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A Mixed Precision jacobi SVD algorithm

ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE 2025年第1期51卷 1-33页

作者： Gao, Weiguo Ma, Yuxin Shao, Meiyue Fudan Univ Sch Math Sci Shanghai Peoples R China Fudan Univ Sch Data Sci Shanghai Peoples R China Charles Univ Prague Fac Math & Phys Dept Numer Math Prague Czech Republic Fudan Univ Sch Data Sci Shanghai Peoples R China Fudan Univ MOE Key Lab Computat Phys Sci Shanghai Peoples R China

We propose a mixed precision jacobi algorithm for computing the singular value decomposition (SVD) of a dense matrix. After appropriate preconditioning, the proposed algorithm computes the SVD in a lower precision as an initial guess and then performs one-sided jacobi rotations in the working precision as iterative refinement. By carefully transforming a lower precision solution to a higher precision one, our algorithm achieves about 2x speedup on the x86-64 architecture compared to the usual one-sided jacobi SVD algorithm in LAPACK, without sacrificing the accuracy. CCS Concepts: center dot Mathematics of computing -> Computations on matrices

关键词： Singular value decomposition jacobi algorithm mixed precision algorithm high relative accuracy

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Hardware Architecture for Eigenvalues Computation using the Modified jacobi algorithm on FPGA 5

Hardware Architecture for Eigenvalues Computation using the ...

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5th IEEE International Conference on Signal Processing, Computing and Control (ISPCC)

作者： Gupta, Krishna Deep Wajid, Mohd Muzammil, Rehan Arif, Syed Javed Aligarh Muslim Univ ZHCET Dept Elect Engn Aligarh Uttar Pradesh India

ISBN: (纸本)9781728139883

The calculation of eigenvalues (EVs) is important in the domain of complex computation and processing of matrices of a higher order. For real-time applications, to get superior performance, the architecture of EV computation comes in handy. In this paper, the authors have investigated an efficient architecture for EV computation using the jacobi algorithm on the Xilinx Zed-Board FPGA evaluation platform, Artix-7 family. The iterative algorithm proves to be faster and more efficient in terms of area consumed in the FPGA architecture and also in terms of accuracy of the EV computation. The proposed architecture can take up to matrices of order 20 x 20 using the jacobi algorithm. The algorithm also works for odd order matrices with few modifications in the input matrix. The architecture was implemented on Zynq-7000 xc7z020clg484-1 FPGA and it takes around 4216 LUTs out of 53200 LUTs for matrices of dimensions 4 x 4.

关键词： FPGA Eigenvalues jacobi algorithm Parallel Iterative algorithm

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On the Convergence of a Regularized jacobi algorithm for Convex Optimization

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2018年第4期63卷 1113-1119页

作者： Banjac, Goran Margellos, Kostas Goulart, Paul J. Univ Oxford Dept Engn Sci Oxford OX1 3PJ England

In this paper, we consider the regularized version of the jacobi algorithm, a block coordinate descent method for convex optimization with an objective function consisting of the sum of a differentiable function and a block-separable function. Under certain regularity assumptions on the objective function, this algorithm has been shown to satisfy the so-called sufficient decrease condition, and consequently, to converge in objective function value. In this paper, we revisit the convergence analysis of the regularized jacobi algorithm and show that it also converges in iterates under very mild conditions on the objective function. Moreover, we establish conditions under which the algorithm achieves a linear convergence rate.

关键词： Block coordinate descent methods decentralized optimization jacobi algorithm linear convergence

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FPGA, GPU, and CPU implementations of jacobi algorithm for eigenanalysis

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JOURNAL OF PARALLEL AND DISTRIBUTED COMPUTING 2016年 96卷 172-180页

作者： Torun, Mustafa U. Yilmaz, Onur Akansu, Ali N. New Jersey Inst Technol Dept Elect & Comp Engn Newark NJ 07102 USA

Parallel implementations of jacobi algorithm for eigenanalysis of a matrix on most commonly used high performance computing (HPC) devices such as central processing unit (CPU), graphics processing unit (GPU), and field-programmable gate array (FPGA) are discussed in this paper. Their performances are investigated and compared. It is shown that CPU, even with multi-threaded implementation, is not a feasible option for large dense matrices. For the GPU implementation, performance impact of the global memory access patterns on the GPU board and the memory coalescing are emphasized. Three memory access methods are proposed. It is shown that one of them achieves 81.6% computational performance improvement over the traditional GPU methods, and it runs 68.5 times faster than a single-threaded CPU for a dense symmetric square matrix of size 1,024. Furthermore, FPGA implementation is presented and its performance running on chips from two major manufacturers are reported. A comparison of GPU and FPGA implementations is quantified and ranked. It is reported that FPGA design delivers the best performance for such a task while GPU is a strong competitor requiring less development effort with superior scalability. We predict that emerging big data applications will benefit from real-time and high performance computing implementations of eigenanalysis for information inference and signal analytics in the future. (C) 2016 Elsevier Inc. All rights reserved.

