Dynamical low-rank approximation for the semiclassical Schrodinger equation with uncertainties
Liu Liu, Limin Xu, Zhenyi Zhu
TL;DR
This work addresses numerical challenges in the semiclassical Schrödinger equation with uncertainties, where both spatial and random-space oscillations are present. It introduces a dynamical low-rank (DLR) framework that evolves the wave function on a low-rank manifold, using a low-rank ansatz $\psi_A=\sum_{i,j=1}^r X_i S_{ij} W_j$ and tangent-space projections to derive evolution equations for $S$, $X$, and $W$. The authors extend projector-splitting and unconventional integrators to this stochastic setting and provide detailed algorithmic steps (K-step, S-step, L-step) along with complexity analyses, showing substantial computational and storage savings compared to stochastic Galerkin methods. Numerical experiments across multiple quantum dynamics scenarios demonstrate that DLR achieves high fidelity with a modest rank (e.g., $r=46$) and that the unconventional integrator often yields the best accuracy, highlighting the method’s potential for high-dimensional uncertainty quantification in the semiclassical regime. The results indicate that the rank dynamics are primarily driven by the randomness in the potential, suggesting practical guidance for applying DLR to uncertain quantum systems and motivating future work on rank-adaptive strategies.
Abstract
In this paper, we propose a dynamical low-rank (DLR) approximation framework for solving the semiclassical Schrodinger equation with uncertainties. The primary numerical challenges arise from the dual nature of the oscillations: the spatial oscillations inherent in the semiclassical limit and the high-frequency oscillations in the random space induced by uncertainties. We extend two robust integrators -- the projector-splitting integrator and the unconventional integrator -- to the semiclassical regime to evolve the solution on a low-rank manifold. Through extensive numerical experiments, we demonstrate that the DLR method is significantly more computationally efficient than the standard stochastic Galerkin method, as it captures the essential quantum dynamics using a much smaller number of basis functions. Our findings reveal that despite the complex oscillatory patterns of the wave function, its evolution remains concentrated in a low-rank subspace for the cases investigated. Specifically, we observe that the DLR method achieves high fidelity with a remarkably small numerical rank, which remains robust even as the semiclassical parameter $\varepsilon$ decreases. Within our problem settings, the results further suggest that the rank growth is primarily driven by the randomness and regularity of the potential. These results provide practical insights into the low-rank structure of uncertain quantum systems and offer an efficient approach for high-dimensional uncertainty quantification in the semiclassical regime.