arXiv subjects

Zhimin Zhang

Publications and source records attributed to Zhimin Zhang.

At least 19 recordsLinked to original sources

arXiv2026

Natural superconvergence points and asymptotic expansions for spline finite elements in one dimension

We study the natural superconvergence points and asymptotic expansions of one-dimensional spline finite element approximations. For a spline space of degree $k$ and any smoothness $0\le\mu\le k-1$, we prove that the $s$-th derivative of the error exhibits enhanced convergence of order $O(h^{k+2-s})$ at points where $k-s$ is even, provided the mesh is symmetric within a region of size $Ch|\ln h|$ around the point. This condition is known to be optimal for the cases of low derivative order $s=0,1$; the present analysis shows that the same local condition is sufficient for all admissible $s$. Moreover, by expanding the error in Legendre polynomials, a closure theorem determines the leading-order Legendre coefficients (the asymptotic expansion of the error) by combining the Galerkin orthogonality with the superconvergence conditions. For $\mu=k-1$ (B-splines) and $\mu=k-2$, the Galerkin orthogonality conditions vanish and the coefficients are determined solely by the superconvergence conditions. The asymptotic expansion can be expressed through a simple antiderivative recurrence on Legendre polynomials. The resulting polynomial's zeros encode the complete set of superconvergence points for all derivative orders. Numerical experiments for selected $(k,\mu)$ pairs confirm the theoretical predictions.

math.NA↗

arXiv2026

Motion Beyond Morphology: Bootstrapping Cross-Category Motion Transfer from Abstract Motion Representations

Video motion transfer aims to animate a target object using dynamics from a reference video. Existing formulations largely rely on fixed structural correspondence, which becomes ill-defined when reference and target objects differ substantially in morphology, articulation, or deformation mechanisms. We introduce Motion Beyond Morphology, a perspective that seeks to transfer motion beyond fixed structural correspondence, by preserving dynamics that remain meaningful across different target morphologies. To realize this, we propose a two-stage framework. Stage~I learns complementary multi-granularity abstract motion views and uses them to bootstrap cross-category video pairs that preserve transferable dynamics across diverse morphologies. Stage~II internalizes this supervision into direct reference-video-conditioned generation, removing the need for explicit motion extraction at inference. We further introduce OpenVMT-Dataset and OpenVMT-Bench for training and evaluating image- and text-conditioned motion transfer across Same, Near, and Far category gaps. Extensive experiments demonstrate state-of-the-art motion fidelity and target preservation. Project page: https://miniz233.github.io/MotionBeyondMorphology/

cs.CV↗

arXiv2026

Artificial Intelligence and Innovation Ecosystem: Evolutionary Developments, Challenges, and Future Directions

The development of the Innovative Ecosystem (IE) presents a new paradigm for economic integration, collaborative advancement, and shared achievements. The rise of Artificial Intelligence (AI) has significantly accelerated the global processes of digitization, informatization, and intelligence. Exploring how AI can leverage inherent characteristics to influence the development trajectory of IE is a topic that warrants further investigation. Given AI's increasing prominence and role within IE, the paper analyzes this new form, examining both AI's unique contributions to IE and its potential challenges. Firstly, the paper synthesizes the conceptual frameworks surrounding IE, decomposing them into manifestations in physical, social, and thinking spaces. Furthermore, the concept of Artificial Intelligence IE (AIIE) is introduced from a spatial perspective, with an exploration of the characteristics AI contributes to IE. Subsequently, the paper employs an evolutionary perspective to analyze the roles provided by AI during different development periods of AIIE. The paper then verifies the feasibility, effectiveness, and rationality of the AIIE's definition and analyzes AIIE development from an evolutionary perspective using enterprise development examples. Finally, acknowledging AI's inherent limitations, the paper examines potential challenges facing AIIE in the future from four perspectives, aiming to identify new research avenues for the further development of AIIE.

cs.AI↗

arXiv2026

OmniDirector: General Multi-Shot Camera Cloning without Cross-Paired Data

Cloning camera motion from reference videos is an important task in video generation, as videos provide intuitive and precise control. Existing methods either directly use parametric representations that fail to handle multi-shot generation or synthesize cross-paired data, which suffer from data scarcity, resulting in poor performance in complicated camera motion cloning. To address these issues, we introduce a general camera motion representation that encodes cameras as grid motion videos. This camera grid represents the camera parameters visually and supports the integration of diverse trajectories for multi-shot video generation. Building upon this, we propose OmniDirector, a unified framework trained on a million-scale camera grid-video pairs that coordinates characters, actions, and cameras to provide director-level control for multimodal diffusion transformers. Furthermore, we design a novel hierarchical prompt expansion agent that harmoniously integrates different control signals by systematically describing camera motion and visual content through understanding signal relationships. Extensive experiments demonstrate the superior performance and outstanding controllability of our framework. Project page: https://ymlinfeng.github.io/OmniDirector.github.io/

cs.CV↗

arXiv2026

A Decoupled Low-Order Conforming Mixed Finite Element Method for a Three-Dimensional Fourth-Order Singularly Perturbed Problem