关键词： Eigenanalysis Principal component analysis Karhunen-Loeve transform jacobi algorithm Chess tournament CORDIC Memory coalescing FPGA GPU CPU

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A jacobi Decomposition algorithm for Distributed Convex Optimization in Distributed Model Predictive Control

A Jacobi Decomposition Algorithm for Distributed Convex Opti...

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20th World Congress of the International-Federation-of-Automatic-Control (IFAC)

作者： Doan, Minh Dang Diehl, Moritz Keviczky, Tames De Schutter, Bart Univ Freiberg Dept Microsyst Engn Syst Control & Optimizat Lab Freiburg Germany Univ Freiberg Dept Math Freiburg Germany Delft Univ Technol Delft Ctr Syst & Control Delft Netherlands

In this paper we introduce an iterative distributed jacobi algorithm for solving convex optimization problems, which is motivated by distributed model predictive control (MPC) for linear time-invariant systems. Starting from a given feasible initial guess, the algorithm iteratively improves the value of the cost function with guaranteed feasible solutions at every iteration step, and is thus suitable for MPC applications in which hard constraints are important. The proposed iterative approach involves solving local optimization problems consisting of only few subsystems, depending on the flexible choice of decomposition and the sparsity structure of the couplings. This makes our approach more applicable to situations where the number of subsystems is large, the coupling is sparse, and local communication is available. We also provide a method for checking a posteriori centralized optimality of the converging solution, using comparison between Lagrange multipliers of the local problems. Furthermore, a theoretical result on convergence to optimality for a particular distributed setting is also provided. (C) 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

关键词： distributed optimization jacobi algorithm distributed model predictive control cooperative optimization algorithm

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A jacobi Decomposition algorithm for Distributed Convex Optimization in Distributed Model Predictive Control

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IFAC-PapersOnLine 2017年第1期50卷 4905-4911页

作者： Dang Doan M. Diehl M. Keviczky T. De Schutter B. Systems Control and Optimization Laboratory Department of Microsystems Engineering and Department of Mathematics University of Freiburg Germany Delft Center for Systems and Control Delft University of Technology Netherlands

关键词： cooperative optimization algorithm distributed model predictive control distributed optimization jacobi algorithm

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Revisiting the (block) jacobi subspace rotation method for the symmetric eigenvalue problem

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NUMERICAL algorithmS 2023年第1期92卷 917-944页

作者： Saad, Yousef Univ Minnesota Comp Sci & Engn Minneapolis MN 55455 USA

The paper revisits the topic of block-jacobi algorithms for the symmetric eigenvalue problem by proposing a few alternative versions. The main advantage of a block jacobi method is that it is built entirely from computations with small dense matrices. The proposed mehod is based on a sequence of subspace rotations whose determination requires to solve small Riccati-like correction equation. The paper discusses theoretical and algorithmic aspects of the algorithm, and illustrates its behavior on a few simple examples.

关键词： Symmetric eigenvalue problem jacobi algorithm Riccati equations

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Any-Time Feasible Coordination for Multivehicle Systems

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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 2022年第5期30卷 1807-1820页

作者： Kneissl, Maximilian Molin, Adam Esen, Hasan Hirche, Sandra DENSO Automot Deutschland GmbH Dept Corp Res & Dev D-85386 Eching Germany Tech Univ Munich Inst Informat Oriented Control D-80290 Munich Germany

