arXiv subjects

Zhonghua Qiao

Publications and source records attributed to Zhonghua Qiao.

At least 19 recordsLinked to original sources

arXiv2026

A Thermodynamically Consistent Model for Multicomponent Vesicles

We develop a thermodynamically consistent diffuse-interface model for multicomponent membranes. The proposed model is derived from a coupled free energy functional that incorporates protein-dependent bending elasticity, diffuse surface tension, a volume penalty, and a membrane-associated Ohta--Kawasaki energy. Applying the Onsager variational principle, we derive a coupled $L^2$ gradient flow system and its energy dissipation law. We then construct a stabilized alternating ETD1 scheme and a stabilized alternating ETDRK2 scheme with Strang-type composition (alternating Strang-ETDRK2). To the best of our knowledge, the proposed alternating Strang-ETDRK2 scheme has not previously been developed and analyzed for coupled phase field systems. We further prove the discrete energy dissipation for both schemes under some regularity assumptions on the numerical solutions. Numerical experiments in two and three dimensions validate the discrete energy dissipation law, and present the protein segregation and membrane deformation produced by the proposed model.

math.NA↗

arXiv2026

A new randomized CholeskyQR based on LU decomposition with partial pivoting

CholeskyQR has received considerable attention in recent years for its efficiency and simplicity in computing QR decomposition of the tall-skinny $X \in \mathbb{R}^{m\times n}$ with $m \ge n$ and $\mbox{rank}(X)=n$. Leveraging matrix sketching from randomized linear algebra, randomized CholeskyQR (RCholeskyQR) has been proposed to accelerate the computation by reducing the dimension of the problems. In this work, we propose RCLUPP, a new randomized CholeskyQR-type algorithm based on LU decomposition with partial pivoting (LUP decomposition). By taking LUP decomposition and the thin HouseholderQR on the sketched matrix, RCLUPP significantly improves the applicability and efficiency compared with LU-CholeskyQR2 (LC2). We present a rigorous rounding error analysis of RCLUPP, with a sharper bound of residual compared with those in the existing works. Comparative studies demonstrate that RCLUPP outperforms CholeskyQR2, Shifted CholeskyQR3 (SCholeskyQR3), and LC2 in terms of applicability while maintaining competitive accuracy and efficiency. A variant, RCLUPPr, performs LUP decomposition directly on $X \in \mathbb{R}^{m\times n}$, offering exceptional robustness and numerical stability for the ill-conditioned scenarios, which exceeds that of RCLUPP and RCholeskyQR. Numerical experiments on the synthetic and real-world matrices validate the theoretical results.

math.NA↗

arXiv2026

Finite Element Approximation of Nonlocal Problems with Heterogeneous Localization and Local Boundary Conditions

This paper studies the finite element approximation of a one-dimensional nonlocal Poisson problem with heterogeneous localization and homogeneous local Dirichlet boundary conditions. These local boundary conditions induce localization kernels with spatially varying interaction neighborhoods, which lead to substantial numerical challenges for the assembly of the singular nonlocal stiffness matrix. An asymptotically compatible conforming finite element method is developed for the variational formulation, together with an exact geometric decomposition for the singular stiffness matrix assembly. Under additional smoothness assumptions on the localization profile, second-order operator consistency is established and error estimates are derived with the corresponding convergence orders. Numerical experiments confirm the theoretical convergence behavior and demonstrate the improved boundary behavior of the heterogeneous localization model.

math.NA↗

arXiv2026

An SPH model with physically prescribed parameters for droplet dynamics on complex surfaces

Numerical simulation of droplet dynamics on complex surfaces with varying wettability is of great significance to both engineering applications and fundamental research. However, existing numerical methods still face challenges in accurately capturing interfacial interactions while preserving physical consistency and computational efficiency. In this work, a physically grounded and efficient smoothed particle hydrodynamics (SPH) model is developed for droplet dynamics simulation. To reduce computational cost, a single-phase droplet modeling strategy is employed. At the interface, long-range interactions are approximated using the SPH kernel function, whereas short-range interactions are represented through pressure. Based on this treatment, an explicit relationship between the intermolecular potential energy and the macroscopic surface tension coefficient is further established, thereby reducing reliance on empirical parameter calibration. The proposed method is first validated through static wetting simulations, where the equilibrium contact angles agree well with the Young--Dupré equation. Further simulations of wetting and droplet impact demonstrate that the method is capable of capturing complex dynamic wetting behaviors.

