A hierarchically blocked Jacobi SVD algorithm for single and multiple graphics processing units

We present a hierarchically blocked one-sided Jacobi algorithm for the singular value decomposition (SVD), targeting both single and multiple graphics processing units (GPUs). The blocking structure reflects the levels of GPU's memory hierarchy. The algorithm may outperform MAGMA's dgesvd, while retaining high relative accuracy. To this end, we developed a family of parallel pivot strategies on GPU's shared address space, but applicable also to inter-GPU communication. Unlike common hybrid approaches, our algorithm in a single GPU setting needs a CPU for the controlling purposes only, while utilizing GPU's resources to the fullest extent permitted by the hardware. When required by the problem size, the algorithm, in principle, scales to an arbitrary number of GPU nodes. The scalability is demonstrated by more than twofold speedup for sufficiently large matrices on a Tesla S2050 system with four GPUs vs. a single Fermi card.Comment: Accepted for publication in SIAM Journal on Scientific Computin

Novel Modifications of Parallel Jacobi Algorithms

We describe two main classes of one-sided trigonometric and hyperbolic Jacobi-type algorithms for computing eigenvalues and eigenvectors of Hermitian matrices. These types of algorithms exhibit significant advantages over many other eigenvalue algorithms. If the matrices permit, both types of algorithms compute the eigenvalues and eigenvectors with high relative accuracy. We present novel parallelization techniques for both trigonometric and hyperbolic classes of algorithms, as well as some new ideas on how pivoting in each cycle of the algorithm can improve the speed of the parallel one-sided algorithms. These parallelization approaches are applicable to both distributed-memory and shared-memory machines. The numerical testing performed indicates that the hyperbolic algorithms may be superior to the trigonometric ones, although, in theory, the latter seem more natural.Comment: Accepted for publication in Numerical Algorithm

The Impact of Precision Tuning on Embedded Systems Performance: A Case Study on Field-Oriented Control

Field Oriented Control (FOC) is an industry-standard strategy for controlling induction motors and other kinds of AC-based motors. This control scheme has a very high arithmetic intensity when implemented digitally - in particular it requires the use of trigonometric functions. This requirement contrasts with the necessity of increasing the control step frequency when required, and the minimization of power consumption in applications where conserving battery life is paramount such as drones. However, it also makes FOC well suited for optimization using precision tuning techniques. Therefore, we exploit the state-of-the-art FixM methodology to optimize a miniapp simulating a typical FOC application by applying precision tuning of trigonometric functions. The FixM approach itself was extended in order to implement additional algorithm choices to enable a trade-off between execution time and code size. With the application of FixM on the miniapp, we achieved a speedup up to 278%, at a cost of an error in the output less than 0.1%

GPU implementation of block transforms

Traditionally, intensive floating-point computational ability of Graphics Processing Units (GPUs) has been mainly limited for rendering and visualization application by architecture and programming model. However, with increasing programmability and architecture progress, GPUs inherent massively parallel computational ability have become an essential part of today\u27s mainstream general purpose (non-graphical) high performance computing system. It has been widely reported that adapted GPU-based algorithms outperform significantly their CPU counterpart. The focus of the thesis is to utilize NVIDIA CUDA GPUs to implement orthogonal transforms such as signal dependent Karhunen-Loeve Transform and signal independent Discrete Cosine Transform. GPU architecture and programming model are examined. Mathematical preliminaries of orthogonal transform, eigen-analysis and algorithms are re-visited. Due to highly parallel structure, GPUs are well suited to such computation. Further, the thesis examines multiple implementations schemes and configuration, measurement of performance is provided. A real time processing display application frame is developed to visually exhibit GPU compute capability