Speaker
Description
Multivariate polynomial approximations to Hamilton-Jacobi-Bellman equations can be expressed using Kronecker products leading to very large, but structured, linear systems. Their structure appears as n-way generalizations of Lyapunov or generalized Lyapunov equations. For monomial terms of degree d, their dimension scales as the number of state dimensions n raised to the d. For feedback control problems with modest dimension n=1000 and relatively low degree approximations of d=3 or 4, matrix-free iterative solvers using scalable preconditioners are essential. This talk will review several options for preconditioners, then present a number of examples arising from discretizations of nonlinear partial differential equation control problems that demonstrate the performance of these linear solvers. As expected, there can be a strong connection between the discretization and the preconditioner choices. We conclude with applications to fluid flow control problems modeled through approximations to Navier-Stokes equations.