Matrix-vector Product Operations Research Articles

A Low-Complexity Beam Selection Algorithm for Two-Dimensional Beamspace Massive MIMO

To mitigate the huge investment in radio-frequency devices and associated high power consumption of modern base stations, the beamspace multiple-input multiple-output (B-MIMO) has become a promising solution at an expense of acceptable performance degradation. Facing the increasingly high data throughput demand, two-dimensional (2D) massive B-MIMO systems with beams steering over both elevation and azimuth angles have been developed recently. However, with such a large number of beams available for wireless communication, effective beam selection for multiple users in three-dimensional (3D) space remains technically challenging. In this letter, low-complexity suboptimal beam selection schemes are investigated to overcome this difficulty. By avoiding massive and repetitive matrix-vector product operations in the beam selection procedure through specifically designed linear algebraic calculation strategy, a lower complexity is achieved. Numerical simulations and theoretical analyses demonstrate the proposed methods have around <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K^{2}$ </tex-math></inline-formula> times speedup compared to state-of-the-art beam selection approaches, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> is the number of users.

On the Implementation of Lattice-Based Identification Schemes

Identification is one of the important concerns of information security, that is widely used in our daily e-systems to approve authorised users. With the advent of quantum computers, development of quantum secure identification schemes is essential. In this paper, we give the implementation details of quantum secure Kawachi’s and Cayrel’s identification schemes performed in JavaScript. The hardness of these schemes is based on lattice-based problem SIS in post-quantum cryptography, which requires matrix-vector product operations for its execution. It’s important that for efficient implementation choosing an algorithm with low complexity needs more careful. Therefore, in identification schemes chosen for this study, we use algorithms specific to those schemes’ parameter properties. Then, we carry out matrix by sparse vector and sparse matrix by vector product operations.We provide experimental results of both standard and property-specific algorithms’ execution with their comparison. According to the experimental results, we receive improvements in the specific implementations.

An Efficient Preconditioner Based on Adaptive Grouping Technique for Multiscale Problems

It is well known that the matrix condition number of discretized formulation for analyzing multiscale objects is very large, because the multiscale objects include the interaction between discrete unknowns in different sizes. The traditional efficient preconditioners for analysis of homogeneous objects are not efficient anymore or cannot be directly extended for the solution of multiscale objects. A good quality compressed block decomposition preconditioner is constructed to speed up the convergence rate of Krylov iterations in this communication by using the adaptive grouping technique. Meanwhile, the multilevel mymargin QR algorithm is utilized to accelerate the matrix-vector product operations. A series of numerical results show that this novel preconditioner is very effective in electromagnetic scattering calculation and can reduce simulation time significantly.

Trainable Projected Gradient Detector for Massive Overloaded MIMO Channels: Data-Driven Tuning Approach

This paper presents a deep learning-aided iterative detection algorithm for massive overloaded multiple-input multiple-output (MIMO) systems where the number of transmit antennas $n$ is larger than that of receive antennas $m$. Since the proposed algorithm is based on the projected gradient descent method with trainable parameters, it is named the trainable projected gradient-detector (TPG-detector). The trainable internal parameters, such as the step-size parameter, can be optimized with standard deep learning techniques, i.e., the back propagation and stochastic gradient descent algorithms. This approach is referred to as data-driven tuning, and ensures fast convergence during parameter estimation in the proposed scheme. The TPG-detector mainly consists of matrix-vector product operations whose computational cost is proportional to $m n$ for each iteration. In addition, the number of trainable parameters in the TPG-detector is independent of the number of antennas. These features of the TPG-detector result in a fast and stable training process and reasonable scalability for large systems. Numerical simulations show that the proposed detector achieves a comparable detection performance to those of existing algorithms for massive overloaded MIMO channels, e.g., the state-of-the-art IW-SOAV detector, with a lower computation cost.

Compressed representation of matrix decomposition algorithm-singular value decomposition for full-wave analysis microstrip problems

In order to efficiently solve large dense complex linear systems arising from electric field integral equations (EFIE) of electromagnetic problems, matrix decomposition algorithm-singular value decomposition (MDA-SVD) is used to accelerate the matrix-vector product (MVP) operations and decrease memory usage. Based on the symmetry of the impedance matrix resulting from the discretization of the EFIE, we introduce a compressed representation of MDA-SVD in this paper. We obtain a sparse representation of the far-field interaction parts of impedance matrix and perform a fast MVP operation. Numerical experiments demonstrate that the compressed representation of MDA-SVD can reduce both the MVP time and memory usage by around 50% with similar accuracy.

