Hang Qi

Hang Qi, Changqing Ye, Xiaofei Guan

The Landau--Lifshitz--Gilbert equation poses significant challenges for numerical simulation due to its nonlinearity, nonconvex unit-length constraint, and nonlocal field contributions. Existing implicit or semi-implicit schemes exhibit unconditional stability but require repeated solution of nonlinear or linearized systems, resulting in high computational costs. In this work, we revisit the tangent plane formulation and show that the resulting discrete problem at each time step can be written as a generalized saddle-point system. Exploiting this structure, we develop a matrix-free preconditioner for the implicit Euler scheme by combining FFT-based techniques with a splitting iteration method. We show that each component in the splitting method can be efficiently implemented in a direct fashion via fast Poisson solvers, enabling the design of an efficient preconditioner for iterative solvers. To enhance geometric consistency of the tangent vector and reduce projection steps, a Crank--Nicolson scheme is further investigated, retaining the same algorithmic structure as the Euler scheme, thus allowing direct reuse of preconditioners and solvers. Extensive 2D and 3D numerical experiments are conducted to validate the framework, demonstrating accurate constraint preservation, improved computational efficiency, and robust solver performance. The proposed approach is shown to scale effectively for large-scale micromagnetic simulations, making it suitable for practical applications in complex magnetic systems.

Zhiyuan Wang, Hang Qi, Xiaofei Guan et al.

Bayesian inference methods have been developed to address inverse problems in function spaces where the unknown parameters are of infinite dimension. However, conventional Gaussian priors remain inadequate for reconstructing discontinuous or sharply varying target functions encountered in practical applications like obstacle reconstruction. Although hybrid priors have emerged as a promising solution, significant challenges remain in developing theoretically rigorous and computationally tractable frameworks in engineering applications. To address these issues, we propose a persistent-homology-Gaussian (PHG) prior for solving the acoustic obstacle scattering inverse problem in the infinite-dimensional Bayesian setting, which combines a weighted persistence-based regularization term with a periodic Gaussian reference measure through a Gibbs tilt. Then, the complex boundary is represented by a log-radial function on the unit circle, so that the reconstruction from far-field data is formulated as a function-space inverse problem. The well-posedness of the resulting posterior measure is established in the Hellinger, total variation, and Wasserstein-\(p\) metrics. Furthermore, the convergence of finite-dimensional posterior approximations is obtained, and posterior sampling is performed by a preconditioned Crank--Nicolson (pCN) method. Numerical experiments show that the proposed PHG prior yields accurate and stable reconstructions under more extensive noisy conditions, providing explicit control of multiscale topological features and better performance compared to other conventional priors.