arXiv subjects

Shengyue Wang

Publications and source records attributed to Shengyue Wang.

4 recordsLinked to original sources

arXiv2026

An Iterative Method with Asymptotic Orthogonality for Simultaneous Eigenpair Computation

The simultaneous computation of a cluster of eigenpairs with mutually orthogonal eigenvectors is a basic task in scientific computing. We develop a predictor--corrector discretization of the quasi-Grassmannian gradient flow for simultaneous eigenpair computation. The proposed iteration requires neither orthogonal initial data nor any orthogonalization operation: the predictor preserves the current Gram matrix, whereas the corrector reduces the orthogonality error, so that the iterates approach orthogonality asymptotically. We establish well-posedness of the discrete scheme and prove invariant-subspace nonexpansion, asymptotic orthogonality, energy decrease, and convergence to the target eigenspace. Numerical experiments with the discrete equations solved approximately show that the numerical iterates converge to eigenpair approximations whose eigenvectors are numerically orthogonal to high accuracy, while the energy and gradient norm decrease over the iterations.

math.NA↗

arXiv2026

A quasi-Grassmannian gradient flow model for eigenvalue problems

We propose a quasi-Grassmannian gradient flow model for eigenvalue problems of linear operators, aiming to efficiently address many eigenpairs. Our model inherently ensures asymptotic orthogonality: without the need for initial orthogonality, the solution naturally evolves toward being orthogonal over time. We establish the well-posedness of the model, and provide the analytic representation of solutions. Through asymptotic analysis, we show that the gradient converges exponentially to zero and that the energy converges exponentially to its minimum. This implies that the solution of the quasi-Grassmannian gradient flow model converges to the solution of the eigenvalue problems as time progresses. These results provide a continuous-flow framework in which the Stiefel constraint is recovered asymptotically rather than imposed on the initial data.

math.NA↗

arXiv2026

A quasi-orthogonal method based on the inverse operator for Schr{ö}dinger eigenvalue problems

Computing many eigenpairs of the Schr{ö}dinger operator presents a computational bottleneck in large-scale quantum simulations due to the global communication overhead of explicit orthogonalization. To address this issue, we propose a quasi-orthogonal evolution model utilizing inverse operators and develop a corresponding discrete numerical scheme. Instead of forcing explicit orthogonalization, the proposed framework confines the numerical approximations within a quasi-Stiefel set, ensuring the iterates maintain full column rank without requiring $\left\langle U, U \right\rangle=I_N$. Moreover, the method naturally absorbs orthogonality errors and asymptotically converges to the exact eigenfunctions, even when initialized with non-orthogonal random data. The scheme guarantees monotonic dissipation of the target energy functional, with exponential convergence rates rigorously established for the discrete energy, gradient, and eigenfunction approximations. Furthermore, infinite-dimensional analysis proves that the admissible time step size is independent of the spatial discretization. This property overcomes the mesh-dependent stability constraints typical of conventional explicit or semi-implicit schemes, permitting larger time increments to accelerate global convergence. Numerical experiments validate the theoretical findings.

math.NA↗

arXiv2025

An orthogonality-preserving approach for eigenvalue problems

Solving large-scale eigenvalue problems poses a significant challenge due to the computational complexity and limitations on the parallel scalability of the orthogonalization operation, when many eigenpairs are required. In this paper, we propose an intrinsic orthogonality-preserving model, formulated as an evolution equation, and a corresponding numerical method for eigenvalue problems. The proposed approach automatically preserves orthogonality and exhibits energy dissipation during both time evolution and numerical iterations, provided that the initial data are orthogonal, thus offering an accurate and efficient approximation for the large-scale eigenvalue problems with orthogonality constraints. Furthermore, we rigorously prove the convergence of the scheme without the time step size restrictions imposed by the CFL conditions. Numerical experiments not only corroborate the validity of our theoretical analyses but also demonstrate the remarkably high efficiency of the algorithm.

math.NA↗