arXiv subjects

Guanghan Li

Publications and source records attributed to Guanghan Li.

At least 19 recordsLinked to original sources

arXiv2026

A high-order Newton multigrid method with a simplified Jacobian for steady-state shallow water equations

A high-order Newton multigrid method is proposed for steady-state shallow water flows in open channels with regular and irregular geometries. The method integrates a finite volume discretization with third-order weighted essentially non-oscillatory (WENO) reconstruction and a Newton multigrid framework with an efficient approximation of the Jacobian matrix for solving the resulting discrete system. In high-order schemes, the computational cost of Jacobian construction becomes dominant due to the wide stencil. Meanwhile, only a small fraction of the non-zero Jacobian entries exhibit large magnitudes. Based on this observation, a simplified Jacobian approximation is introduced using reduced stencils, in which selected off-stencil contributions are neglected, thereby achieving a substantial reduction in computational cost. The proposed approach is verified numerically to show significant efficiency improvement while maintaining comparable convergence behavior to that obtained with the full Jacobian approach. To further enhance performance, a geometric multigrid method incorporating a successive over-relaxation iteration as the smoother is applied to solve the linear systems arising in each Newton step. A variety of numerical experiments, including a one-dimensional smooth subcritical flow, flows over a hump, and a two-dimensional hydraulic jump over a wedge, are carried out to illustrate the third-order accuracy, efficiency, and robustness of the proposed method.

math.NA↗

arXiv2026

The Anisotropic Capillary $L_p$-Minkowski Problem

This paper introduces the \textit{anisotropic $ω_0$-capillary $p$-sum} of two hypersurfaces in $\mathbb{R}_+^{n+1}$, and establishes a theory for anisotropic capillary convex bodies. For a smooth convex hypersurface $Σ$ with anisotropic $ω_0$-capillary boundary, we compute the variation of its anisotropic capillary $k$-th quermassintegral via this $p$-sum, thereby defining the associated anisotropic $ω_0$-capillary $k$-th $p$-surface area measure on the capillary Wulff shape $\mathcal{C}_{ω_{0}}$. This motivates us to propose and solve the anisotropic capillary $L_{p}$-Minkowski problem for $p\geq1$.

math.DG↗

arXiv2026

Capillary Orlicz-Minkowski flow in the upper half-space

In this paper, we study the long-time existence and asymptotic behavior of an anisotropic capillary Gauss curvature flow. By studying this flow and proving its convergence to a stationary solution, we establish a new existence result for the capillary Orlicz-Minkowski problem without the evenness assumption, and provide a flow approach to the existence of smooth solutions.

math.DG↗

arXiv2025

A novel fast iterative moment method for near-continuum flows

We develop a novel fast iterative moment method for the steady-state simulation of near-continuum flows, which are modeled by the high-order moment system derived from the Boltzmann-BGK equation. The fast convergence of the present method is mainly achieved by alternately solving the moment system and the hydrodynamic equations with consistent constitutive relations and boundary conditions. To be specific, the consistent hydrodynamic equations are solved in each alternating iteration to obtain improved predictions of macroscopic quantities, which are subsequently utilized to expedite the evolution of the moment system. Additionally, a semi-implicit scheme treating the collision term implicitly is introduced for the moment system. The resulting alternating iteration can be further accelerated by employing the Gauss-Seidel method with a cell-by-cell sweeping strategy. It is also noteworthy that such an alternating iteration works well with the nonlinear multigrid method. Numerical experiments for planar Couette flow, shock structure, and lid-driven cavity flow are carried out to investigate the performance of the proposed fast iterative moment method. All results show impressive efficiency and robustness.

math.NA↗

arXiv2025

Alexandrov-Fenchel inequalities for convex anisotropic capillary hypersurfaces in the half-space

In this paper, the results of Mei, Wang, Weng and Xia [Math. Z., 2025, MR4911815] on capillary convex bodies are extended to the anisotropic setting. We develop a theory for anisotropic capillary convex bodies in the half-space and establish a general Alexandrov-Fenchel inequality for mixed volumes of anisotropic capillary convex bodies. Thus, this weakens the conditions of the inequality in [Ding-Gao-Li,arXiv:2408.10740, Theorem 1.4] and extends it to a more general case.

math.DG↗

arXiv2025

Generalized Minkowski formulas and rigidity results for anisotropic capillary hypersurfaces

