CRM CAMP in Nonlinear Analysis

Computer-Assisted Mathematical Proofs in Nonlinear Analysis

February 23, 2021 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

Beyond Exponential Complexity of Newton-Galerkin Validation Methods: A Polynomial-Time Newton-Picard Validation Algorithm for linear ODEs

Seminar presented by Florent Bréhard (Université de Lille, France)

A wide range of techniques have been developed to compute validated numerical solutions to various kind of equations (e.g., ODE, PDE, DDE) arising in computer-assisted proofs. Among them are Newton-Galerkin a posteriori validation techniques, which provide error bounds for approximate solutions by using the contraction map principle in a suitable coefficient space (e.g., Fourier or Chebyshev). More precisely, a contracting Newton-like operator is constructed by truncating and inverting the inverse Jacobian of the equation.

While these techniques were extensively used in cutting-edge works in the community, we show that they suffer from an exponential running time w.r.t. the input equation. We illustrate this shortcomings on simple linear ODEs, where a "large" parameter in the equation leads to an intractable instance for Newton-Galerkin validation algorithms.

From this observation, we build a new validation scheme, called Newton-Picard, which breaks this complexity barrier. The key idea consists in replacing the inverse Jacobian not by a finite-dimensional truncated matrix as in Newton-Galerkin, but by an integral operator with a polynomial approximation of the so-called resolvent kernel. Moreover, this method is also less basis-dependent and more suitable to be formalized in a computer proof assistant towards a fully certified implementation in the future.