arXiv subjects

Boying Wu

Publications and source records attributed to Boying Wu.

At least 19 recordsLinked to original sources

arXiv2026

Hessian-recovery-based C0 finite element methods for non-divergence form elliptic equations

A Hessian-recovery-based C0 finite element framework is proposed for second-order elliptic equations in non-divergence form. The construction is based on a direct approximation of the strong non-divergence operator: the Hessian D2u is replaced by a recovered Hessian Hhuh, so that A : D2u is approximated by A : Hhuh. The resulting discretizations include a nodal formulation and a Galerkin-type formulation for general Lagrange finite element spaces, as well as a biorthogonal Petrov-Galerkin formulation for linear elements. The analysis focuses on the recovered nodal matrix and identifies two verifiable algebraic solvability mechanisms. The first is a globally monotone regime leading to a discrete maximum principle, and the second is a localized Schur-complement criterion for sign-violating rows. A uniform inverse bound and a condition-number estimate are derived in the globally monotone case. Residual consistency estimates are obtained from the Hessian recovery error. In the globally monotone regime, these estimates combine with the uniform inverse bound to give a nodal L-error estimate for the nodal formulation. Numerical experiments with nonsmooth and discontinuous coefficients support the predicted algebraic diagnostics and show the accuracy of the proposed recovered-residual discretizations. A Monge-Ampere type test further illustrates the use of the recovered Hessian in a Newton iteration for a fully nonlinear problem.

math.NA↗

arXiv2026

Latent Geometric Chords for Query-Efficient Decision-Based Adversarial Attacks

While decision-based black-box adversarial attacks present a severe security threat, current methodologies suffer from fundamental limitations. Pixel-wise attacks frequently introduce unnatural, high-frequency visual artifacts, while latent-space frameworks are confined by the limited search space of low-dimensional manifolds and inherent reconstruction flaws. To resolve these limitations, we propose Latent Geometric Chords (LGC) for Query-Efficient Decision-Based Adversarial Attacks alongside a variant, LGC-H. At its core, LGC navigates decision boundaries by executing a curvature-aware geometric search within a compressed semantic manifold. To guarantee high visual fidelity and circumvent dimensionality bottlenecks, we introduce a Residual-based Adversarial Generation (RAG) mechanism. RAG isolates semantic perturbations as geometric chords and superimposes them directly onto the original source image. RAG substantially resolves baseline reconstruction flaws and effectively doubles the permissible search space dimensions. Experimental results demonstrate that LGC achieves robust cross-dataset transferability and substantially outperforms state-of-the-art baselines. Notably, our method, LGC, minimizes perturbation magnitudes while achieving state-of-the-art visual fidelity--with a Structural Similarity Index Measure (SSIM) exceeding 0.99 and a Learned Perceptual Image Patch Similarity (LPIPS) below 0.01 at 5000 queries--and sustaining high attack success rates under stringent perceptual constraints, successfully compromising adversarially trained robust models. The source code is available at: https://github.com/eihmuekhine/Latent-Geometric-Chords.

cs.CV↗

arXiv2025

TA-LSDiff:Topology-Aware Diffusion Guided by a Level Set Energy for Pancreas Segmentation

Pancreas segmentation in medical image processing is a persistent challenge due to its small size, low contrast against adjacent tissues, and significant topological variations. Traditional level set methods drive boundary evolution using gradient flows, often ignoring pointwise topological effects. Conversely, deep learning-based segmentation networks extract rich semantic features but frequently sacrifice structural details. To bridge this gap, we propose a novel model named TA-LSDiff, which combined topology-aware diffusion probabilistic model and level set energy, achieving segmentation without explicit geometric evolution. This energy function guides implicit curve evolution by integrating the input image and deep features through four complementary terms. To further enhance boundary precision, we introduce a pixel-adaptive refinement module that locally modulates the energy function using affinity weighting from neighboring evidence. Ablation studies systematically quantify the contribution of each proposed component. Evaluations on four public pancreas datasets demonstrate that TA-LSDiff achieves state-of-the-art accuracy, outperforming existing methods. These results establish TA-LSDiff as a practical and accurate solution for pancreas segmentation.