physics.flu-dyn↗

arXiv2026

Exponential Time Differencing Schemes for a Phase-Field Model of Multicomponent Membranes

In this paper, we develop and analyze exponential time differencing (ETD) schemes for a phase-field model of multicomponent membranes proposed in our previous work \cite{luo2025ohta}, in which membrane deformation is governed by a force-balance phase-field equation and protein segregation is described by a membrane-associated Ohta-Kawasaki (OK) dynamics. For a fixed phase-field membrane, we introduce a geometry-adapted operator splitting method based on the localization function, which reformulates the surface OK dynamics into a form suitable for ETD integration. The resulting first- and second-order ETD schemes, combined with finite-difference spatial discretization, are rigorously proved to satisfy a discrete maximum-bound principle and unconditional energy stability. For the coupled system, we construct stabilized ETD schemes in an FFT-based spectral framework, treating stiff linear terms exactly and nonlinear mechanochemical couplings explicitly. A narrow-band implementation further reduces the computational cost by restricting surface calculations to the diffuse membrane region. Numerical experiments confirm the predicted temporal accuracy, maximum-bound preservation, and energy decay for the fixed-membrane OK problem, and demonstrate stable and efficient three-dimensional simulations of protein-driven pattern formation and membrane deformation.

math.NA↗

arXiv2026

An efficient and stable diffusion generated method for quadrilateral mesh generation in general domains

This paper introduces a novel, robust, and computationally efficient framework for high-quality quadrilateral mesh generation on general two-dimensional domains. The core of the proposed approach is a novel method for computing cross fields by minimizing a modified and relaxed Ginzburg--Landau-type energy functional. A key innovation is the extension of the problem from the original, potentially complex domain to a larger regular computational domain. This extension transforms the central computational procedure into an iterative scheme that requires only two straightforward and efficient operations: linear diffusion solved globally via the Fast Fourier Transform (FFT) and point-wise normalization. Notably, our method eliminates the conventional need for generating an intermediate triangular mesh or solving complex nonlinear optimization problems on the irregular domain. We provide a rigorous theoretical analysis, proving that the proposed iterative algorithm guarantees unconditional monotonic decay of the objective functional. Comprehensive numerical experiments demonstrate the method's robustness across a wide range of complex geometries, its significant computational efficiency afforded by the FFT-based diffusion, and its consistent generation of high-quality quadrilateral meshes. This work presents a reliable and theoretically sound alternative to existing mesh generation techniques, with strong potential for practical applications in scientific computing.

math.NA↗

arXiv2026

Adaptive Coordinate Transforms for Neural Operators

Neural operators have achieved promising performance on partial differential equations (PDEs), but most existing models are built on fixed Eulerian coordinates. This mismatch between evolving physical structures and static coordinates creates spatial misalignment, leading to unnecessarily non-local operator mappings and reinforcing a smoothness preference near sharp transitions. Inspired by adaptive coordinate transformations in classical PDE analysis, we propose the Adaptive Coordinate Transform (ACT) block, a plug-and-play module for data-driven geometric adaptation in neural operators. ACT blocks resolve this structural limitation by learning adaptive coordinate systems within the operator learning pipeline. Specifically, given an input feature, the ACT block learns a coordinate transformation and represents the same feature under the transformed coordinates via differentiable sampling. This operation preserves the underlying signal while changing its spatial representation, equivalent to expressing the same physical quantity in different coordinate systems. By adapting the coordinate system to the data, ACT allows the network to better track evolving structures, reduce operator complexity, and dynamically focus on critical features to improve learning. We evaluate the proposed approach across diverse PDE benchmarks and multiple neural operator architectures. Experimental results demonstrate consistent and significant improvements in predictive accuracy, indicating that learning coordinate systems provides a powerful mechanism for enhancing operator learning.

cs.CE↗

arXiv2026

Generalized Transferable Neural Networks for Steady-State Partial Differential Equations