In this paper, we obtain a new Hsiung-Minkowski integral formula for anisotropic capillary hypersurfaces in the half-space, which includes the weighted Hsiung-Minkowski formula and classical anisotropic Minkowski identity for closed hypersurfaces as special cases. As applications, we prove some anisotropic Alexandrov-type theorems and rigidity results for anisotropic capillary hypersurfaces. Specially, the uniqueness of the solution to the anisotropic Orlicz-Christoffel-Minkowski problem is obtained, and thus a new proof is provided for the uniqueness of the solution to $L_p$-Minkowski problem with $p\geq 1$ in the Euclidean capillary convex bodies geometry.

math.DG↗

arXiv2025

Anisotropic mean curvature type flow and capillary Alexandrov-Fenchel inequalities

In this paper, an anisotropic volume-preserving mean curvature type flow for star-shaped anisotropic $ω_0$-capillary hypersurfaces in the half-space is studied, and the long-time existence and smooth convergence to a capillary Wulff shape are obtained. If the initial hypersurface is strictly convex, the solution of this flow remains to be strictly convex for all $t>0$ by adopting a new approach applicable to anisotropic capillary setting. In analogy with closed hypersurfaces, if the $ω_0$-capillary Wulff shape is a $θ$-capillary hypersurface with constant contact angle $θ$, the quermassintegrals for anisotropic capillary hypersurfaces match the mixed volume of two $θ$-capillary convex bodies. Thus, generalized quermassintegrals for anisotropic capillary hypersurfaces with general Wulff shapes (i.e., the $ω_0$-capillary Wulff shape has a variable contact angle) can be defined, which satisfy certain monotonicity properties along the flow. As applications, we establish an anisotropic capillary isoperimetric inequality for star-shaped anisotropic capillary hypersurfaces and a family of new Alexandrov-Fenchel inequalities for strictly convex anisotropic capillary hypersurfaces. In particular, we provide a flow's method to derive the Alexandrov-Fenchel inequalities for two $θ$-capillary hypersurfaces, demonstrated in [30] (arXiv:2408.13655) from the view of point in convex geometry.

math.DG↗

arXiv2024

Semantic Convergence: Harmonizing Recommender Systems via Two-Stage Alignment and Behavioral Semantic Tokenization

Large language models (LLMs), endowed with exceptional reasoning capabilities, are adept at discerning profound user interests from historical behaviors, thereby presenting a promising avenue for the advancement of recommendation systems. However, a notable discrepancy persists between the sparse collaborative semantics typically found in recommendation systems and the dense token representations within LLMs. In our study, we propose a novel framework that harmoniously merges traditional recommendation models with the prowess of LLMs. We initiate this integration by transforming ItemIDs into sequences that align semantically with the LLMs space, through the proposed Alignment Tokenization module. Additionally, we design a series of specialized supervised learning tasks aimed at aligning collaborative signals with the subtleties of natural language semantics. To ensure practical applicability, we optimize online inference by pre-caching the top-K results for each user, reducing latency and improving effciency. Extensive experimental evidence indicates that our model markedly improves recall metrics and displays remarkable scalability of recommendation systems.

cs.IR↗

arXiv2024

Locally constrained flows and geometric inequalities in sphere

In this paper, we uncover an intriguing algebra property of an element symmetric polynomial. By this property, we establish the longtime existence and convergence of a locally constrained flow, thereby some families of geometric inequalities in sphere can be derived. Meanwhile, a new family of ``three terms'' geometric inequalities involving two weighted curvature integrals and one quermassintegral are proved. Unlike hyperbolic spaces, we also obtain an inverse weighted geometric inequality in sphere.

math.DG↗

arXiv2024

Anisotropic flows without global terms and dual Orlicz Christoffel-Minkowski type problem

In this paper, we study the long-time existence and asymptotic behavior for a class of anisotropic non-homogeneous curvature flows without global forcing terms. By the stationary solutions of such anisotropic flows, we obtain existence results for a class of dual Orlicz Christoffel-Minkowski type problems, which is equivalent to solve the PDE $G(x,u_K,Du_K)F(D^2u_K+u_KI)=1$ on $\mathbb S^n$ for a convex body $K$, where $D$ is the covariant derivative with respect to the standard metric on $\mathbb S^n$ and $I$ is the unit matrix of order $n$. This result covers many previous known solutions to $L^p$ dual Minkowski problem, $L^p$ dual Christoffel-Minkowski problem, and some dual Orlicz Minkowski problem etc.. Meanwhile, the variational formula of some modified quermassintegrals and the corresponding prescribed area measure problem (Orlicz Christoffel-Minkowski type problem) are considered, and inequalities involving modified quermassintegrals are also derived. As corollary, this gives a partial answer about the general prescribed curvature problem raised in Guan-Ren-Wang (CPAM, 2015).

