Davide Pradovera: Adaptive data-driven surrogate modeling of parametric nonlinear eigenproblems

Abstract:

When dealing with complex dynamical systems, the solution of eigenvalue problems is pivotal in the analysis of frequency responses and in controller design. Eigenvalue nonlinearities can easily arise, e.g., when dealing with controller delays. Moreover, parameters are often included to model uncertainties or design variables.

In this context, we propose a strategy for solving parametric nonlinear eigenproblems. For a wider applicability of our method, we make no assumptions on how the problem depends on the eigenvalue or on the parameters. Our method piggybacks on a reliable contour-integration-based eigensolver for dealing with non-parametric versions of the target problem, obtained by "freezing" all parameters at some collocation parameter values. Then the collection of obtained eigenvalues is used to synthesize the eigenvalue manifolds, i.e., to understand how the eigenvalues vary as the parameters change.

Several issues arise, mostly due to the possible irregularity of the eigenvalue manifolds: (i) manifold crossings may happen if the target spectrum is not well isolated, (ii) bifurcations may reduce the smoothness of the manifolds, and (iii) the target manifolds may "appear and disappear" as parameters vary, since eigenvalues may migrate outside the integration contour. We describe a strategy to flag these undesirable effects and, to some extent, to circumvent them.