Accelerating Model Reduction of Large Linear Systems with Graphics Processors

Abstract

Model order reduction of a dynamical linear time-invariant system appears in many applications from science and engineering. Numerically reliable SVD-based methods for this task require in general O(n 3 ) floating-point arithmetic operations, with n being in the range 10 3 − 10 5 for many practical applications. In this paper we investigate the use of graphics processors (GPUs) to accelerate model reduction of large-scale linear systems by off-loading the computationally intensive tasks to this device. Experiments on a hybrid platform consisting of state-of-the-art general-purpose multi-core processors and a GPU illustrate the potential of this approach.

HPC en simulación y control a gran escala

“…Posteriormente, el trabajo fue extendido [18] para la resolución de ecuaciones de Lyapunov al utilizar una GPU y explotando estrategias de precisión mixta para alcanzar niveles de precisión altos a un costo computacional bajo. También es interesante el artículo [17], donde se acelera la resolución del problema de reducción de modelos generalizado, mediante el método BT sobre arquitecturas híbridas GPU-CPU. El trabajo se basa en el uso de implementaciones eficientes propietarias, y particularmente desarrolladas sobre GPU, de las principales operaciones matriciales requeridas por el método: la factorización LU, la resolución de sistemas triangulares y la multiplicación de matrices.…”

Extending lyapack for the solution of band Lyapunov equations on hybrid CPU–GPU platforms

“…In recent years, Graphics Processing Units (GPUs) have shown remarkable performance in the computation of large-scale matrix operations, and particularly, in the solution of matrix equations; see [4,5] among others. In addition to its performance, GPUs present other interesting properties, such as a low Watt-per-floating-point arithmetic operation ratio and an afforable price.…”

Solving Matrix Equations on Multi-Core and Many-Core Architectures

“…A number of efforts have demonstrated the remarkable speed-up that these systems provide for the solution of dense and sparse linear algebra problems. Among these, a few works targeted the numerical solution of matrix equations, which basically can be decomposed into primitive linear algebra problems, using this class of hardware [5][6][7][8][9]. In this paper, we review the rapid solution of Lyapunov equations, AREs and DREs, on multicore processors, as well as GPUs, making the following specific contributions:…”

“…• We collect and update a number of previous results distributed in the literature [5][6][7][8][9], which show that the matrix sign function provides a common and crucial building block for the efficient parallel solution of these three types of matrix equations on multicore CPUs, hybrid CPU-GPU systems and hybrid platforms with multiple GPUs. • We extend the single-precision (SP) experiments previously reported in [5][6][7] with double-precision (DP) data. This is especially relevant, as DP solutions are required in practical control theory applications.…”

“…In particular, the NVIDIA K20 features a high number of cores (2496) and low frequency (706 MHz), while each GPU in the NVIDIA S2050 presents a lower number of cores (448), but operates at higher frequency (1150 MHz). None of our previous work [5][6][7][8][9] reported data with the newer NVIDIA K20. • We follow the general design framework for the solution of Lyapunov equations and DREs on multi-GPU platforms, extending these ideas to implement and evaluate a new ARE solver that leverages the presence of multiple GPUs in a hybrid platform.…”