General Framework and Error Estimates for ROM-accelerated Fixed Point Iterations

Whether it is for solving nonlinear equations, optimization problems, or autonomous dynamical systems, fixed-point-type iterations are widely used in numerical sciences. On-the-fly reduced-order modelling (ROM) enables the construction of a low-dimensional, self-correcting approximation of the solution to this system during the iterative process, while removing the need to do an offline training phase and any dependence on a precomputed reduced basis (e.g., a fixed geometry or mesh). This technique has been used in specific fields before, including fluid-structure interactions and topology optimization, but no general study of this method has been done to the knowledge of the authors. A general method for accelerating fixed point schemes will be presented. We show that when the iteration mapping is contractive, the error of the approximate solution is guaranteed to be within the user-defined tolerance using inexact fixed-point theory. This methodology is then applied to the solution of systems of PDEs with a block Gauss-Seidel scheme. Errors due to the ROM are propagated through each iteration with respect to the computational graph of the system, which allows one to estimate whether the current iteration is still within the user-defined tolerance. Some working hypotheses necessary to observe a significant speedup and the limitations of the method are explored as well. As a numerical illustration, the methodology is applied to a multiphysics lid-driven cavity flow in two and three dimensions.