Modified Gram-Schmidt Algorithm Research Articles

Data-driven sensitivity indices for models with dependent inputs using polynomial chaos expansions

Uncertainties exist in both physics-based and data-driven models. Variance-based sensitivity analysis characterizes how the variance of a model output is propagated from the model inputs. The Sobol index is one of the most widely used sensitivity indices for models with independent inputs. For models with dependent inputs, different approaches have been explored to obtain sensitivity indices in the literature. Typical approaches are based on procedures of transforming the dependent inputs into independent inputs. However, such transformation requires additional information about the inputs, such as the dependency structure or the conditional probability density functions. In this paper, data-driven sensitivity indices are proposed for models with dependent inputs. We first construct ordered partitions of linearly independent polynomials of the inputs. The modified Gram-Schmidt algorithm is then applied to the ordered partitions to generate orthogonal polynomials with respect to the empirical measure based on observed data of model inputs and outputs. Using the polynomial chaos expansion with the orthogonal polynomials, we obtain the proposed data-driven sensitivity indices. The sensitivity indices provide intuitive interpretations of how the dependent inputs affect the variance of the output without a priori knowledge on the dependence structure of the inputs. Four numerical examples are used to validate the proposed approach.

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.

Moving force identification based on modified preconditioned conjugate gradient method

This paper develops a modified preconditioned conjugate gradient (M-PCG) method for moving force identification (MFI) by improving the conjugate gradient (CG) and preconditioned conjugate gradient (PCG) methods with a modified Gram-Schmidt algorithm. The method aims to obtain more accurate and more efficient identification results from the responses of bridge deck caused by vehicles passing by, which are known to be sensitive to ill-posed problems that exist in the inverse problem. A simply supported beam model with biaxial time-varying forces is used to generate numerical simulations with various analysis scenarios to assess the effectiveness of the method. Evaluation results show that regularization matrix L and number of iterations j are very important influence factors to identification accuracy and noise immunity of M-PCG. Compared with the conventional counterpart SVD embedded in the time domain method (TDM) and the standard form of CG, the M-PCG with proper regularization matrix has many advantages such as better adaptability and more robust to ill-posed problems. More importantly, it is shown that the average optimal numbers of iterations of M-PCG can be reduced by more than 70% compared with PCG and this apparently makes M-PCG a preferred choice for field MFI applications.

A Comparison of MVDR and LCMV Beamformers’ Floating Point Implementations on FPGAs

Adaptive beamformers are designed with the aim of detection of noise and intentional destructive interference and then removing them from the desired signal. This is done by placing high attenuation in the direction of the destructive signal in the radiation pattern of antenna arrays without attenuating the signal from a known direction. Minimum variance distortionless response and linearly constrained minimum variance are among such algorithms in mobile communications. A short explanation of the theory and weights formulae of these beamformers will be given. Two designs for weights calculations in Simulink with the added library of DSP-Builder tool from Altera will be presented. Quadrature Rectangular decomposition with Modified Gram-Schmidt algorithm is used instead of the direct matrix inversion. All calculations are done in single floating point mode for the required high accuracy. Modelsim has been used for hardware simulation and measuring the required clock cycles. The target FPGA is Aria10 from Altera which has floating point DSP blocks for high-performance computations. Hardware resources usage, power consumption, maximum clock frequency and update rate for different matrix sizes of these two beamformers are compared and discussed. The results are also compared with some other relevant reported works.

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.

Iterative <formula formulatype="inline"><tex Notation="TeX">$QR$</tex> </formula> Decomposition Architecture Using the Modified Gram–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.

Analysis and Architecture Design of a Downlink $M$-Modification MC-CDMA System Using the Tomlinson–Harashima Precoding Technique

This paper investigates the feasibility of applying Tomlinson-Harashima precoding (THP) to multicarrier code-division multiple-access (MC-CDMA) system downlinks, in which multiple-access interference and possible complexity in the mobile terminal (MT) are major burdens. A THP based on the minimum mean-square error (MMSE) criterion is also presented, making it possible to realize a low-complexity receiver at the MT. The hardware architecture of the MMSE THP with modified QR decomposition at the transmitter is presented, along with the required word length analyzed. Pipeline and parallel schemes are adopted to reduce the execution time. For a generic square matrix of order for RQ decomposition, the required latency is time units, which is based on the modified Gram-Schmidt algorithm.

