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Parameters Inference for Nonlinear Wave Equations with Markovian Switching

Yi Zhang, Zhikun Zhang, Xiangjun Wang

Abstract

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.

Parameters Inference for Nonlinear Wave Equations with Markovian Switching

Abstract

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.
Paper Structure (14 sections, 3 theorems, 80 equations, 3 figures, 3 tables)

This paper contains 14 sections, 3 theorems, 80 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Wave equation with Markovian switching
  3. Definition of Wave equation with Markovian switching
  4. Parameter inference for the wave equation with Markovian switching
  5. Convergence and error bound of DSBL using in the wave systems with Markovian switching
  6. Numerical Experiment
  7. Case 1: Sine-Gordon Equation with Markovian switching
  8. Case 2: Klein-Gordon Equation with Markovian switching
  9. Case 3: A two-dimensional nonlinear wave equation with Markovian switching
  10. Conclusion and future work
  11. Appendix
  12. A proof of Theorem \ref{['bounded sequence']}
  13. A proof of Theorem \ref{['convergence']}
  14. A proof of Theorem \ref{['Error bound']}

Key Result

Theorem 2.1

$\{\mathcal{L}(\theta^{(k)}, \gamma^{(k)})\}_{k=0}^{\infty}$ is a bounded, monotone non-increasing sequence.

Figures (3)

  • Figure 1: Numerical Solution of SG equation with Markovian Switching
  • Figure 4: Numerical Solution, Inferred Solution and Error Heat Map of SG Equation with Markovian Switching
  • Figure 6: Numerical vs. Inferred Solution and Corresponding Error Heat Map for Mixture Model and Single Model

Theorems & Definitions (3)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3