Automatic differentiation for Fourier series and the radii polynomial approach

Abstract

In this work we develop a computer-assisted technique for proving existence of periodic solutions of nonlinear differential equations with non-polynomial nonlinearities. We exploit ideas from the theory of automatic differentiation in order to formulate an augmented polynomial system. We compute a numerical Fourier expansion of the periodic orbit for the augmented system, and prove the existence of a true solution nearby using an a-posteriori validation scheme (the radii polynomial approach). The problems considered here are given in terms of locally analytic vector fields (i.e. the field is analytic in a neighborhood of the periodic orbit) hence the computer-assisted proofs are formulated in a Banach space of sequences satisfying a geometric decay condition. In order to illustrate the use and utility of these ideas we implement a number of computer-assisted existence proofs for periodic orbits of the Planar Circular Restricted Three-Body Problem (PCRTBP).

Fourier–Taylor parameterization of unstable manifolds for parabolic partial differential equations: Formalism, implementation and rigorous validation

“…This is not as restrictive as it might seem upon first glance, as transcendental nonlinearities given by elementary functions can be treated using methods of automatic differentiation. A thorough discussion of automatic differentiation as a tool for semi-numerical computations and computer assisted proof is beyond the scope of the present work, and we refer the interested reader to the books [58,83] and also to the work of [43,63] as an entry point to the literature. In terms of the present discussion the relevant point is that automatic differentiation allows us to develop formal series evaluation of non-polynomial nonlinearities by appending additional differential equations.…”

Rigorous validation of stochastic transition paths

“…This polynomial reformulation is based on ideas from automatic differentiation, and was introduced in the context of a posteriori validation in [49]. See [40,Section 4.2] for a discusion about this type of polynomial reformulation in a more general framework, and also [42,35] for detailed algorithmic descriptions.…”

“…Estimate (49) comes from a meticulous but rather straightforward analysis of the various terms appearing in (48) for the columns of indices l > 4K − 2, using that Estimates similar to (49) were obtained previously in [68,72].…”

Validated computations for connecting orbits in polynomial vector fields

“…The idea is to solve (1.1) by computing Taylor expansions for charts on the local (un)stable manifolds via the parameterization method [6,30], which are used to supplant the boundary conditions in (1.1) with explicit equations, and to use Chebyshev series and domain decomposition techniques [32] to parameterize the orbit in between. We remark that the assumption that g is polynomial is not as restrictive as it initially might seem, since many nonlinearities which consist of elementary functions can be brought into polynomial form by using automatic differentiation techniques, see [19] for instance. Before we proceed with a more detailed description of our method, a few remarks concerning the development of numerical methods for connecting orbits are in order.…”