This paper develops a decoupled low-order conforming finite element method for a fourth-order elliptic singular perturbation problem in three dimensions. By means of a generalized Helmholtz decomposition, the problem is reduced to two second-order elliptic problems and a system of generalized singularly perturbed Stokes-type equations subject to a curl-free constraint. The former are discretized by standard linear finite elements. For the latter, we employ the MINI element and show that, after adding an $L^2$ term involving a Lagrange multiplier, the resulting discretization becomes robust with respect to the perturbation parameter. We further establish an error estimate of order $h^{1/2}$ uniform with respect to the perturbation parameter. Numerical experiments are included to support the theory.

math.NA↗

arXiv2026

Optimal convergence of local discontinuous Galerkin methods for convection-diffusion equations

The $hp$ local discontinuous Galerkin (LDG) method proposed by Castillo et al. [Math. Comp.,~71 (238): 455-478, 2002] has been shown to be an efficient approach for solving convection-diffusion equations. However, theoretical analysis indicates that, for solutions with limited spatial regularity, the method exhibits suboptimal convergence in $p$, suffering a loss of one order, comparing to numerical experiments. The purpose of this paper is to close the gap between theoretical estimates and numerical evidence. This is accomplished by establishing new approximation results for the associated Gauss-Radau projections of functions in suitable function spaces that can optimally characterize the regularity of singular solutions. We show that such a framework arises aturally and enables the study of various types of singular solutions, with full consistency between theoretical analysis and numerical results. This investigation sheds light on the resolution of the suboptimality in $p$ observed in the literature for several other types of DG schemes in different settings.

math.NA↗

arXiv2026

Natural superconvergence points for splines

This paper develops a unified theory of natural superconvergence points for polynomial spline approximations to second-order elliptic problems. Beginning with the one-dimensional case, we establish that when a point $x_0$ is a local symmetric center of the partition, the numerical error $(u-u_h)^{(s)}(x_0)$ exhibits superconvergence whenever the polynomial degree $k$ and the derivative order $s$ share the same parity. In particular, for the smoothest spline (B-spline) solution, the abundance of superconvergence points allows us to construct asymptotic expansion of the error within the element that fully characterize all superconvergence points, for both function values and derivatives. The theoretical framework is then extended to higher-dimensional settings on simplicial and tensor-product meshes, and the essential conclusions are preserved, with one-dimensional derivatives generalized to mixed derivatives. Numerical experiments demonstrate that superconvergence persists even in extremely localized symmetric regions, revealing that superconvergence points are both readily attainable and follow systematic distribution patterns.

math.NA↗

arXiv2026

Element-based B-spline basis function spaces: construction and application in isogeometric analysis

This paper develops a unified theoretical framework for constructing B-spline basis function spaces with structural equivalence to finite element spaces. The theory rigorously establishes that these bases emerge as explicit linear combinations of B-spline element bases. For any prescribed smoothness requirements, this element-wise formulation enables the Hermite interpolation at nodes, which directly utilizes function values and derivatives without solving global linear systems. By focusing on explicit interpolation properties, element-wise analysis establishes optimal approximation errors, even when the space smoothness attains its theoretical maximum for the space degree. In isogeometric analysis (IgA), the construction naturally decomposes geometric mappings into element-level representations, allowing efficient computations across elements regardless of node distribution. Notably, the same Hermite interpolation framework simultaneously handles domain parameterization and IgA solutions, allowing direct imposition of boundary conditions through function and derivative matching. Numerical tests demonstrate optimal convergence rates and superconvergence properties in 2D IgA under uniform knot configurations, and improved computational efficiency in 3D IgA with non-uniform knot distributions.

math.NA↗

arXiv2025

Novel superconvergence and ultraconvergence structures for the finite volume element method

