Fractal Scaling Laws and Asymptotic Behavior for Schwarz Domain Decomposition of Parabolic Hamilton-Jacobi-Bellman with AI-Driven Adaptivity

We develop a unified analytical and computational framework that integrates fractal geometry, scaling laws, and artificial intelligence into the study of parabolic quasi-variational inequalities associated with Hamilton-Jacobi-Bellman equations. The analysis is done for irregular domains with boundaries having a Hausdorff dimension Formula: see text, which enables the influence of the geometric complexity of the fractal interface on the numerical approximation. We develop a generalized overlapping domain decomposition method in a fractal Sobolev space setting. The geometric convergence rate is dictated by a scaling law, with the contraction factor given by Formula: see text where Formula: see text denotes the overlap width. This result reveals that interface roughness directly controls the decay rate of the iterative error. Additionally, we prove a multifractal maximum norm error estimate of the form Formula: see text which shows that the spatial convergence rates depend on the boundary’s fractal dimension. In order to improve the computational efficiency, we have proposed an AI-assisted adaptive overlap method, which relies on the prediction of local spectral radii, and we have shown that it accelerates the convergence significantly. The above results link, in a rigorous way, the fractal geometric structure, the scaling properties of domain decomposition algorithms, and data-driven computational optimization, which provides a novel interdisciplinary tool for the analysis of nonlinear PDEs.