Multiple interval methods for ODEs with an optimization constraint

Abstract

We are interested in numerical methods for the optimization constrained second order ordinary differential equations arising in biofilm modelling. This class of problems is challenging for several reasons. One of the reasons is that the underlying solution has a steep slope, making it difficult to resolve. We propose a new numerical method with techniques such as domain decomposition and asynchronous iterations for solving certain types of ordinary differential equations more efficiently. In fact, for our class of problems after applying the techniques of domain decomposition with overlap we are able to solve the ordinary differential equations with a steep slope on a larger domain than previously possible. After applying asynchronous iteration techniques, we are able to solve the problem with less time.~We provide theoretical conditions for the convergence of each of the techniques. The other reason is that the second order ordinary differential equations are coupled with an optimization problem, which can be viewed as the constraints. We propose a numerical method for solving the coupled problem and show that it converges under certain conditions. An application of the proposed methods on biofilm modeling is discussed. The numerical method proposed is adopted to solve the biofilm problem, and we are able to solve the problem with larger thickness of the biofilm than possible before as is shown in the numerical experiments.