Efficient recursive–iterative solution for EM scattering problems

An efficient recursive–iterative solution is proposed for electromagnetic (EM) scattering problems. The scheme involves dividing the impedance matrix of the method of moments into strong and weak interaction parts. The induced scattered current can be expanded in terms of the Taylor series. Then it can be solved recursively by the equations of the strong interaction part with multi-right-vectors that are related to the weak interaction part. The multilevel fast multipole method is used to accelerate the matrix–vector product operations of both strong and weak interaction parts. Moreover, the sparse approximate inverse preconditioner is applied to speed up the iterative convergence for equations of the strong interaction part. Numerical results confirm the accuracy and efficiency of the proposed method.

An Improved Fast Iterative Shrinkage Thresholding Algorithm for Image Deblurring

An improved fast iterative shrinkage thresholding algorithm (IFISTA) for image deblurring is proposed. The IFISTA algorithm uses a positive definite weighting matrix in the gradient function of the minimization problem of the known fast iterative shrinkage thresholding (FISTA) image restoration algorithm. A convergence analysis of the IFISTA algorithm shows that due to the weighting matrix, the IFISTA algorithm has an improved convergence rate and improved restoration capability of the unknown image over that of the FISTA algorithm. The weighting matrix is predetermined and fixed, and hence, like the FISTA algorithm, the IFISTA algorithm requires only one matrix vector product operation in each iteration. As a result, the computational burden per iteration of the IFISTA algorithm remains the same as in the FISTA algorithm. Numerical examples are presented that demonstrate the improved performance of the IFISTA algorithm over that of the FISTA and iterative shrinkage thresholding (ISTA) algorithms in terms...

An IE‐ODDM Scheme Combined With Efficient Direct Solver for 3D Scattering Problems

ABSTRACTAn improved scheme of integral equation‐based overlapped domain decomposition method combined with an efficient direct solver based on hierarchical LU (H‐LU) decomposition is proposed for the three‐dimensional (3D) scattering problems. In this scheme, the H‐LU direct method is utilized to solve the self‐impedance matrix equation in each extended subdomain; simultaneously, the multilevel fast multipole algorithm is applied to complete the matrix vector product operation of mutual‐impedance matrix between different subdomains. In the H‐LU operation, a recompressed adaptive cross approximation technique is used to generate the low‐rank approximations of the far‐field matrices. After integrating all the techniques, the proposed scheme is very efficient especially for 3D electromagnetic scattering problems with multiple incident fields such as the computation of the monostatic radar cross section. Some examples are provided to demonstrate its accuracy and efficiency. © 2013 Wiley Periodicals, Inc. Microwave Opt Technol Lett 55:2027–2033, 2013

The Cost of Continuity: Performance of Iterative Solvers on Isogeometric Finite Elements

In this paper we study how the use of a more continuous set of basis functions affects the cost of solving systems of linear equations resulting from a discretized Galerkin weak form. Specifically, we compare performance of linear solvers when discretizing using $C^0$ B-splines, which span traditional finite element spaces, and $C^{p-1}$ B-splines, which represent maximum continuity. We provide theoretical estimates for the increase in cost of the matrix-vector product as well as for the construction and application of black-box preconditioners. We accompany these estimates with numerical results and study their sensitivity to various grid parameters such as element size $h$ and polynomial order of approximation $p$ in addition to the aforementioned continuity of the basis. Finally, we present timing results for a range of preconditioning options for the Laplace problem. We conclude that the matrix-vector product operation is at most $\left.{33p^2}\middle/{8}\right.$ times more expensive for the more continuous space, although for moderately low $p$, this number is significantly reduced. Moreover, if static condensation is not employed, this number further reduces to at most a value of 8, even for high $p$. Preconditioning options can be up to $p^3$ times more expensive to set up, although this difference significantly decreases for some popular preconditioners such as incomplete LU factorization.

An improved multilevel simple sparse method with adaptive cross approximation for scattering from target above lossy half space

AbstractThe adaptive cross approximation (ACA) algorithm has been used to reduce memory requirements and matrix‐vector product (MVP) time since it was proposed to solve electric field integral equation of electromagnetic problems. In this letter, we firstly introduce an improved multilevel simple sparse method (MLSSM). Based on the improved MLSSM, the impedance matrix resulting from ACA algorithm can be represented very sparsely. In the sparse representation, the far‐field interaction parts of impedance matrix can be represented by multiplication of three sparse matrices in recursive manner. Moreover, an efficient MVP operation is implemented by using the sparse representation. Numerical experiments demonstrate that our improved MLSSM with ACA algorithm outperforms original ACA algorithm in terms of memory requirement of far‐field interactions and the time per MVP operation. © 2012 Wiley Periodicals, Inc. Microwave Opt Technol Lett 54:573–577, 2012; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.26606

