Novel iterative methods for solving matrix equations arising from partial differential equations

In this thesis, we investigate the numerical solution of large-scale linear matrix equations arising from the discretisation of diffusion and convection-diffusion partial differential equations (PDEs). The matrix equations which arise can be Lyapunov, Sylvester or generalised Sylvester equations. Both Lyapunov and Sylvester equations can be solved using a rational Krylov approach, where orthonormal bases of rational Krylov subspaces are used to project the matrix equations. The projected matrix equation is then solved, and a low rank solution is computed. As part of the generation of the orthonormal basis, the rational Krylov subspace requires a selection of poles, which, in this thesis, are chosen a priori. For Lyapunov equations arising from the discretisation of diffusion PDEs, we derive an explicit rational approximation for the solution for both 1- and 2-sided projections and provide a comparison of the two approaches. For both Lyapunov and Sylvester equations, we adapt the iterative rational Krylov algorithm (IRKA) from model order reduction to generate efficient pole choices for rational Krylov subspaces, which we then use in 2-sided projections. We perform thorough comparisons of pole choice approaches to solve the matrix equations from the practical point of view by considering convergence rates, computational costs and scalability. As a result, we can see that the IRKA poles can be competitive choices for a large set of realistic problems. For the generalised Sylvester equation we derive a stationary iterative method, determine a convergence criterion, and evaluate the performance via numerical results.In this thesis, we investigate the numerical solution of large-scale linear matrix equations arising from the discretisation of diffusion and convection-diffusion partial differential equations (PDEs). The matrix equations which arise can be Lyapunov, Sylvester or generalised Sylvester equations. Both Lyapunov and Sylvester equations can be solved using a rational Krylov approach, where orthonormal bases of rational Krylov subspaces are used to project the matrix equations. The projected matrix equation is then solved, and a low rank solution is computed. As part of the generation of the orthonormal basis, the rational Krylov subspace requires a selection of poles, which, in this thesis, are chosen a priori. For Lyapunov equations arising from the discretisation of diffusion PDEs, we derive an explicit rational approximation for the solution for both 1- and 2-sided projections and provide a comparison of the two approaches. For both Lyapunov and Sylvester equations, we adapt the iterative rational Krylov algorithm (IRKA) from model order reduction to generate efficient pole choices for rational Krylov subspaces, which we then use in 2-sided projections. We perform thorough comparisons of pole choice approaches to solve the matrix equations from the practical point of view by considering convergence rates, computational costs and scalability. As a result, we can see that the IRKA poles can be competitive choices for a large set of realistic problems. For the generalised Sylvester equation we derive a stationary iterative method, determine a convergence criterion, and evaluate the performance via numerical results