Adjoint-based variational methods for computing invariant solutions in spatio-temporally chaotic PDEs

One approach for describing spatio-temporally chaotic dynamical systems, including fluid turbulence, is to study non-chaotic but unstable invariant solutions embedded within the chaotic attractor of the system. These include steady, periodic, and quasi-periodic solutions, as well as connecting orbits between them. Individual invariant solutions are able to capture essential features of the observed spatio-temporal structures, and collectively, they promise a framework for quantitatively predicting the statistics of the system. However, computing invariant solutions for high-dimensional chaotic systems remains challenging. The standard shooting-based methods exhibit poor convergence properties since they rely on time-marching the chaotic dynamics, resulting in exponential error amplification.

We develop a family of adjoint-based variational methods for computing unstable invariant solutions. Our methods eliminate the need for any time-marching of the chaotic dynamics. Hence, they are not affected by the exponential error amplification associated with time-marching a chaotic system and converge robustly from inaccurate initial guesses. These methods are formulated such that no construction and inversion of Jacobian matrices is required. As a consequence, these methods show favorable memory scalings and can thus be applied to very high-dimensional problems including 3D fluid flows. As a proof of concept, we apply the introduced methods to the 1D Kuramoto-Sivashinsky equation. We demonstrate the robustness of the methods by computing multiple periodic and connecting orbits in a spatio-temporally chaotic regime of this model system.

The primary challenge in applying these methods to 3D wall-bounded flows lies in handling the pressure field in the presence of solid walls. We formulate the variational dynamics in a way that explicit computation of pressure is circumvented. Instead, our formulation employs the influence matrix method. We demonstrate the method by computing multiple equilibria for plane Couette flow starting from inaccurate initial guesses extracted from a turbulent time series. We also introduce a data-driven technique based on dynamic mode decomposition (DMD) to accelerate the convergence of the variational method.

We employ our family of adjoint-based variational methods to formalize the phenomenon of saddle-node bifurcation ghosts by defining representative state-space structures for this phenomenon. Our methods provide a unifying framework for computing invariant solutions and their ghosts as the global and local minima of a suitably defined cost function, respectively. We show the family of methods for a range of dynamical systems of various complexities, including the 3D Rayleigh-Bénard convection.