Kernel-based Discretisation for Solving Matrix-Valued PDEs
This work addresses the challenge of solving matrix-valued PDEs for dynamical systems, but the method is incremental, extending existing kernel-based techniques to a more complex setting.
The paper develops a kernel-based meshfree discretization scheme for solving matrix-valued PDEs, which arise in constructing Riemannian contraction metrics for dynamical systems. It provides error estimates and demonstrates the method on a typical example.
In this paper, we discuss the solution of certain matrix-valued partial differential equations. Such PDEs arise, for example, when constructing a Riemannian contraction metric for a dynamical system given by an autonomous ODE. We develop and analyse a new meshfree discretisation scheme using kernel-based approximation spaces. However, since these approximation spaces have now to be matrix-valued, the kernels we need to use are fourth order tensors. We will review and extend recent results on even more general reproducing kernel Hilbert spaces. We will then apply this general theory to solve a matrix-valued PDE and derive error estimates for the approximate solution. The paper ends with a typical example from dynamical systems.