Mathematics of computational modelling: some challenges of computing nonlinear phenomena

摘要

The prediction of various phenomena through models and computation is a necessity in numerous real-life problems. The computation of singular phenomena (interfaces, shocks, defects, cracks, etc.) arises in many complex systems and poses many challenges. For computing such phenomena, it is natural to seek methods that are able to detect them and to devote the necessary computational recourses to their accurate resolution. These phenomena are very interesting for their importance and applicability, but also for the challenge they pose in mathematical research. From a computational perspective, we need to design algorithms that are fast but also reliable. In this paper, we consider examples of such phenomena modelled by nonlinear partial differential equations and discrete microscopic systems. Since the often weak solutions of PDEs related to these problems are not unique, we demonstrate that "natural ad hoc" computational methods might predict irrelevant solutions. Since numerical methods perturb the mathematical model, mathematical analysis emerges as a necessary tool which ensures that our computational methods approximate physically relevant solutions. We discuss in more detail three nonlinear problems: (i) a problem related to design and analysis of approximate atomistic-continuum energies to atomistic models arising in crystalline materials; (ii) a problem of cell interaction within a fibrin medium. Motivated by experiments, we demonstrate that the combination of sophisticated mathematical modelling and numerical analysis leads to reliable computational predictions and reveals the real mechanisms of the observed interactions; and (iii) a problem arising in statistical inference of solutions to nonlinear hyperbolic systems. To compute measure-valued solutions, we propose new discrete kinetic models, and we study corresponding kinetic formulations of viscous and inviscid conservation laws.