Abstract

Many problems in applied mathematics involve simulating the evolution of a system using differential equations with known initial conditions. But what if one records observations and seeks to determine the causal factors which produced them? This is known as an inverse problem. Some prominent inverse problems include data assimilation and parameter recovery, which use partial observations of a system of evolutionary, dissipative partial differential equations to estimate the state of the system and relevant physical parameters (respectively). Recently a set of procedures called nudging algorithms have shown promise in performing simultaneous data assimilation and parameter recovery for the Lorentz equations and the Kuramoto-Sivashinsky equation. This work applies these algorithms and extensions of them to the case of Rayleigh-B\'enard convection, one of the most ubiquitous and commonly-studied examples of turbulent flow. The performance of various parameter update formulas is analyzed through direct numerical simulation. Under appropriate conditions and given the correct parameter update formulas, convergence is also established, and in one case, an analytical proof is obtained.

Degree

College and Department

Physical and Mathematical Sciences; Mathematics

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