cs.CV↗

arXiv2025

Coupling local and nonlocal total variation flow for image despeckling

Nonlocal equations effectively preserve textures but exhibit weak regularization effects in image denoising, whereas local equations offer strong denoising capabilities yet fail to protect textures. To integrate the advantages of both approaches, this paper investigates a coupled local-nonlocal total variation flow for image despeckling. We establish the existence and uniqueness of the weak solution for the proposed equation. Several properties, including the equivalent forms of the weak solution and its asymptotic behavior, are derived. Furthermore, we demonstrate that the weak solutions of the proposed equation converge to the weak solution of the classical total variation flow under kernel rescaling. The importance of coupling is highlighted through comparisons with local and nonlocal models for image despeckling.

math.AP↗

arXiv2025

Incorporating Local H\"older Regularity into PINNs for Solving Elliptic PDEs

In this paper, local H\"older regularization is incorporated into a physics-informed neural networks (PINNs) framework for solving elliptic partial differential equations (PDEs). Motivated by the interior regularity properties of linear elliptic PDEs, a modified loss function is constructed by introducing local H\"older regularization term. To approximate this term effectively, a variable-distance discrete sampling strategy is developed. Error estimates are established to assess the generalization performance of the proposed method. Numerical experiments on a range of elliptic problems demonstrate notable improvements in both prediction accuracy and robustness compared to standard physics-informed neural networks.

math.NA↗

arXiv2025

Progressive Alignment Degradation Learning for Pansharpening

Deep learning-based pansharpening has been shown to effectively generate high-resolution multispectral (HRMS) images. To create supervised ground-truth HRMS images, synthetic data generated using the Wald protocol is commonly employed. This protocol assumes that networks trained on artificial low-resolution data will perform equally well on high-resolution data. However, well-trained models typically exhibit a trade-off in performance between reduced-resolution and full-resolution datasets. In this paper, we delve into the Wald protocol and find that its inaccurate approximation of real-world degradation patterns limits the generalization of deep pansharpening models. To address this issue, we propose the Progressive Alignment Degradation Module (PADM), which uses mutual iteration between two sub-networks, PAlignNet and PDegradeNet, to adaptively learn accurate degradation processes without relying on predefined operators. Building on this, we introduce HFreqdiff, which embeds high-frequency details into a diffusion framework and incorporates CFB and BACM modules for frequency-selective detail extraction and precise reverse process learning. These innovations enable effective integration of high-resolution panchromatic and multispectral images, significantly enhancing spatial sharpness and quality. Experiments and ablation studies demonstrate the proposed method's superior performance compared to state-of-the-art techniques.

cs.CV↗

arXiv2025

A class of thin-film equations with space-time dependent gradient nonlinearity and its application to image sharpening

We introduce a class of thin-film equations with space-time dependent gradient nonlinearity and apply them to image sharpening. By modifying the potential well method, we overcome the challenges arising from variable exponents and the "moving" Nehari manifold, thereby providing a classification of global existence and finite time blow-up of solutions under different initial conditions. Decay rates for global solutions and lifespan estimates for blow-up solutions are also established. Numerical experiments demonstrate the effectiveness of this class of equations as image sharpening models.

math.AP↗

arXiv2025

On the solutions to variable-order fractional p-Laplacian evolution equation with L^1-data

This study investigates Dirichlet boundary condition related to a class of nonlinear parabolic problem with nonnegative $L^1$-data, which has a variable-order fractional $p$-Laplacian operator. The existence and uniqueness of renormalized solutions and entropy solutions to the equation is proved. To address the significant challenges encountered during this process, we use approximation and energy methods. In the process of proving, the well-posedness of weak solutions to the problem has been established initially, while also establishing a comparative result of solutions.

