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

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

The stochastic linear quadratic optimal control problem in Hilbert spaces: A polynomial chaos approach

“…Characterization of optimality. The optimality of our approach (34) can be characterized in terms of the solution of the stochastic Riccati equation (5). The following theorem summarizes our result.…”

“…with a positive self-adjoint operator P(t) solving the stochastic Riccati equation (5). Since the state equations (1) and (25) are equivalent, we are going to interpret the optimal solution (33), involving the Riccati operator P(t) in terms of chaos expansions.…”

“…Condition (36) which characterizes optimality represents the action of the stochastic Riccati operator in each level of the noise. Note that the stochastic Riccati equation (5) and the deterministic one (8) differ only in the term C P (t) C, i.e. the operator C P (t) C captures the stochasticity of the equation.…”

A GPU‐aware mixed‐precision solver for low‐rank algebraic Riccati equations

“…Different studies have demonstrated the benefits of using GPUs to accelerate the computation of matrix equations() and matrix Riccati equations in particular. ()…”

“…Different studies have demonstrated the benefits of using GPUs to accelerate the computation of matrix equations 6-8 and matrix Riccati equations in particular. 9,10 Additionally, energy consumption has become one of the major restrictions for the design of future supercomputers because of the economic costs of electricity, the negative effect of heat on the reliability of hardware components, and the negative environmental impact. While the advances in the performance of the hardware platforms in the Top500 list 11 show that an Exascale system may be available in the next quinquennium 12-14 ; a system of that capacity built over current technology would dissipate ridiculously large amounts of energy.…”