Deep learning has emerged as a compelling framework for scientific and engineering computing, motivating growing interest in neural network-based solvers for partial differential equations (PDEs). Within this landscape, network architectures with deterministic feature construction have become an appealing approach, offering both high accuracy and computational efficiency in practice. Among them, the transferable neural network (TransNet) is a special class of shallow neural networks (i.e., single-hidden-layer architectures), whose hidden-layer parameters are predetermined according to the principle of uniformly distributed partition hyperplanes. Although TransNet has demonstrated strong performance in solving PDEs with relatively smooth solutions, its accuracy and stability may deteriorate in the presence of highly oscillatory solution structures, where activation saturation and system conditioning issues become limiting factors. In this paper, we propose a generalized transferable neural network (GTransNet) for solving steady-state PDEs, which augments the original TransNet design with additional hidden layers while preserving its interpretable feature-generation mechanism. In particular, the first hidden layer of GTransNet retains TransNet's parameter sampling strategy but incorporates an additional symmetry constraint on the neuron biases, while the subsequent hidden layers omit bias terms and employ a variance-controlled sampling strategy for selecting neuron weights.

math.NA↗

arXiv2026

Convergence analysis of SPH method on irregular particle distributions for the Poisson equation

The numerical accuracy of particle-based approximations in Smoothed Particle Hydrodynamics (SPH) is significantly affected by the spatial uniformity of particle distributions, especially for second-order derivatives. This study aims to enhance the accuracy of SPH method and analyze its convergence with irregular particle distributions. By establishing regularity conditions for particle distributions, we ensure that the local truncation error of traditional SPH formulations, including first and second derivatives, achieves second-order accuracy. Our proposed method, the volume reconstruction SPH method, guarantees these regularity conditions while preserving the discrete maximum principle. Benefiting from the discrete maximum principle, we conduct a rigorous global error analysis in the $L^\infty$-norm for the Poisson equation with variable coefficients, achieving second-order convergence. Numerical examples are presented to validate the theoretical findings.

math.NA↗

arXiv2025

An Effective Model for Droplet Impact Dynamics on Micro-Structured Surfaces: Nonlocal Theory and SPH Simulation of Pancake Bouncing

The accurate mathematical modeling of droplet impact dynamics on micro-structured surfaces is fundamental to understanding and predicting complex fluid behaviors relevant to a wide range of engineering and scientific applications. In particular, the pancake bouncing phenomenon--systematically studied by Liu et al. (Nature Physics, 2014)--on superhydrophobic micro-structured substrates presents significant theoretical challenges. Central to these challenges is the need to construct effective mathematical models that capture the intricate influence of substrate micro/nanostructures on droplet dynamics. This requires the development of robust formulations for surface tension, contact line dynamics, and the interaction forces between fluid and solid structures. In this work, we formulate a nonlocal mathematical framework for the simulation of 3D pancake bouncing on superhydrophobic micro-cone arrays. The model incorporates intermolecular attractive forces to represent droplet surface tension, and we provide a strict theoretical derivation linking these forces quantitatively to the macroscopic surface tension coefficient, thereby circumventing the reliance on empirical parameter tuning. The complex geometry of micro-cone arrays introduces fundamental difficulties in defining local normal directions for contact algorithms. To overcome this, we develop a nonlocal contact repulsion force model that governs fluid-solid interactions and ensures numerical stability under high Weber number conditions. Based on this mathematical foundation, we implement the model using smoothed particle hydrodynamics (SPH), enabling high-precision 3D simulations. Computational experiments, validated against empirical data, confirm the model's accuracy and robustness, while underscoring the key role of numerical simulation in elucidating droplet-microstructure interactions.

physics.flu-dyn↗

arXiv2025

Dynamic-stabilization-based linear schemes for the Allen-Cahn equation with degenerate mobility: MBP and energy stability

In this paper, we investigate linear first- and second-order numerical schemes for the Allen--Cahn equation with a general (possibly degenerate) mobility. Compared with existing numerical methods, our schemes employ a novel dynamic stabilization approach that guarantees unconditional preservation of the maximum bound principle (MBP) and energy stability. A key advance is that the discrete energy stability remains valid even in the presence of degenerate mobility-a property we refer to as mobility robustness. Rigorous maximum-norm error estimates are also established. In particular, for the second-order scheme, we introduce a new prediction strategy with a cut-off preprocessing procedure on the extrapolation solution, and only one linear system needs to be solved per time level. Representative numerical examples are provided to validate the theoretical findings and performance of the proposed schemes.

