A Reduced Basis Method for the Stochastic Landau-Lifshitz-Gilbert Equation

TL;DR

The paper tackles efficient surrogates for the stochastic Landau-Lifshitz-Gilbert equation by applying Reduced Basis (RB) methods to the parametric reformulation obtained via a Doss–Sussmann transform. An offline POD builds compact reduced spaces from high-fidelity trajectory data generated by a Tangent Plane Scheme (TPS); two online strategies are developed: (i) Galerkin POD-Tangent Plane Scheme (POD-TPS) with supremizer stabilization to ensure inf-sup stability, and (ii) Sparse Grid–RB Projection (SG-RBP) that interpolates reduced coefficients to avoid online Galerkin solves. Numerical results show that stabilized POD-TPS yields accurate, energy-consistent online reduced dynamics and that SG-RBP provides a complementary, more stable data-driven surrogate, with performance sensitive to parameter-dimension and switching dynamics. Together, the methods enable significant cost reductions for parameter studies and uncertainty quantification in stochastic micromagnetics, with practical impact for design and optimization of ferromagnetic devices. The work highlights both the strengths and limitations of linear MOR in transport-dominated, high-dimensional stochastic PDEs and suggests directions for adaptive time stepping and nonlinear ROM extensions.

Abstract

In this work, we consider the construction of efficient surrogates for the stochastic version of the Landau-Lifshitz-Gilbert (LLG) equation using model order reduction techniques, in particular, the Reduced Basis (RB) method. The Stochastic LLG (SLLG) equation is a widely used phenomenological model for the time evolution of the magnetization field confined to a ferromagnetic body while taking into account the effect of random heat perturbations. This phenomenon is mathematically formulated as a nonlinear parabolic problem, where the stochastic component is represented as a parameter-dependent datum depending on a non-compact and high-dimensional parameter. In an $\textit{offline}$ phase, we use Proper Orthogonal Decomposition (POD) on high-fidelity samples of the unbounded parameter space. To that end, we use the so-called $\textit{tangent plane scheme}$. For the $\textit{online}$ phase of the RB method, we again employ the tangent plane scheme in the RB space. This is possible due to our particular construction that reduces both spaces of the magnetization and of its time derivative. Due to the saddle-point nature of this scheme, a stabilization that appropriately enriches the RB space is required. Numerical experiments show a clear advantage over earlier approaches using sparse grid interpolation. In a complementary approach, we test a sparse grid approximation of the reduced coefficients in a purely data-driven method, exhibiting the weaknesses of earlier sparse grid approaches, but benefiting from increased stability.

Figures (14)

• Figure 1: Left: The domain (2D unit square in the $xy$ plane) and an example of mesh used for the numerical tests. Center and Right: A snapshot of the random magnetization respectively at times $t=0$ and $t=1$ and colored with the $z$ component.
• Figure 2: Left: First 100 singular values of velocities ($\sigma_{\boldsymbol{v}}$), Lagrange multipliers ($\sigma_{\lambda}$), and magnetizations ($\sigma_{\bm}$) based on $N_S = 128$ snapshots. Right: Projection errors onto the reduced spaces for velocities, Lagrange multipliers, and magnetizations with respect to the number of reduced basis functions.
• Figure 3: First three magnetization reduced basis functions. The three vector fields are color-coded with the magnitude. Each reduced basis function is defined only up to a constant. Therefore, they are scaled for ease of readability.
• Figure 4: Comparison of the four online Galerkin POD-TPS strategies: OG-1x, OG-3x, and the respective versions with supremizer stabilization, SS-OG-1x and SS-OG-3x. Left: Galerkin POD errors $\varepsilon^{\text{G-POD}}_{\bm,J}$ and $\varepsilon^{\text{G-POD,Stab}}_{\bm,J,K,R}$ of the magnetization field as a function of the total number of reduced basis functions. Right: Minimum inf-sup constant over time-steps as a function of the total number of reduced basis functions. In black, the minimum inf-sup constant over the time steps of the high fidelity solver.
• Figure 5: Study of OG-3x for different time step sizes (used only in online phase). Left: Error as a function of the time step size. Right: Minimum inf-sup constant over time-steps as a function of the time step size.

Theorems & Definitions (4)

• Remark 1: Criterion to select $J$
• Remark 2
• Proposition 4.1
• proof