Reduction in Degrees of Freedom for Large-Scale Nonlinear Systems in Computer-Assisted Proofs

Abstract

In many physical systems, it is important to know the exact trajectory of a solution. Relevant applications include celestial mechanics, fluid mechanics, robotics, etc. For cases where analytical methods cannot be applied, one can use computer-assisted proofs or rigorous computations. One can obtain a guaranteed bound for the solution trajectory in the phase space. The application of rigorous computations poses few problems for low-dimensional systems of ordinary differential equations (ODEs) but is a challenging problem for large-scale systems, for example, systems of ODEs obtained from the discretization of the PDEs. A large-scale system size for rigorous computations can be as small as about a hundred ODE equations because computational complexity for rigorous algorithms is much larger than that for simple computations. We are interested in the application of rigorous computations to the problem of proving the existence of a periodic orbit in the Kolmogorov problem for the Navier–Stokes equations. One of the key issues, among others, is the computation complexity, which increases rapidly with the growth of the problem dimension. In previous papers, we showed that 79 degrees of freedom are needed in order to achieve convergence of the rigorous algorithm only for the system of ordinary differential equations. Here, we wish to demonstrate the application of the proper orthogonal decomposition (POD) in order to approximate the attracting set of the system and reduce the dimension of the active degrees of freedom.