Abstract
We address the numerical solution of Lyapunov, algebraic and differential Riccati equations, via the matrix sign function, on platforms equipped with general-purpose multicore processors and, optionally, one or more graphics processing units (GPUs). In particular, we review the solvers for these equations, as well as the underlying methods, analyze their concurrency and scalability and provide details on their parallel implementation. Our experimental results show that this class of hardware provides sufficient computational power to tackle large-scale problems, which only a few years ago would have required a cluster of computers.
Highlights
Matrix equations are frequently encountered in control theory applications, like model-order reduction or linear-quadratic optimal control problems, involving dynamical linear systems that represent a variety of physical phenomena or chemical processes [1]
We review the rapid solution of Lyapunov equations, AREs and differential Riccati equation (DRE), on multicore processors, as well as graphics processing units (GPUs), making the following specific contributions:
GECLNC MGPU shows that the multi-GPU kernel offers high performance for smaller problems, which decays for larger problems
Summary
Matrix equations are frequently encountered in control theory applications, like model-order reduction or linear-quadratic optimal control problems, involving dynamical linear systems that represent a variety of physical phenomena or chemical processes [1]. We address the solution of Lyapunov equations and algebraic/differential Riccati equations (AREs/DREs), involving dense coefficient matrices, via the matrix sign function [2] This iterative method exhibits a number of appealing properties, such as numerical reliability, high concurrency and scalability and a moderate computational cost of O(n3 ) floating-point arithmetic operations (flops) per iteration, where n denotes the number of the states of the dynamical linear system. Overall, we provide a clear demonstration that commodity hardware, available in current desktop systems, offers sufficient computational power to solve large-scale matrix equations, with n ≈ 5,000–10,000 These methods are revealed as appealing candidates to replace and complement more cumbersome message-passing libraries for control theory applications.
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