arXiv subjects

Luhao Xue

Publications and source records attributed to Luhao Xue.

3 recordsLinked to original sources

arXiv2026

Parametric Sensitivity of POD Reduced-Order Models for Semilinear Evolution Equations with Applications to G-Equations

Proper Orthogonal Decomposition (POD) provides low-dimensional surrogate models of evolution equations from solution snapshots. In parameterized problems, however, a basis computed at one parameter value need not remain accurate at another, while recomputing the basis for every parameter in a many-query study is costly. We develop a sensitivity analysis for POD reduced-order models of a class of parameterized semilinear evolution equations that includes the viscous G-equation and a viscous strain G-equation in their mean-free formulations. The analysis is based on a cross-space Lipschitz condition for the nonlinear operator, which accommodates the nonsmooth, gradient-dependent nonlinearities of these flame-propagation models, and on continuity moduli quantifying the parameter dependence of the bilinear form and of the nonlinearity. We prove that when a POD basis constructed at a reference parameter is applied to nearby query parameters, the resulting error variation is controlled by the corresponding parameter modulus, with constants independent of the POD dimension, the number of time steps, and the perturbation magnitude. Numerical experiments are consistent with the predicted modulus-dependent sensitivity behavior, and illustrate the practical robustness of basis reuse for nearby parameters.

math.NA↗

arXiv2026

Optimization on Affine-Transversal Hilbert Submanifolds: Part I -- Theoretical Foundations

In this paper, we establish the theoretical foundations for the generic optimization problem in a Hilbert space whose feasible set is an affine-transversal Hilbert submanifold given by the intersection of a nonlinear manifold and an affine subspace. We develop the geometric and analytical toolkit, such as the tangent-space characterization, projection operators, and implicit retraction operators, to essentially ensure the feasibility of iterates for algorithmic design. We also derive weak-form expressions for the derivatives of lifted objective functionals. In particular, we introduce a projection-induced Riemannian metric whose induced norm is uniformly equivalent to the ambient norm, under which the projection-induced gradient becomes an exact Riemannian gradient with an explicit formula. This construction replaces the implicit tangent-space Riesz representation underlying classical Riemannian optimization with directly computable operator evaluations, yielding a practical variable-metric framework for algorithmic design while preserving the geometric structure required for convergence analysis. With these theoretical foundations, it becomes possible to apply standard techniques in Euclidean spaces to design algorithms with strictly feasible iterates for the optimization problem on an affine-transversal Hilbert submanifold. We also propose the theoretical frameworks for algorithmic design on affine-transversal Hilbert submanifolds by showcasing the Riemannian line-search and trust-region algorithms with rigorous convergence analysis.

math.OC↗

arXiv2023

Eisenstein series part of the primitive representations for even rank quadratic forms

In this paper, we first investigate the relationship between the number of primitive representations of $n$ by quadratic forms and the number of non-primitive ones. We hence obtain a theorem to deal with the Eisenstein series part with quadratic Dirichlet character when deriving the formula for the number of primitive representations of an integer $n$ by even rank quadratic forms from the number of non primitive ones. Formulas for special cases are given as examples.

math.NT↗