Quantitative Structure−Activity Relationship Modeling of Juvenile Hormone Mimetic Compounds for Culex Pipiens Larvae, with a Discussion of Descriptor-Thinning Methods

Quantitative structure-activity relationship (QSAR) modelers often encounter the problem of multicollinearity owing to the availability of large numbers of computable molecular descriptors. Sparsity of the variables while using descriptors such as atom pairs increases the complexity. Three different predictor-thinning methods, namely, a modified Gram-Schmidt algorithm, a marginal soft thresholding algorithm, and LASSO (least absolute shrinkage and selection operator), were utilized to reduce the number of descriptors prior to developing linear models. Juvenile hormone (JH) activity of 304 compounds on Culex pipiens larvae was taken as the model data set, and predictor trimming of a large number of diverse descriptors comprising 268 global molecular descriptors (topostructural, topochemical, and geometrical), 13 quantum chemical descriptors, and 915 atom pairs (substructural counts) was applied prior to linear regression by the ridge regression method. The data set (N = 304) was split into five calibration data sets of random samples of sizes 60/110/160/210/260, and the remaining 244/194/144/94/44 compounds were used for validations. LASSO was not found to be a very effective method in handling a large set of descriptors because the number of predictors retained could not exceed the number of observations. The results indicated that the modified Gram-Schmidt algorithm could be used to trim the number of predictors in the global molecular descriptor set where collinearity of the descriptors was the major concern. On the contrary, the soft thresholding approach was found to be an effective tool in subset selection from a diverse set of descriptors having both sparsity and multicollinearity, as in the case of the combined set of atom pairs and global molecular descriptors. The final model developed after variable selection was dominated more by atom pairs, which indicated the important structural moieties that affect JH activity of the compounds. The success of the method reiterates the fact that QSAR or quantitative structure-property relationship (QSPR) models can be developed for a diverse set of compounds using properly parametrized and diverse sets of descriptors, of course, with the selection of the appropriate statistical tools.

When modified Gram-Schmidt generates a well-conditioned set of vectors

In this paper, we show why the modified Gram-Schmidt algorithm generates a well-conditioned set of vectors. This result holds under the assumption that the initial matrix is not 'too ill-conditioned' in a way that is quantified. As a consequence we show that if two iterations of the algorithm are performed, the resulting algorithm produces a matrix whose columns are orthogonal up to machine precision. Finally, we illustrate through a numerical experiment the sharpness of our result.

The prediction of EEG signals using a feedback-structured adaptive rational function filter.

In this article, we present a feedback-structured adaptive rational function filter based on a recursive modified Gram-Schmidt algorithm and apply it to the prediction of an EEG signal that has nonlinear and nonstationary characteristics. For the evaluation of the prediction performance, the proposed filter is compared with other methods, where a single-step prediction and a multi-step prediction are considered for a short-term prediction, and the prediction performance is assessed in normalized mean square error. The experimental results show that the proposed filter shows better performance than other methods considered for the short-term prediction of EEG signals.

Self-scaling fast rotations for stiff and equality-constrained linear least squares problems

We present algorithms which apply self-scaling fast plane rotations to the QR decomposition for stiff least squares problems. We show that both fast and standard Givens rotation-based algorithms produce accurate results, regardless of row sorting and even with extremely large weights, when equality-constrained least squares problems are solved by the weighting method. Numerical test results show that the Householder QR decomposition algorithm is sensitive to row sorting and produces less accurate results when the weights are large, and that the modified Gram-Schmidt algorithm is less sensitive to row sorting. This makes the fast plane rotation a method of choice for the QR decomposition of stiff matrices, since it is also competitive in computational complexity. Based on the above results, an efficient algorithm is also derived for the application where the least squares solutions are required for various constrained matrices for each fixed data matrix.

VLSI array to speed up the computation of TV ghost canceler parameters

One of the most well-known adaptive algorithms that adjust conveniently the parameters of the TV ghost canceler is the least mean square (LMS) algorithm. The convergence rate of this algorithm is slow. In this paper a parallel processing technique to speed up the computation of the least square (LS) algorithm is studied. The technique is based on mapping the recursive modified Gram-Schmidt algorithm into two levels of linear systolic arrays. The arrays structure as well as the computation time are described. The performance of the system in terms of computational saving and the array utilization are evaluated. The system is shown to be composed of identical modules which makes it suitable for VLSI technology implementation. The great saving in computation will qualify the system for real-time applications.

