Parameters Inference for Nonlinear Wave Equations with Markovian Switching
This work addresses a gap in parameter estimation for PDEs with Markovian switching, which is incremental but important for modeling abrupt changes in real-world phenomena like wave dynamics.
The paper tackles parameter inference for wave equations with Markovian switching, proposing a Bayesian framework with discrete sparse Bayesian learning that achieves strong performance in estimating variable coefficients, as demonstrated in three numerical cases.
Traditional partial differential equations with constant coefficients often struggle to capture abrupt changes in real-world phenomena, leading to the development of variable coefficient PDEs and Markovian switching models. Recently, research has introduced the concept of PDEs with Markov switching models, established their well-posedness and presented numerical methods. However, there has been limited discussion on parameter estimation for the jump coefficients in these models. This paper addresses this gap by focusing on parameter inference for the wave equation with Markovian switching. We propose a Bayesian statistical framework using discrete sparse Bayesian learning to establish its convergence and a uniform error bound. Our method requires fewer assumptions and enables independent parameter inference for each segment by allowing different underlying structures for the parameter estimation problem within each segmented time interval. The effectiveness of our approach is demonstrated through three numerical cases, which involve noisy spatiotemporal data from different wave equations with Markovian switching. The results show strong performance in parameter estimation for variable coefficient PDEs.