Applied & Computational Mathematics seminar: Validated numerics via the FFT

Speaker:  Jay Mireles James, Florida Atlantic University

Title:  Validated numerics via the FFT

Abstract:  Invariant manifolds of smooth dynamical systems are typically described as solutions of appropriate conjugacy equations.  These equations involve the composition of the unknown solution with the known diffeomorphism or vector field generating the dynamics.  Depending on how we choose to represent the solution (Taylor series, Fourier series, Chebyshev series, splines, etcetera) computing this composition can be more or less challenging.  If the dynamics are generated by polynomial maps or vector fields, then formulas for the composition can usually be worked out in terms of Cauchy products/discrete convolutions.  For simple enough non-polynomial nonlinearities, one can often append polynomial differential equations describing the nonlinear terms.  However, for complicated enough nonlinearities even this is cumbersome, and it’s natural to consider methods based on interpolation.  In each of the Taylor, Fourier, and Chebyshev series cases this leads to the discrete Fourier transform (DFT) and its fast implementation the FFT.  I'm especially interested in computer assisted proofs involving solutions of conjugacy equations, so that it is necessary to bound all errors rigorously.  In the case of the DFT there are three kinds of errors: rounding, truncation, and also aliasing.  After introducing these ideas in a little more detail, I'll discuss a simple approach using a little complex analysis which allows one to manage the second two (rounding is managed using interval arithmetic).  This is joint work with J.P. Lessard, J.B. van den Berg, and Maxime Breden.  

Time:  Friday, November 10, 2023 - 1:30pm-2:30pm

Place:  Exploratory Hall, Room 4106 and Zoom