This article proposes a distributed control methodology for the coordination of multiple connected and automated vehicles (CAVs). Each vehicle computes its local trajectory based on a model predictive control (MPC) law and communicates the result with other relevant vehicles. In an iterative negotiation process, planned trajectories are optimized within a sampling time step. Inspired by the jacobi over-relaxation (JOR) algorithm, we develop a distributed jacobi over-relaxation algorithm (DJOR) for vehicle coordination. The modified algorithm exploits the structure of the distributed problem setting in which coupling occurs only in a bilateral way. Besides being able to guarantee any-time feasibility that implies collision-freeness, the algorithm scales well, unlike the standard JOR algorithm. The DJOR algorithm allows for significantly less conservatism in the choice of the update weightings. As a result, much faster convergence rates can be expected. Furthermore, the collision avoidance guarantee is extended for unforeseen scenarios such as emergency braking. Using an exact penalty function formulation ensures that the distributed optimization problem remains feasible even in previously unforeseen cases. Numerical simulations of an intersection crossing scenario illustrate the presented approach and show its benefits in comparison with standard traffic rules and a centralized computation.

关键词： Trajectory Optimization Safety jacobian matrices Heuristic algorithms Scalability Roads Distributed model predictive control (MPC) environmental uncertainty feasibility jacobi algorithm multivehicle systems scalability

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Hardware Acceleration of MUSIC algorithm for Sparse Arrays and Uniform Linear Arrays

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I-REGULAR PAPERS 2022年第7期69卷 2941-2954页

作者： Li, Zeying Wang, Weijiang Jiang, Rongkun Ren, Shiwei Wang, Xiaohua Xue, Chengbo Beijing Inst Technol BIT Sch Integrated Circuits & Elect Beijing 100081 Peoples R China Beijing Inst Technol Chongqing Ctr Microelect & Microsyst Chongqing 401332 Peoples R China Beijing Inst Technol Chongqing Innovat Ctr Chongqing 401135 Peoples R China

Multiple Signal Classification (MUSIC) is a high-performance Direction of Arrival (DOA) estimation algorithm, which has been widely used. The algorithm needs to calculate the covariance matrix, eigenvalue decomposition and spectral peak search. In the paper, the hardware structure of the existing jacobi algorithm for Hermitian matrices is proposed. On this basis, a novel hardware acceleration of the MUSIC algorithm for sparse arrays and uniform linear arrays is proposed, and the sparse array is a nested array. There are two designs, Design 1 supports 1 similar to 10 nested array elements or 1 similar to 32 uniform linear array elements, distinguishes 1 similar to 32 sources, configures snapshots 1 similar to 2048, and the maximum number of iterations and iteration accuracy of the complex jacobi algorithm. Design 2 only needs 101.8 mu s to complete a DOA estimation when the number of array elements is 8, the number of sources is 1, and the snapshots is 128. In more detail, the Root Mean Squared Error (RMSE) of both can reach 0.03 degrees. The logic resources on the Zynq-7000 development board are 14,761 and 28,305 Look-Up Tables (LUTs), respectively.

关键词： MUSIC algorithm FPGA hardware implementation sparse arrays uniform linear arrays DOA estimation jacobi algorithm

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An SSD-Based Accelerator for Singular Value Decomposition Recommendation algorithm on Edge

An SSD-Based Accelerator for Singular Value Decomposition Re...

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IEEE High Performance Extreme Computing Virtual Conference (HPEC)

作者： Wu, Wei Zhao, Letian Wu, Qizhe Wang, Xiaotian Tian, Teng Jin, Xi Univ Sci & Technol China State Key Lab Particle Detect & Elect Hefei Peoples R China Univ Sci & Technol China Inst Microelect Dept Phys Hefei Peoples R China

ISBN: (数字)9781665497862

ISBN: (纸本)9781665497862

Recommender system (RS) is widely used in social networks, computational advertising, video platforms and many other Internet applications. Most RSs are based on the cloud-to-edge framework. Recommended item lists are computed in the cloud server and then transmitted to the edge device. Network bandwidth and latency between cloud server and edge may cause the delays in recommendation. Edge computing could help obtain user's real-time preferences and thus improve the performance of recommendation. However, the increasing complexity of recommendation algorithms and data scales cause challenges to real-time recommendation on edge. To solve these problems, in this paper, we mainly focus on the jacobi-based singular value decomposition (SVD) algorithm because of its high parallel processing potential and cost effective NVM-storage. We propose an SSD-based accelerator for the one-sided jacobi transformation algorithm. We implement a hardware prototype on a real Xilinx FPGA development board. Experimental results show that the proposed SVD engine can achieve 3.4x speedup to 5.8x speedup compared with software SVD solvers such as MATLAB running on a high-performance CPU.

关键词： Singular Value Decomposition FPGA jacobi algorithm Hardware Acceleration

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