A memory-efficient Broyden method to compute fixed points of non-linear maps arising in periodically forced processes

In this paper we propose a limited memory method, called the Broyden rank reduction (BRR) method, to efficiently compute fixed points of high-dimensional non-linear maps. The method is based on the singular value decomposition and has good convergence properties. We discuss some basic properties of the BRR method and prove convergence for linear systems. The method is developed for a concrete problem in engineering. In chemical engineering, periodically forced processes in packed bed reactors are generally simulated using pseudo-homogeneous one-dimensional models on a coarse grid discretization. In these simulations, one typically uses Broyden's method because of its simplicity and the fact that appropriate initial conditions can be chosen in a natural way. A disadvantage of Broyden's method is the memory usage to store the Broyden matrix. The BRR method resolves this issue and allows us to consider more complicated and accurate models. We show that the BRR method developed to simulate periodically forced processes is in certain situations a good option to compute fixed points of high-dimensional non-linear maps and illustrate this using examples from integral equations and parabolic partial differential equations. To benchmark our method, we conclude the paper with a comparison of the BRR method with the generally preferred Newton–GMRES(m) method.