Solution of large-scale Lyapunov Differential Equations
Large-scale Lyapunov differential equations (LDEs) arise in many fields like: model reduction, damping optimization, optimal control, numerical solution of stochastic partial differential equations (SPDEs), etc. In particular, LDEs are the key ingredient to perform a simulation of systems governed by certain SPDEs. For example, El Niño–Southern Oscillation or El Niño is modeled by this type of equations. After discretizing the LDE, a (algebraic) Lyapunov equation (LE) with special structure has to be solved in every step. If the structure of the matrix coefficients is not exploited, then it is not possible to solve LDEs of high dimension arising in applications due to memory requirements and computational power. Although, many methods for solving large-scale LE have been proposed in recent years, there has been no attempt in the literature for solving the differential case. In this project we develop new integrators for solving efficiently large-scale LDEs. We will follow three approaches: low-rank approximations, exponential integrators and splitting methods. We will investigate error estimates and step size and order control strategies for each integrator.