency required is 2m - 1 time units, which is based on the MGS algorithm. Thus, the total clock latency is only 10 m - 5 cycles.

On direct elimination methods for solving the equality constrained least squares problem

Two closely related methods for solving the least squares problem with equality constraints (LSE) are considered. The first is the direct elimination (DE) method that is implemented using Modified Gram-Schmidt (MGS) for minimizing ‖Ax − c‖2 subject to the constraints ‖Bx − d‖2 = min. The second is the MGS for solving the following weighted least squares problem where we take the limit τ → ∞ analytically. The two methods are proved to be equivalent. We provide the backward and forward round-off error analysis for the DE method via MGS, and illustrate our results with numerical experiments.

Modified Gram-Schmidt-based antenna selection for MIMO systems in correlated channels

A modified Gram-Schmidt (MGS)-based antenna selection scheme is proposed for multiple-input multiple-output (MIMO) systems in correlated channels. This MGS antenna selection strategy is developed based on an auxiliary parameter formulated by the projection component that is obtained from the GS orthogonalisation process. Consequently, the use of the MGS antenna selection algorithm is capable of achieving an improvement in system capacity and a substantial saving in system hardware of the multiple radio frequency chains associated with multiple antennas.

Stability of the MGS-like elimination method for equality constrained least squares problems

This paper proves that the weighting method via modified Gram-Schmidt(MGS) for solving the equality constrained least squares problem in the limit is equivalent to the direct elimination method via MGS(MGS-elimination method). By virtue of this equivalence, the backward and forward roundoff error analysis of the MGS-elimination method is proved. Numerical experiments are provided to verify the results.

A Useful Form of Unitary Matrix Obtained from Any Sequence of Unit 2-Norm n-Vectors

Charles Sheffield pointed out that the modified Gram–Schmidt (MGS) orthogonalization algorithm for the QR factorization of $B\!\in\!\R^{n\times k}$ is mathematically equivalent to the QR factorization applied to the matrix B augmented with a $k\times k$ matrix of zero elements on top. This is true in theory for any method of QR factorization, but for Householder's method it is true in the presence of rounding errors as well. This knowledge has been the basis for several successful but difficult rounding error analyses of algorithms which in theory produce orthogonal vectors but significantly fail to do so because of rounding errors. Here we show that the same results can be found more directly and easily without recourse to the MGS connection. It is shown that for any sequence of k unit 2-norm n-vectors there is a special $(n\!+\!k)$-square unitary matrix which we call a unitary augmentation of these vectors and that this matrix can be used in the analyses without appealing to the MGS connection. We describe the connection of this unitary matrix to Householder matrices. The new approach is applied to an earlier analysis to illustrate both the improvement in simplicity and advantages for future analyses. Some properties of this unitary matrix are derived. The main theorem on orthogonalization is then extended to cover the case of biorthogonalization.

Fast frequency-domain equalization for single-carrier V-BLAST systems over multipath channel

A low complex minimum mean-square error frequency-domain decision feedback (MMSE-FDDF) equalization algorithm is proposed in this paper for the single-carrier V-BLAST systems. Exploiting the factor that the discrete Fourier transform (DFT) is unitary, the proposed receiver can equalize the signals by the MMSE detecting to the spectrums in the frequency domain instead of the waveforms in the time domain. In order to obtain the right decisions, the detector must be able to equalize the overall spectrum with regard to each layer. This work can be performed very efficiently since the system matrix has been designed as a special block-circulant-block matrix. Similar to other V-BLAST-like systems, the detecting order has strong impact on the performance of MMSE-FDDF. Therefore, we further give a fast optimally sorting scheme for the MMSE-FDDF architecture. By using the newly constructed matrix, the coefficients computation and the sorting can be combined into one process, and then we employ the modified Gram-Schmidt (MGS) to simplify the process. The simulation results and the computational complexity analysis show that the proposed MMSE-FDDF has better tradeoff between the performance and the complexity than the existing algorithms. In addition, MMSE-FDDF can avoid the performance floor caused by the overlap-and-save technique in the severe dispersive channel.