## 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.

(Image source: Wikipedia)

We will develop an integrator based on the low-rank factorization of the solution in a way that the whole iteration will be performed directly on the factors of the solution. After that, we will investigate the application of a more general approach the so-called dynamical low-rank approximation. Moreover, we will investigate exponential integrators for solving LDEs. Due to the dimension of the equation, we will focus on keeping the computational cost and memory requirements as low as possible. Specially, for computing the action of matrix functions which is the most computationally demanding operation using an exponential integrators approach. We will also consider splitting methods and use the same ideas as for the exponential integrator approach for computing the action of matrix functions. In this way, we will develop an algorithm that can take advantage of the structure of the system. We will also investigate higher order splitting methods to improve the accuracy of the numerical solution.

Finally, we will develop a state-of-the-art implementation for a hybrid CPU-GPU platform that efficiently exploits all the available computational resources in the hardware; namely, the multicore processor(s) and the graphics processor(s). As an application, we will perform the whole simulation of El Niño phenomena with real data. The results of this project will allow computing the simulation of El Niño more accurately and in that way contributing for a better understanding of the phenomena. Moreover, the integrators for solving large-scale LDEs will be tested in specific problems arising in: model reduction of linear time-varying systems, damping optimization in mechanical systems and control of shear flows subject to stochastic excitations.

### Funded by :

FWF Austrian Science Fundation