This paper develops novel natural superconvergence and ultraconvergence structures for the bi-$k$-order finite volume element (FVE) method on rectangular meshes. These structures furnish tunable and possibly asymmetric superconvergence and ultraconvergence points. We achieve one-order-higher superconvergence for both derivatives and function values, and two-orders-higher ultraconvergence for derivatives--a phenomenon that standard bi-$k$-order finite elements do not exhibit. Derivative ultraconvergence requires three conditions: a diagonal diffusion tensor, zero convection coefficients, and the FVE scheme satisfying tensorial $k$-$k$-order orthogonality (imposed via dual mesh constraints). This two-dimensional derivative ultraconvergence is not a trivial tensor-product extension of the one-dimensional phenomena; its analysis is also considerably more complex due to directional coupling. Theoretically, we introduce the asymmetric-enabled M-decompositions (AMD-Super and AMD-Ultra) to rigorously prove these phenomena. Numerical experiments confirm the theory.

math.NA↗

arXiv2025

Polynomial preserving recovery for PHT-splines

We propose a polynomial preserving recovery method for PHT-splines within isogeometric analysis to obtain more accurate gradient approximations. The method fully exploits the local interpolation properties of PHT-splines and avoids the need for information on gradient superconvergent points. By leveraging the superconvergence argument of difference quotients and the interior error estimate, we establish the superconvergence property of the recovered gradient on translation invariant meshes. As a byproduct, a recovery-based a posteriori error estimator is developed for adaptive refinement. Numerical results confirm the theoretical findings and demonstrate the effectiveness of the proposed method.

math.NA↗

arXiv2025

Superconvergence points of Hermite spectral interpolation

Hermite spectral method plays an important role in the numerical simulation of various partial differential equations (PDEs) on unbounded domains. In this work, we study the superconvergence properties of Hermite spectral interpolation, i.e., interpolation at the zeros of Hermite polynomials in the space spanned by Hermite functions. We identify the points at which the convergence rates of the first- and second-order derivatives of the interpolant converge faster. We further extend the analysis to the Hermite spectral collocation method in solving differential equations and identify the superconvergence points both for function and derivative values. Numerical examples are provided to confirm the analysis of superconvergence points.

math.NA↗

arXiv2025

Behave Your Motion: Habit-preserved Cross-category Animal Motion Transfer

Animal motion embodies species-specific behavioral habits, making the transfer of motion across categories a critical yet complex task for applications in animation and virtual reality. Existing motion transfer methods, primarily focused on human motion, emphasize skeletal alignment (motion retargeting) or stylistic consistency (motion style transfer), often neglecting the preservation of distinct habitual behaviors in animals. To bridge this gap, we propose a novel habit-preserved motion transfer framework for cross-category animal motion. Built upon a generative framework, our model introduces a habit-preservation module with category-specific habit encoder, allowing it to learn motion priors that capture distinctive habitual characteristics. Furthermore, we integrate a large language model (LLM) to facilitate the motion transfer to previously unobserved species. To evaluate the effectiveness of our approach, we introduce the DeformingThings4D-skl dataset, a quadruped dataset with skeletal bindings, and conduct extensive experiments and quantitative analyses, which validate the superiority of our proposed model.

cs.CV↗

arXiv2025

Monolayer Two-dimensional Materials Database (ML2DDB) and Applications

The discovery of two-dimensional (2D) materials with tailored properties is critical to meet the increasing demands of high-performance applications across flexible electronics, optoelectronics, catalysis, and energy storage. However, current 2D material databases are constrained by limited scale and compositional diversity. In this study, we introduce a scalable active learning workflow that integrates deep neural networks with density functional theory (DFT) calculations to efficiently explore a vast set of candidate structures. These structures are generated through physics-informed elemental substitution strategies, enabling broad and systematic discovery of stable 2D materials. Through six iterative screening cycles, we established the creation of the Monolayer 2D Materials Database (ML2DDB), which contains 242,546 DFT-validated stable structures-an order-of-magnitude increase over the largest known 2D materials databases. In particular, the number of ternary and quaternary compounds showed the most significant increase. Combining this database with a generative diffusion model, we demonstrated effective structure generation under specified chemistry and symmetry constraints. This work accomplished an organically interconnected loop of 2D material data expansion and application, which provides a new paradigm for the discovery of new materials.

cond-mat.mtrl-sci↗

arXiv2025

Fast Maxwell Solvers Based on Exact Discrete Eigen-Decompositions I. Two-Dimensional Case