math.NA↗

arXiv2025

A linear unconditionally structure-preserving L1 scheme for the time-fractional Allen-Cahn equation

As a variational phase-field model, the time-fractional Allen-Cahn (TFAC) equation enjoys the maximum bound principle (MBP) and a variational energy dissipation law. In this work, we develop and analyze linear, structure-preserving time-stepping schemes for TFAC, including first-order and $\min\{1+α, 2-α\}$-order L1 discretizations, together with fast implementations based on the sum-of-exponentials (SOE) technique. A central feature of the proposed linear schemes is their unconditional preservation of both the discrete MBP and the variational energy dissipation law on general temporal meshes, including graded meshes commonly used for these problems. Leveraging the MBP of the numerical solutions, we establish sharp error estimates by employing the time-fractional Gronwall inequality. Finally, numerical experiments validate the theoretical results and demonstrate the effectiveness of the proposed schemes with an adaptive time-stepping strategy.

math.NA↗

arXiv2025

Ohta-Kawasaki Model Reveals Patterns on Multicomponent Vesicles

We present a new mechanochemical modeling framework to explore the shape deformation and pattern formation in multicomponent vesicle membranes. In this framework, the shape of the membrane is described by an elastic bending model, while phase separation of membrane-bound activator proteins is determined by an Ohta-Kawasaki (OK) model. The coupled dynamics consist of an overdamped force-balanced equation for the membrane geometry and an OK-type advection-reaction-diffusion equation on the deformable membrane. We implement efficient spectral methods to simulate these dynamics in both two- and three-dimensions. Numerical experiments show that the model successfully reproduces a wide range of experimentally observed membrane morphologies \cite{baumgart2003imaging}. Taken together, the framework unifies curvature mechanics, microphase separation, and active forcing, providing new insight into membrane-bounded multicomponent vesicle dynamics and a practical platform for studying multicomponent biomembrane morphology.

math-ph↗

arXiv2025

IELDG: Suppressing Domain-Specific Noise with Inverse Evolution Layers for Domain Generalized Semantic Segmentation

Domain Generalized Semantic Segmentation (DGSS) focuses on training a model using labeled data from a source domain, with the goal of achieving robust generalization to unseen target domains during inference. A common approach to improve generalization is to augment the source domain with synthetic data generated by diffusion models (DMs). However, the generated images often contain structural or semantic defects due to training imperfections. Training segmentation models with such flawed data can lead to performance degradation and error accumulation. To address this issue, we propose to integrate inverse evolution layers (IELs) into the generative process. IELs are designed to highlight spatial discontinuities and semantic inconsistencies using Laplacian-based priors, enabling more effective filtering of undesirable generative patterns. Based on this mechanism, we introduce IELDM, an enhanced diffusion-based data augmentation framework that can produce higher-quality images. Furthermore, we observe that the defect-suppression capability of IELs can also benefit the segmentation network by suppressing artifact propagation. Based on this insight, we embed IELs into the decoder of the DGSS model and propose IELFormer to strengthen generalization capability in cross-domain scenarios. To further strengthen the model's semantic consistency across scales, IELFormer incorporates a multi-scale frequency fusion (MFF) module, which performs frequency-domain analysis to achieve structured integration of multi-resolution features, thereby improving cross-scale coherence. Extensive experiments on benchmark datasets demonstrate that our approach achieves superior generalization performance compared to existing methods.

cs.CV↗

arXiv2025

A Structure-Preserving Framework for Solving Parabolic Partial Differential Equations with Neural Networks

Solving partial differential equations (PDEs) with neural networks (NNs) has shown great potential in various scientific and engineering fields. However, most existing NN solvers mainly focus on satisfying the given PDE formulas in the strong or weak sense, without explicitly considering some intrinsic physical properties, such as mass and momentum conservation, or energy dissipation. This limitation may result in nonphysical or unstable numerical solutions, particularly in long-term simulations. To address this issue, we propose ``Sidecar'', a novel framework that enhances the physical consistency of existing NN solvers for solving parabolic PDEs. Inspired by the time-dependent spectral renormalization approach, our Sidecar framework introduces a small network as a copilot, guiding the primary function-learning NN solver to respect the structure-preserving properties. Our framework is highly flexible, allowing the preservation of various physical quantities for different PDEs to be incorporated into a wide range of NN solvers. Experimental results on some benchmark problems demonstrate significant improvements brought by the proposed framework to both accuracy and structure preservation of existing NN solvers.