math.AP↗

arXiv2024

A Tunable Despeckling Neural Network Stabilized via Diffusion Equation

The removal of multiplicative Gamma noise is a critical research area in the application of synthetic aperture radar (SAR) imaging, where neural networks serve as a potent tool. However, real-world data often diverges from theoretical models, exhibiting various disturbances, which makes the neural network less effective. Adversarial attacks can be used as a criterion for judging the adaptability of neural networks to real data, since adversarial attacks can find the most extreme perturbations that make neural networks ineffective. In this work, the diffusion equation is designed as a regularization block to provide sufficient regularity to the whole neural network, due to its spontaneous dissipative nature. We propose a tunable, regularized neural network framework that unrolls a shallow denoising neural network block and a diffusion regularity block into a single network for end-to-end training. The linear heat equation, known for its inherent smoothness and low-pass filtering properties, is adopted as the diffusion regularization block. In our model, a single time step hyperparameter governs the smoothness of the outputs and can be adjusted dynamically, significantly enhancing flexibility. The stability and convergence of our model are theoretically proven. Experimental results demonstrate that the proposed model effectively eliminates high-frequency oscillations induced by adversarial attacks. Finally, the proposed model is benchmarked against several state-of-the-art denoising methods on simulated images, adversarial samples, and real SAR images, achieving superior performance in both quantitative and visual evaluations.

cs.CV↗

arXiv2024

Adversarial Transferability in Deep Denoising Models: Theoretical Insights and Robustness Enhancement via Out-of-Distribution Typical Set Sampling

Deep learning-based image denoising models demonstrate remarkable performance, but their lack of robustness analysis remains a significant concern. A major issue is that these models are susceptible to adversarial attacks, where small, carefully crafted perturbations to input data can cause them to fail. Surprisingly, perturbations specifically crafted for one model can easily transfer across various models, including CNNs, Transformers, unfolding models, and plug-and-play models, leading to failures in those models as well. Such high adversarial transferability is not observed in classification models. We analyze the possible underlying reasons behind the high adversarial transferability through a series of hypotheses and validation experiments. By characterizing the manifolds of Gaussian noise and adversarial perturbations using the concept of typical set and the asymptotic equipartition property, we prove that adversarial samples deviate slightly from the typical set of the original input distribution, causing the models to fail. Based on these insights, we propose a novel adversarial defense method: the Out-of-Distribution Typical Set Sampling Training strategy (TS). TS not only significantly enhances the model's robustness but also marginally improves denoising performance compared to the original model.

cs.CV↗

arXiv2024

Superconvergence of the local discontinuous Galerkin method with generalized numerical fluxes for one-dimensional linear time-dependent fourth-order equations

In this paper, we concentrate on the superconvergence of the local discontinuous Galerkin method with generalized numerical fluxes for one-dimensional linear time-dependent fourth-order equations. The adjustable numerical viscosity of the generalized numerical fluxes is beneficial for long time simulations with a slower error growth. By using generalized Gauss--Radau projections and correction functions together with a suitable numerical initial condition, we derive, for polynomials of degree $k$, $(2k+1)$th order superconvergence for the numerical flux and cell averages, $(k+2)$th order superconvergence at generalized Radau points, and $(k+1)$th order for error derivative at generalized Radau points. Moreover, a supercloseness result of order $(k+2)$ is established between the generalized Gauss--Radau projection and the numerical solution. Superconvergence analysis of mixed boundary conditions is also given. Equations with discontinuous initial condition and nonlinear convection term are numerically investigated, illustrating that the conclusions are valid for more general cases.

math.NA↗

arXiv2024

On a nonlinear nonlocal reaction-diffusion system applied to image restoration

This paper deals with a novel nonlinear coupled nonlocal reaction-diffusion system proposed for image restoration, characterized by the advantages of preserving low gray level features and textures.The gray level indicator in the proposed model is regularized using a new method based on porous media type equations, which is suitable for recovering noisy blurred images. The well-posedness, regularity, and other properties of the model are investigated, addressing the lack of theoretical analysis in those existing similar types of models. Numerical experiments conducted on texture and satellite images demonstrate the effectiveness of the proposed model in denoising and deblurring tasks.

math.AP↗

arXiv2024

Variable-order fractional 1-Laplacian diffusion equations for multiplicative noise removal

In this paper, we study a class of fractional $1$-Laplacian diffusion equations with variable orders, proposed as a model for multiplicative noise removal. The existence and uniqueness of the weak solution are proven. To overcome the difficulties in the approximation process,we place particular emphasis on studying the density propertiesof the variable-order fractional Sobolev spaces. Numerical experiments demonstrate that our model exhibits favorable performance across the entire image.

math.AP↗

arXiv2023

PINNsFailureRegion Localization and Refinement through White-box AdversarialAttack