An Improved Implementation of Preconditioned Conjugate Gradient Method on GPU

An improved implementation of the Preconditioned Conjugate Gradient method on GPU using CUDA (Compute Unified Device Architecture) is proposed. It aims to solving the Poisson equation arising in liquid animation with high efficiency. We consider the features of the linear system obtained from the Poisson equation and propose an optimization method to solve it. First, a novel storage format called mDIA (modified diagonal storage format) is presented to improve the efficiency of the Sparse Matrix-Vector product (SpMV) operation. Second, a parallel Jacobi iterative method is proposed when using the Incomplete Cholesky preconditioner to explore inherent parallelism. Third, CUDA streams are also introduced to overlap computations among separate streams. The proposed optimization technique is embedded into our GPU based PCG algorithm. Results on Geforce G100 show that our SpMV kernel yields an improvement of nearly 100% for large sparse matrix with more than 30, 0000 rows. Also, a speedup of more than 7 is obtained for PCG method, making the real-time physics engine possible.

GPU-accelerated fem solver for three dimensional electromagnetic analysis

A new Graphics Processing Unit (GPU) parallelization strategy is proposed to accelerate sparse finite element computation for three dimensional electromagnetic analysis. The parallelization strategy is employed based on a new compression format called sliced ELL Four (sliced ELL-F). The sliced ELL-F format-based parallelization strategy is designed for hastening many addition, dot product, and Sparse Matrix Vector Product (SMVP) operations in the Conjugate Gradient Norm (CGN) calculation of finite element equations. The new implementation of SMVP on GPUs is evaluated. The proposed strategy executed on a GPU can efficiently solve sparse finite element equations, especially when the equations are huge sparse (size of most rows in a coefficient matrix is less than 8). Numerical results show the sliced ELL-F format-based parallelization strategy can reach significant speedups compared to Compressed Sparse Row (CSR) format.

Fast iterative solution of integral equation with matrix decomposition algorithm and multilevel simple sparse method

In order to efficiently solve large dense complex linear systems arising from the electric field integral equations formulation of electromagnetic problems, matrix decomposition algorithm–singular value decomposition (MDA–SVD) is used to accelerate the matrix–vector product (MVP) operation. Based on the symmetry of the impedance matrix, a modified multilevel simple sparse method (M-MLSSM) is proposed. Combining MDA and M-MLSSM, the authors obtain a sparse representation of the impedance matrix and construct a fast MVP operation. A perfectly electrically conducting sphere and the series-fed microstrip antenna array are simulated to demonstrate the validity of the proposed method. Moreover, numerical experiments demonstrate that combined MDA and M-MLSSM can accelerate the MVP operation and reduce memory requirements by at least half an order of magnitude compared to MDA–SVD.

A modified complex shifted preconditioner combined with sparse approximate inversion preconditioner for electromagnetic scattering

AbstractTo efficiently solve large, dense, and complex linear systems arising from electric field integral equations formulation of electromagnetic scattering problems, the multilevel fast multipole algorithm is used to accelerate the matrix–vector product operations. This article presents a modified complex shifted preconditioner (MCSP), which is combined with sparse approximate inversion (SAI) preconditioner can greatly improve the convergence of restart generalized minimal residual iterative method for large, dense, complex linear systems. Numerical experiments demonstrate that the MCSP combined with SAI preconditioner is very effective and can reduce the iteration number and the computational time significantly. © 2010 Wiley Periodicals, Inc. Microwave Opt Technol Lett 53:55–58, 2011; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.25651

A Novel High-Capacity Electromagnetic Compression Technique Based on a Direct Matrix Solution

Electromagnetic solvers based on integral equations in conjunction with the method of moments or the partial element equivalent circuit method (PEEC) proved to be popular because of their efficiency and accuracy. There is one serious drawback of the integral equation approach: it often leads to a linear system involving a full matrix. Many efficient approaches have been proposed to overcome this, largely based on compressing the matrix-vector product operation and using an iterative solver. Iterative EM solvers, however, suffer from slow convergence, which does not have a totally reliable method to address; further, large multiple right-hand sides significantly increase the solving time. In this paper, we present a novel method to compress low rank sub-block matrixes into sparse matrix to be used with a direct sparse matrix solver to obtain an efficient high-capacity electromagnetic solver based on an integral equation formulation. The full-rank system matrix is represented in a hierarchical matrix format that has its sub-matrixes compressed with numerically controllable accuracy; it is then analytically converted to a sparse matrix which is further solved by a direct sparse matrix solver. Analytically this method results in O(N (log N) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) complexity for computing the inverse of a hierarchical matrix presented in Fig. 2 where N is the number of unknowns.