math.DG↗

arXiv2023

Cross-Camera Human Motion Transfer by Time Series Analysis

With advances in optical sensor technology, heterogeneous camera systems are increasingly used for high-resolution (HR) video acquisition and analysis. However, motion transfer across multiple cameras poses challenges. To address this, we propose an algorithm based on time series analysis that identifies motion seasonality and constructs an additive model to extract transferable patterns. Validated on real-world data, our algorithm demonstrates effectiveness and interpretability. Notably, it improves pose estimation in low-resolution videos by leveraging patterns derived from HR counterparts, enhancing practical utility. Code is available at: https://github.com/IndigoPurple/TSAMT

cs.CV↗

arXiv2023

A flow method for curvature equations

We consider a general curvature equation $F(κ)=G(X,ν(X))$, where $κ$ is the principal curvature of the hypersurface $M$ with position vector $X$. It includes the classical prescribed curvature measures problem and area measures problem. However, Guan-Ren-Wang \cite{GRW} proved that the $C^2$ estimate fails usually for general function $F$. Thus, in this paper, we pose some additional conditions of $G$ to get existence results by a suitably designed parabolic flow. In particular, if $F=σ_{k}^\frac{1}{k}$ for $\forall 1\le k\le n-1$, the existence result has been derived in the famous work \cite{GLL} with $G=ψ(\frac{X}{|X|})\langle X,ν\rangle^{\frac1k}{|X|^{-\frac{n+1}{k}}}$. This result will be generalized to $G=ψ(\frac{X}{|X|})\langle X,ν\rangle^\frac{1-p}{k}|X|^\frac{q-k-1}{k}$ with $p>q$ for arbitrary $k$ by a suitable auxiliary function. The uniqueness of the solutions in some cases is also studied.

math.AP↗

arXiv2022

A class of anisotropic inverse Gauss curvature flows and dual Orlicz Minkowski type problem

In this paper, we study the long-time existence and asymptotic behavior for a class of anisotropic inverse Gauss curvature flows. By the stationary solutions of anisotropic flows, we obtain some new existence results for the dual Orlicz Minkowski type problem and even dual Orlicz Minkowski type problem for smooth measures, which is the most reasonable extension of the $L^p$ dual Minkowski problem from the dual point of view. The results of corresponding $L^p$ versions are $L^p$ dual Minkowski problem for $p>q$; and even $L^p$ dual Minkowski problem for $p>-1$, or $q<1$, or some ranges of $p<0<q$, which contain all existence results for smooth measures up to now except $p=q$ or $q=n+1$ ($L^p$ Minkowski problem).

math.AP↗

arXiv2022

An efficient nonlinear multigrid solver for the simulation of rarefied gas cavity flow

We study efficient simulation of steady state for rarefied gas flow, which is modeled by the Boltzmann equation with BGK-type collision term. A nonlinear multigrid solver is proposed to resolve the efficiency issue by the following approaches. The unified framework of numerical regularized moment method is first adopted to derive the high-quality discretization of the underlying problem. A fast sweeping iteration is introduced to solve the derived discrete problem more efficiently than the usual time-integration scheme on a single level grid. Taking it as the smoother, the nonlinear multigrid solver is then established to significantly improve the convergence rate. The OpenMP-based parallelization is applied in the implementation to further accelerate the computation. Numerical experiments for two lid-driven cavity flows and a bottom-heated cavity flow are carried out to investigate the performance of the resulting nonlinear multigrid solver. All results show the efficiency and robustness of the solver for both first- and second-order spatial discretization.