cs.LG↗

arXiv2025

Overcoming logarithmic singularities in the Cahn-Hilliard equation with Flory-Huggins potential: An unconditionally convergent ADMM approach

The Cahn-Hilliard equation with Flory-Huggins potential serves as a fundamental phase field model for describing phase separation phenomena. Due to the presence of logarithmic singularities at $u=\pm 1$, the solution $u$ is constrained within the interval $(-1,1)$. While convex splitting schemes are commonly employed to preserve this bound and guarantee unconditional unique solvability, their practical implementation requires solving nonlinear systems containing singular logarithmic terms at each time step. This introduces significant challenges in both ensuring convergence of iterative solvers and maintaining the solution bounds throughout the iterations. Existing solvers often rely on restrictive conditions -- such as the strict separation property or small time step sizes -- to ensure convergence, which can limit their applicability. In this work, we introduce a novel iterative solver that is specifically designed for singular nonlinear systems, with the use of a variant of the alternating direction method of multipliers (ADMM). By developing a tailored variable splitting strategy within the ADMM framework, our method efficiently decouples the challenging logarithmic nonlinearity, enabling effective handling of singularities. Crucially, we rigorously prove the unconditional convergence of our ADMM-based solver, which removes the need for time step constraints or strict separation conditions. This allows us to fully leverage the unconditional solvability offered by convex splitting schemes. Comprehensive numerical experiments demonstrate the superior efficiency and robustness of our ADMM variant, strongly validating both our algorithmic design and theoretical results.

math.NA↗

arXiv2025

Intrinsic local Gauss's law preserving PIC method: A self-consistent field-particle update scheme for plasma simulations

In order to perform physically faithful particle-in-cell (PIC) simulations, the Gauss's law stands as a critical requirement, since its violation often leads to catastrophic errors in long-term plasma simulations. This work proposes a novel method that intrinsically enforces the Gauss's law for the Vlasov-Ampère/Vlasov-Poisson system without requiring auxiliary field corrections or specialized current deposition techniques. The electric field is managed to get updated locally and consistently with the motion of particles via splitting the motion into sub-steps along each dimension of the computational mesh. To further obtain a curl-free electric field, a local update scheme is developed to relax the electric-field free energy subject to the Gauss's law. The proposed method avoids solving the Poisson's or Ampère's equation, resulting in a local algorithm of linear complexity for each time step which can be flexibly combined with various temporal discretization for particle motion in PIC simulations. Theoretical analysis verifies that the proposed method indeed maintains the discrete Gauss's law exactly. Numerical tests on classical benchmarks, including the Landau damping, two-stream instability and Diocotron instability, demonstrate the key advantages of the proposed method. It is expected that the local nature of the proposed method makes it a promising tool in parallel simulations of large-scale plasmas.

physics.plasm-ph↗

arXiv2025

An improved Shifted CholeskyQR based on columns

Among all the deterministic CholeskyQR-type algorithms, Shifted CholeskyQR3 is specifically designed to address the QR factorization of ill-conditioned matrices. This algorithm introduces a shift parameter $s$ to prevent failure during the initial Cholesky factorization step, making the choice of this parameter critical for the algorithm's effectiveness. Our goal is to identify a smaller $s$ compared to the traditional selection based on $\norm{X}_{2}$. In this research, we propose a new definition for the input matrix $X$ called $[X]_{g}$, which is based on the column properties of $X$. $[X]_{g}$ allows us to obtain a reduced shift parameter $s$ for the Shifted CholeskyQR3 algorithm, thereby improving the sufficient condition of $κ_{2}(X)$ for this method. We provide rigorous proofs of orthogonality and residuals for the improved algorithm using our proposed $s$. Numerical experiments confirm the enhanced numerical stability of orthogonality and residuals with the reduced $s$. We find that Shifted CholeskyQR3 can effectively handle ill-conditioned $X$ with a larger $κ_{2}(X)$ when using our reduced $s$ compared to the original $s$. Furthermore, we compare CPU times with other algorithms to assess performance improvements.

math.NA↗