QUANTUM DYNAMICS ON MASSIVELY PARALLEL COMPUTERS: EFFICIENT NUMERICAL IMPLEMENTATION FOR PRECONDITIONED LINEAR SOLVERS AND EIGENSOLVERS

The eigenvalue/eigenvector and linear solve problems arising in computational quantum dynamics applications (e.g. rovibrational spectroscopy, reaction cross-sections, etc.) often involve large sparse matrices that exhibit a certain block structure. In such cases, specialized iterative methods that employ optimal separable basis (OSB) preconditioners (derived from a block Jacobi diagonalization procedure) have been found to be very efficient, vis-à-vis reducing the required CPU effort on serial computing platforms. Recently,1,2 a parallel implementation was introduced, based on a nonstandard domain decomposition scheme. Near-perfect parallel scalability was observed for the OSB preconditioner construction routines up to hundreds of nodes; however, the fundamental matrix–vector product operation itself was found not to scale well, in general. In addition, the number of nodes was selectively chosen, so as to ensure perfect load balancing. In this paper, two essential improvements are discussed: (1) new algorithm for the matrix–vector product operation with greatly improved parallel scalability and (2) generalization for arbitrary number of nodes and basis sizes. These improvements render the resultant parallel quantum dynamics codes suitable for robust application to a wide range of real molecular problems, running on massively parallel computing architectures.

SSOR PRECONDITIONED INNER-OUTER FLEXIBLE GMRES METHOD FOR MLFMM ANALYSIS OF SCATTERING OF OPEN OBJECTS

To efficiently solve large dense complex linear system arising from electric field integral equations (EFIE) formulation of electromagnetic scattering problems, the multilevel fast multipole method (MLFMM) is used to accelerate the matrix-vector product operations. The inner-outer flexible generalized minimum residual method (FGMRES) is combined with the symmetric successive over- relaxation (SSOR) preconditioner based on the near-part matrix of the EFIE in the inner iteration of FGMRES to speed up the convergence rate of iterative methods. Numerical experiments with a few electromagnetic scattering problems for open structures are given to demonstrate the efficiency of the proposed method.

Enhanced GMRES method combined with MLFMA for solving electromagnetic wave scattering problems

AbstractFor efficiently solving large dense complex linear systems that arise in the electric field integral equation (EFIE) formulation of electromagnetic wave scattering problems, the multilevel fast multipole algorithm (MLFMA) is used to speed up the matrix vector product operations, and the sparse approximate inverse (SAI) preconditioning technique is employed to accelerate the convergence rate of the generalized minimal residual (GMRES) iterative method. We show that the convergence rate can be greatly improved by augmenting to the GMRES method a few eigenvectors associated with the smallest eigenvalues of the preconditioned system. Numerical experiments indicate that this new variant GMRES method is very effective with the MLFMA and can reduce both the iteration number and the computational time significantly. © 2008 Wiley Periodicals, Inc. Microwave Opt Technol Lett 50: 1433–1439, 2008; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.23385

SCHWARZ-KRYLOV SUBSPACE METHOD FOR MLFMM ANALYSIS OF ELECTROMAGNETIC WAVE SCATTERING PROBLEMS

In this paper, the high-order hierarchical basis functions are used for solving electromagnetic wave scattering problems. The multilevel fast multipole method (MLFMM) is applied to accelerate the matrix-vector product operation and the Schwarz method is employed to speed up the convergence rate of the Krylov subspace iterative methods. The efficiency of the proposed approach is studied on several numerical model problems and the comparison with conventional Kryloviterativ e methods is made. Numerical results demonstrate that the combination of the Schwarz method and the Krylovsubspace iterative method is very effective with MLFMM and can reduce the overall simulation time significantly.

A Spectral Multigrid Method Combined With MLFMM for Solving Electromagnetic Wave Scattering Problems

A new spectral multigrid method (SMG) combined with the multilevel fast multipole method (MLFMM) is proposed for solving electromagnetic wave scattering problems. The MLFMM is used to speed up the matrix-vector product operations and the SMG is employed to accelerate the convergence rate of the Krylov iteration. Unlike traditional algebraic multigrid methods (AMG), the spectral multigrid method is an algebraic two-grid cycle built on a preconditioned Krylov iterative method that is used as the smoother, and the grid transfer operators are defined using the spectral information of the preconditioned matrix. Numerical experiments indicate that this class of multigrid method is very effective with the MLFMM and can reduce both the iteration number and the overall simulation time significantly.