On the modified Gram-Schmidt algorithm for weighted and constrained linear least squares problems

A framework and an algorithm for using modified Gram-Schmidt for constrained and weighted linear least squares problems is presented. It is shown that a direct implementation of a weighted modified Gram-Schmidt algorithm is unstable for heavily weighted problems. It is shown that, in most cases it is possible to get a stable algorithm by a simple modification free from any extra computational costs. In particular, it is not necessary to perform reorthogonalization. Solving the weighted and constrained linear least squares problem with the presented weighted modified Gram-Schmidt algorithm is seen to be numerically equivalent to an algorithm based on a weighted Householder-likeQR factorization applied to a slightly larger problem. This equivalence is used to explain the instability of the weighted modified Gram-Schmidt algorithm. If orthogonality, with respect to a weighted inner product, of the columns inQ is important then reorthogonalization can be used. One way of performing such reorthogonalization is described. Computational tests are given to show the main features of the algorithm.

A recursive modified Gram-Schmidt algorithm based adaptive beamformer

In this paper, we present an adaptive beamformer based on recursive modified Gram-Schmidt (RMGS) algorithm. The beamformer has a highly modular structure, exhibits good numerical properties and is especially suitable for fixed point arithmetic and VLSI implementation. The error feedback form of RMGS algorithm improves the numerical accuracy of the conventional RMGS algorithm. The spatial smoothing scheme has also been implemented in time recursive form using RMGS algorithm to combat coherent jammers. Computer simulations confirm that the proposed beamformer has a superior interference rejection capability over existing adaptive beamformers.

On best least square solutions of two-point boundary value problems via modified Gram-Schmidt process

The best least square solution of the two-point boundary value problem is constructed via the modified Gram-Schmidt algorithm. As a result of this work one has a choice of effective methods for finding solution of the two-point boundary value problems in the non-invertible case.

Givens rotation based least squares lattice and related algorithms

The author presents a general and systematic approach for deriving new LS (least squares) estimation algorithms that are based solely on Givens rotations. In particular, this approach is used to derive efficient Givens-rotation-based LS lattice algorithms-the Givens-lattice algorithms. By exploiting the relationship between the Givens algorithms and the recursive modified Gram-Schmidt algorithm, it is shown that the time and order update of any order-recursive LS estimation algorithm can be realized by employing only Givens rotations. Applying this general conclusion to LS estimation of time-series signals results in the Givens-lattice algorithms. Two Givens-lattice algorithms, one with square roots and the other without, are presented. It is shown that the Givens-lattice algorithms are computationally more efficient than the fast QR algorithm of Cioffi (1987). The derivation of other Givens rotation-based LS estimation algorithms and their systolic array implementations are discussed. >

Parallel QR factorization by householder and modified Gram-Schmidt algorithms

In this paper, the parallel implementation of two algorithms for forming a QR factorization of a matrix is studied. We propose parallel algorithms for the modified Gram-Schmidt and the Householder algorithms on message passing systems in which the matrix is distributed by blocks or rows. The models that predict performance of the algorithms are validated by experimental results on several parallel machines.

Iterative algorithms for Gram-Schmidt orthogonalization

The algorithms that are treated in this paper are based on the classical and the modified Gram-Schmidt algorithms. It is shown that Gram-Schmidt orthogonalization for constructing aQR factorization should be carried out iteratively to obtain a matrixQ that is orthogonal in almost full working precision. In the formulation of the algorithms, the parts that express manipulations with matrices or vectors are clearly identified to enable an optimal implementation of the algorithms on parallel and/or vector machines. An extensive error analysis is presented. It shows, for instance, that the iterative classical algorithm is not inferior to the iterative modified algorithm when full precision ofQ is required. Experiments are reported to support the outcomes of the analysis.

A recursive modified Gram-Schmidt algorithm for least- squares estimation

This paper presents a recursive form of the modified Gram-Schmidt algorithm (RMGS). This new recursive least-squares (RLS) estimation algorithm has a computational complexity similar to the conventional RLS algorithm, but is more robust to roundoff errors and has a highly modular structure, suitable for VLSI implementation. Its properties and features are discussed and compared to other LS estimation algorithms.