In this paper, we propose fast solvers for Maxwell's equations in rectangular domains. We first discretize the simplified Maxwell's eigenvalue problems by employing the lowest-order rectangular Nédélec elements and derive the discrete eigen-solutions explicitly, providing a Hodge-Helmholtz decomposition framework at the discrete level. Based on exact eigen-decompositions, we further design fast solvers for various Maxwell's source problems, guaranteeing either the divergence-free constraint or the Gauss's law at the discrete level. With the help of fast sine/cosine transforms, the computational time grows asymptotically as $\mathcal{O}(n^2\log n)$ with $n$ being the number of grids in each direction. Our fast Maxwell solvers outperform other existing Maxwell solvers in the literature and fully rival fast scalar Poisson/Helmholtz solvers based on trigonometric transforms in either efficiency, robustness, or storage complexity. It is also utilized to perform an efficient pre-conditioning for solving Maxwell's source problems with variable coefficients. Finally, numerical experiments are carried out to illustrate the effectiveness and efficiency of the proposed fast solver.

math.NA↗

arXiv2025

Spectral Method for 1-D Neutron Transport Equation

In this paper, we present an efficient fully spectral approximation scheme for exploring the one-dimensional steady-state neutron transport equation. Our methodology integrates the spectral-(Petrov-)Galerkin scheme in the spatial dimension with the Legendre-Gauss collocation scheme in the directional dimension. The directional integral in the original problem is discretized with Legendre-Gauss quadrature. We furnish a rigorous proof of the solvability of this scheme and, to our best knowledge, conduct a comprehensive error analysis for the first time. Notably, the order of convergence is optimal in the directional dimension, while in the spatial dimension, it is suboptimal and, importantly, non-improvable. Finally, we verify the computational efficiency and error characteristics of the scheme through several numerical examples.

math.NA↗

arXiv2025

YingLong-weather: AI-Based Limited Area Models for Forecasting of Non-precipitation Surface Meteorological Variables

Recently, artificial intelligence-based (AI-based) models for forecasting of global weather have been rapidly developed. Most of the global models are trained on reanalysis datasets with a spatial resolution of 0.25°*0.25°. However, research on AI-based high spatial resolution limited area weather forecasting models remains limited. In this study, YingLong, an AI-based limited area weather forecasting model with a spatial resolution of 3 km * 3 km is developed. YingLong employs a parallel structure of global and local blocks to capture multiscale meteorological features and operates much faster than the dynamical limited area model WRF-ARW. In two selected limited areas (one relatively flat and the other featuring significant mountain ranges), YingLong (with lateral boundary condition imposed by the global AI-based model Pangu-weather) demonstrates superior skill in forecasting surface wind speed compared to WRF-ARW. Additionally, it shows comparable skill in forecasting surface temperature and pressure. The accuracy of surface temperature and humidity forecasts can be further improved by applying better boundary conditions. YingLong also addresses issues related to the lateral boundary conditions of AI-based limited area models, such as selecting the width of the lateral boundary region and combining finer and coarser resolution predictions in this region. Therefore, YingLong has a great potential to generate cost-effective multiyear high-resolution synthetic wind speed that maintain meteorological realism both spatially and temporally, aiding in the planning and operations for wind power generation companies.

physics.ao-ph↗

arXiv2025

Parallel ADMM Algorithm with Gaussian Back Substitution for High-Dimensional Quantile Regression and Classification

In the field of high-dimensional data analysis, modeling methods based on quantile loss function are highly regarded due to their ability to provide a comprehensive statistical perspective and effective handling of heterogeneous data. In recent years, many studies have focused on using the parallel alternating direction method of multipliers (P-ADMM) to solve high-dimensional quantile regression and classification problems. One efficient strategy is to reformulate the quantile loss function by introducing slack variables. However, this reformulation introduces a theoretical challenge: even when the regularization term is convex, the convergence of the algorithm cannot be guaranteed. To address this challenge, this paper proposes the Gaussian Back-Substitution strategy, which requires only a simple and effective correction step that can be easily integrated into existing parallel algorithm frameworks, achieving a linear convergence rate. Furthermore, this paper extends the parallel algorithm to handle some novel quantile loss classification models. Numerical simulations demonstrate that the proposed modified P-ADMM algorithm exhibits excellent performance in terms of reliability and efficiency.

stat.CO↗

arXiv2024

Recovery Techniques for Finite Element Methods

Post-processing techniques are essential tools for enhancing the accuracy of finite element approximations and achieving superconvergence. Among these, recovery techniques stand out as vital methods, playing significant roles in both post-processing and pre-processing. This paper provides an overview of recent developments in recovery techniques and their applications in adaptive computations. The discussion encompasses both gradient recovery and Hessian recovery methods. To establish the superconvergence properties of these techniques, two theoretical frameworks are introduced. Applications of these methods are demonstrated in constructing asymptotically exact {\it a posteriori} error estimators for second-order elliptic equations, fourth-order elliptic equations, and interface problems. Numerical experiments are performed to evaluate the asymptotic exactness of recovery type a posteriori error estimators.

math.NA↗