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In this paper, we leverage the intrinsic data- parallelism of the band matrix-matrix product to accelerate this operation on Graphics Processing Units (GPUs). In par- ticular, we propose a Level-3 BLAS style algorithm to tackle the band matrix-matrix product and implement two GPU-based versions that off-load the most expensive computations —i.e., general dense matrix-matrix multiplication, triangular matrix- matrix multiplication and matrix addition— to the hardware accelerator. Results collected using GPUs for the two most recent generations of NVIDIA ("Fermi" and "Kepler") and a complete set of benchmark cases (which differ in the matrix dimensions and bandwidth) show that the GPU-enabled implementations deliver a notable reduction of the execution time.

La simulación y control de fenómenos que aparecen en microelectrónica, micro-mecánica, electromagnetismo, dinámica de fluidos y en general en muchos procesos industriales, constituye un problema difícil de resolver, debido principalmente al elevado costo computacional de los algoritmos para este propósito. Gran parte de los modelos matemáticos que describen estos fenómenos poseen dimensión grande; por ejemplo, la modelización de microprocesadores desemboca en un sistema dinámico a gran escala que no puede ser resuelto con métodos numéricos tradicionales.En su defecto, son necesarias e incluso obligatorias varias técnicas computacionales de alto desempeño (high performance computing, HPC) para enfrentar este tipo de problemas. En el presente artículo revisamos herramientas de HPC que permiten simular y controlar problemas a gran escala. Concretamente, nos centramos en técnicas para la reducción de modelos vía truncamiento balanceado y la resolución de problemas de control lineal cuadrático, que pueden ser implementadas eficientemente en plataformas multi-núcleo con memoria compartida que, además, utilizan uno o más procesadores gráficos (GPUs).

In this paper we introduce a variant of the Newton iteration for the matrix sign function that results in an efficient numerical solver for a certain class of algebraic Riccati equations (AREs). In particular, when the Hamiltonian matrix associated with the ARE can be composed as A B B T C T C ? A T $\left [\begin {array}{llll}{A}&{BB^{T}}\\{C^{T}C}&{-A^{T}}\end {array}\right ]$ , with B and C T $C^{T}$ having a much larger number of rows than columns, the new algorithm exploits the special structure of the off-diagonal blocks to yield an alternative factored Newton iteration which reduces the cost per iteration by a factor of up to 8 (16 in case A is symmetric negative definite) w.r.t. the conventional iterative scheme. Experiments with a large collection of benchmark examples show that the factored iteration attains numerical accuracy similar to that of the conventional Newton iteration as well as the structure-preserving doubling algorithm. High-performance implementations of these methods, making heavy use of LAPACK linked to a multi-threaded implementation of BLAS, demonstrate the clear advantage of the new iteration on a 48-core AMD-based platform.

Hyperspectral imaging is a growing area in remote sensing in which an imaging spectrometer collects hundreds of images (at different wavelength channels) for the same area on the surface of the Earth. Hyperspectral images are extremely high-dimensional, and require on-board processing algorithms able to satisfy near real-time constraints in applications such as wildland fire monitoring, mapping of oil spills and chemical contamination, etc. One of the most widely used techniques for analyzing hyperspectral images is spectral unmixing, which allows for sub-pixel data characterization. This is particularly important since the available spatial resolution in hyperspectral images is typically of several meters, and therefore it is reasonable to assume that several spectrally pure substances (called endmembers in hyperspectral imaging terminology) can be found within each imaged pixel. There have been several efforts towards the efficient implementation of hyperspectral unmixing algorithms on architectures susceptible of being mounted onboard imaging instruments, including field programmable gate arrays (FPGAs) and graphics processing units (GPUs). While FPGAs are generally difficult to program, GPUs are difficult to adapt to onboard processing requirements in spaceborne missions due to its extremely high power consumption. In turn, with the increase in the number of cores, multi-core platforms have recently emerged as an easier to program platform compared to FPGAs, and also more tolerable radiation and power consumption requirements. However, a detailed assessment of the performance versus energy consumption of these architectures has not been conducted as of yet in the field of hyperspectral imaging, in which it is particularly important to achieve processing results in real-time. In this article, we provide a thoughtful perspective on this relevant issue and further analyze the performance versus energy consumption ratio of different processing chains for spectral unmixing when implemented on multi-core platforms.

Model order reduction is necessary in many complex scientific and engineering applications. Among the methods for model reduction, those based on the SVD are well-known for their beneficial theoretical properties, though they require O(n3) floating-point arithmetic operations, with n being in the range of 10^3 -- 10^4 for many practical applications. In this paper we propose several high performance implementations of the Balanced Stochastic Truncation method for model reduction. The new routines carefully distribute the computations among the computational resources of a hybrid platform composed of one (or more) multicore CPU(s) and a manycore GPU. Our results show that model reduction of a large-scale problem with 9,669 state variables, which previously required the use of a cluster of computers, can now be carried out in the target platform in less than 25 minutes.

We introduce high performance codes for the inversion of a symmetric positive definite matrix. Two alternativesare studied and evaluated, the traditional approach based onthe Cholesky factorization and the Gauss-Jordan elimination algorithm. Several implementations of the two algorithms are developed on a hybrid architecture equipped with a general purpose multi-core processor and a graphics processor. Numerical experiments show the efficiency attained by the proposed implementations on the target architecture.

We describe a hybrid Lyapunov solver based on the matrix sign function, where the intensive parts of the computation are accelerated using a graphics processor (GPU) while executing the remaining operations on a general-purpose multi-core processor (CPU). The initial stage of the iteration operates in single-precision arithmetic, returning a low-rank factor of an approximate solution. As the main computation in this stage consists of explicit matrix inversions, we propose a hybrid implementation of Gausz-Jordan elimination using look-ahead to overlap computations on GPU and CPU. To improve the approximate solution, we introduce an iterative refinement procedure that allows to cheaply recover full double-precision accuracy. In contrast to earlier approaches to iterative refinement for Lyapunov equations, this approach retains the low-rank factorization structure of the approximate solution. The combination of the two stages results in a mixed-precision algorithm, that exploits the capabilities of both general-purpose CPUs and many-core GPUs and overlaps critical computations. Numerical experiments using real-world data and a platform equipped with two Intel Xeon QuadCore processors and an Nvidia Tesla C1060 show a significant efficiency gain of the hybrid method compared to a classical CPU implementation.

In this paper, we present our joint efforts to design and develop parallel implementations of the GNU Scientific Library for a wide variety of parallel platforms. The multilevel software architecture proposed provides several interfaces: a sequential interface that hides the parallel nature of the library to sequential users, a parallel interface for parallel programmers, and a web services based interface to provide remote access to the routines of the library. The physical level of the architecture includes platforms ranging from distributed and shared-memory multiprocessors to hybrid systems and heterogeneous clusters. Several well-known operations arising in discrete mathematics and sparse linear algebra are used to illustrate the challenges, benefits, and performance of different parallelization approaches.

We investigate the solution of linear systems that appear when model reduction via system balancing is applied in circuit simulation and design. We show how the properties and structure of the coefficient matrices allow the development and use of efficient parallel algorithms for the solution of the corresponding linear systems. Experimental results are reported on an Intel SMP multiprocessor.