math.NA↗

arXiv2022

A class of inverse curvature flows and $L^p$ dual Christoffel-Minkowski problem

In this paper, we consider a large class of expanding flows of closed, smooth, star-shaped hypersurface in Euclidean space $\mathbb{R}^{n+1}$ with speed $ψu^αρ^δf^{-β}$, where $ψ$ is a smooth positive function on unit sphere, $u$ is the support function of the hypersurface, $ρ$ is the radial function, $f$ is a smooth, symmetric, homogenous of degree one, positive function of the principal curvatures of the hypersurface on a convex cone. When $ψ=1$, we prove that the flow exists for all time and converges to infinity if $α+δ+β\le1, β>0$ and $α\le0$, while in case $α+δ+β>1,α,δ\le0$, the flow blows up in finite time, and where we assume the initial hypersurface to be strictly convex. In both cases the properly rescaled flows converge to a sphere centered the origin. In particular, the results of Gerhardt \cite{GC,GC3} and Urbas \cite{UJ2} can be recovered by putting $α=δ=0$. Our previous works \cite{DL,DL2} can be recovered by putting $δ=0$. By the convergence of these flows, we can give a new proof of uniqueness theorems for solutions to $L^p$-Minkowski problem and $L^p$-Christoffel-Minkowski problem with constant prescribed data. Similarly, we pose the $L^p$ dual Christoffel-Minkowski problem and prove a uniqueness theorem for solutions to $L^p$ dual Minkowski problem and $L^p$ dual Christoffel-Minkowski problem with constant prescribed data. At last, we focus on the longtime existence and convergence of a class of anisotropic flows (i.e. for general function $ψ$). The final result not only gives a new proof of many previously known solutions to $L^p$ dual Minkowski problem, $L^p$-Christoffel-Minkowski problem, etc. by such anisotropic flows, but also provides solutions to $L^p$ dual Christoffel-Minkowski problem with some conditions.

math.DG↗

arXiv2021

A Class Of Curvature Flows Expanded By Support Function And Curvature Function In The Euclidean Space And Hyperbolic Space

In this paper, we first consider a class of expanding flows of closed, smooth, star-shaped hypersurface in Euclidean space $\mathbb{R}^{n+1}$ with speed $u^αf^{-β}$, where $u$ is the support function of the hypersurface, $f$ is a smooth, symmetric, homogenous of degree one, positive function of the principal curvatures of the hypersurface on a convex cone. For $α\le 0<β\le 1-α$, we prove that the flow has a unique smooth solution for all time, and converges smoothly after normalization, to a sphere centered at the origin. In particular, the results of Gerhardt \cite{GC3} and Urbas \cite{UJ2} can be recovered by putting $α=0$ and $β=1$ in our first result. If the initial hypersurface is convex, this is our previous work \cite{DL}. If $α\le 0<β< 1-α$ and the ambient space is hyperbolic space $\mathbb{H}^{n+1}$, we prove that the flow $\frac{\partial X}{\partial t}=(u^αf^{-β}-ηu)ν$ has a longtime existence and smooth convergence to a coordinate slice. The flow in $\mathbb{H}^{n+1}$ is equivalent (up to an isomorphism) to a re-parametrization of the original flow in $\mathbb{R}^{n+1}$ case. Finally, we find a family of monotone quantities along the flows in $\mathbb{R}^{n+1}$. As applications, we give a new proof of a family of inequalities involving the weighted integral of $k$th elementary symmetric function for $k$-convex, star-shaped hypersurfaces, which is an extension of the quermassintegral inequalities in \cite{GL2}.

math.DG↗

arXiv2020

Zoom in to the details of human-centric videos

Presenting high-resolution (HR) human appearance is always critical for the human-centric videos. However, current imagery equipment can hardly capture HR details all the time. Existing super-resolution algorithms barely mitigate the problem by only considering universal and low-level priors of im-age patches. In contrast, our algorithm is under bias towards the human body super-resolution by taking advantage of high-level prior defined by HR human appearance. Firstly, a motion analysis module extracts inherent motion pattern from the HR reference video to refine the pose estimation of the low-resolution (LR) sequence. Furthermore, a human body reconstruction module maps the HR texture in the reference frames onto a 3D mesh model. Consequently, the input LR videos get super-resolved HR human sequences are generated conditioned on the original LR videos as well as few HR reference frames. Experiments on an existing dataset and real-world data captured by hybrid cameras show that our approach generates superior visual quality of human body compared with the traditional method.

cs.CV↗

arXiv2020

A class of curvature flows expanded by support function and curvature function

In this paper, we consider an expanding flow of closed, smooth, uniformly convex hypersurface in Euclidean \mathbb{R}^{n+1} with speed u^αf^β(α, β\in\mathbb{R}^1), where u is support function of the hypersurface, f is a smooth, symmetric, homogenous of degree one, positive function of the principal curvature radii of the hypersurface. If α\leq 0<β\leq 1-α, we prove that the flow has a unique smooth and uniformly convex solution for all time, and converges smoothly after normalization, to a round sphere centered at the origin.

math.DG↗