Physics-informed neural networks (PINNs) have shown great promise in solving partial differential equations (PDEs). However, vanilla PINNs often face challenges when solving complex PDEs, especially those involving multi-scale behaviors or solutions with sharp or oscillatory characteristics. To precisely and adaptively locate the critical regions that fail in the solving process we propose a sampling strategy grounded in white-box adversarial attacks, referred to as WbAR. WbAR search for failure regions in the direction of the loss gradient, thus directly locating the most critical positions. WbAR generates adversarial samples in a random walk manner and iteratively refines PINNs to guide the model's focus towards dynamically updated critical regions during training. We implement WbAR to the elliptic equation with multi-scale coefficients, Poisson equation with multi-peak solutions, high-dimensional Poisson equations, and Burgers equation with sharp solutions. The results demonstrate that WbAR can effectively locate and reduce failure regions. Moreover, WbAR is suitable for solving complex PDEs, since locating failure regions through adversarial attacks is independent of the size of failure regions or the complexity of the distribution.

cs.LG↗

arXiv2022

Analysis of the local discontinuous Galerkin method with generalized fluxes for 1D nonlinear convection-diffusion systems

In this paper, we present optimal error estimates of the local discontinuous Galerkin method with generalized numerical fluxes for one-dimensional nonlinear convection-diffusion systems. The upwind-biased flux with adjustable numerical viscosity for the convective term is chosen based on the local characteristic decomposition, which is helpful in resolving discontinuities of degenerate parabolic equations without enforcing any limiting procedure. For the diffusive term, a pair of generalized alternating fluxes are considered. By constructing and analyzing generalized Gauss-Radau projections with respect to different convective or diffusive terms, we derive optimal error estimates for nonlinear convection-diffusion systems with the symmetrizable flux Jacobian and fully nonlinear diffusive problems. Numerical experiments including long time simulations, different boundary conditions and degenerate equations with discontinuous initial data are provided to demonstrate the sharpness of theoretical results.

math.NA↗

arXiv2021

Optimal error estimates for Legendre expansions of singular functions with fractional derivatives of bounded variation

We present a new fractional Taylor formula for singular functions whose Caputo fractional derivatives are of bounded variation. It bridges and ``interpolates" the usual Taylor formulas with two consecutive integer orders. This enables us to obtain an analogous formula for the Legendre expansion coefficient of this type of singular functions, and further derive the optimal (weighted) $L^\infty$-estimates and $L^2$-estimates of the Legendre polynomial approximations. This set of results can enrich the existing theory for $p$ and $hp$ methods for singular problems, and answer some open questions posed in some recent literature.

math.NA↗

arXiv2019

Superconvergence of local discontinuous Galerkin methods with generalized alternating fluxes for 1D linear convection-diffusion equations

This paper investigates superconvergence properties of the local discontinuous Galerkin methods with generalized alternating fluxes for one-dimensional linear convection-diffusion equations. By the technique of constructing some special correction functions, we prove the $(2k+1)$th order superconvergence for the cell averages, and the numerical traces in the discrete $L^2$ norm. In addition, superconvergence of order $k+2$ and $k+1$ are obtained for the error and its derivative at generalized Radau points. All theoretical findings are confirmed by numerical experiments.

math.NA↗

arXiv2019

A Generalized-Jacobi-Function Spectral Method for Space-Time Fractional Reaction-Diffusion Equations with Viscosity Terms

In this work, we study a new spectral Petrov-Galerkin approximation of space-time fractional reaction-diffusion equations with viscosity terms built by Riemann-Liouville fractional-order derivatives. The proposed method is reliant on generalized Jacobi functions (GJFs) for our problems. The contributions are threefold: First, thanks to the theoretical framework of variational problems, the well-posedness of the problem is proved. Second, new GJF-basis functions are established to fit weak solutions, which take full advantages of the global properties of fractional derivatives. Moreover, the basis functions conclude singular terms, in order to solve our problems with given smooth source term. Finally, we get a numerical analysis of error estimates to depend on GJF-basis functions. Numerical experiments confirm the expected convergence. In addition, they are given to show the effect of the viscosity terms in anomalous diffusion.

math.NA↗