Incorporating robust transition probabilities into MCMC for initial value problems with variable coefficients in modelling

This paper presents a rigorously constructed Markov Chain Monte Carlo method designed to efficiently solve initial value problems arising in complex physical modelling problems, where the system dynamics are characterized by time-dependent and potentially uncertain parameters. This approach leverages optimized transition probabilities within the Monte Carlo scheme to enhance the robustness, accuracy, and computational efficiency of convergence; it is particularly advantageous for addressing problems that exhibit periodic or structured solutions. The structural potential of this method enables parallelization, making it well-suited for large-scale data processing and computationally intensive environments, such as those encountered in big-data analytics and parallel supercomputing systems. Numerical results, including applications to systems with variable coefficients and matrix structures characteristic of physical models, demonstrate the superior performance of the proposed stochastic approach over conventional deterministic methods—particularly the two-stage waveform relaxation method—thereby offering new avenues for simulating and analyzing complex physical phenomena with enhanced scalability and reliability.