Special Session 46: Theory, Numerical methods, and Applications of Partial Differential Equations

Variational method and its application in medical image registration

Han Huan Wuhan University of Technology Peoples Rep of China Co-Author(s):    Ke Chen, Peng Chen, Zhengping Wang, Daoping Zhang, Yimin Zhang

Abstract: In this talk, I will introduce a class of variational problem in medical image registration. Focus on this variational problem, some recent results on addressing the challenging problems (i.e., mesh folding, large deformation, greedy matching and intensity inhomogeneity ) will be introduced. Furthermore, numerical results will also be showed to validate the theoretical results.

ON THE SECOND BOUNDARY VALUE PROBLEM FOR MEAN CURVATURE FLOW

Rongli Huang Guangxi Normal University Peoples Rep of China Co-Author(s):

Abstract: This is a sequel to our previous work, which study the second boundary value problems for mean curvature flow. We aim to construct convex spacelike translating solitons with prescribed Gauss image over any strictly convex domains by making use of the mean curvature flow. We will exhitbit the asymptotic behavior of the mean curvature flow by doing a series of prior estimates of the solution. Consequently, we obtain the translating solitons with prescribed Gauss image in Minkowski space and Euclidean space.

Total Curvature-Driven Blind Image Deblurring

Qiyu Jin Inner Mongolia University Peoples Rep of China Co-Author(s):    Caiying Wu, Lulu Zhang, Tingting Zhang, Jiawei Lu, Guoqing Chen, Jun Liu, Tieyong Zeng

Abstract: Blind image deblurring is an inherently ill-posed problem, requiring the estimation of both blur kernel and the original image from a single blurred image. To achieve accurate estimation, prior knowledge is crucial. In this paper, we introduce the total curvature on image surface regularization prior, utilizing the image`s geometric features. This prior preserves sharp edges in the intermediate latent image and enhances the restoration of the blur kernel. We then propose a total curvature weighted image surface minimization model. The strong enhancement of edge preservation by total curvature allows for replacing $L_{0}$ norm with $L_p$ norm, ensuring sparsity in our model. This not only enhances our model`s performance but also improves its mathematical properties, enabling us to demonstrate its theoretical convergence. Furthermore, we incorporate inertial technology to enhance the numerical results of our algorithms. Extensive experiments demonstrate the superior performance of our method in diverse image deblurring scenarios compared to state-of-the-art methods. Notably, our method also extends its capabilities to non-uniform deblurring problems, showcasing its versatility and effectiveness in practical settings.

Superconvergence of the local discontinuous Galerkin method with generalized numerical fluxes for fourth-order equations

Linhui Li Harbin Institute of Technology Peoples Rep of China Co-Author(s):    Xiong Meng, Boying Wu

Abstract: In this talk, 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 Navier boundary conditions, Dirichlet boundary conditions, discontinuous initial condition and nonlinear convection term are numerically investigated, illustrating that the conclusions are valid for more general cases.

Robust Image Denoising through Out-of-Distribution Typical Set Sampling

Yao Li Harbin Institute of Technology Peoples Rep of China Co-Author(s):    Jie Ning, Jiebao Sun, Shengzhu Shi, Zhichang Guo, and Yao Li

Abstract: 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\&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 concepts of the 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.

A Fast Minimization Algorithm for the Euler Elastica Model Based on a Bilinear Decomposition

Zhifang Liu Tianjin Normal University Peoples Rep of China Co-Author(s):    Baochen Sun, Xue-Cheng Tai, Qi Wang, and Huibin Chang.

Abstract: Euler elastica (EE), as a regulariser for the curvature and length of the image surface`s level lines, can effectively suppress the staircase artifacts of traditional regulariser and has attracted lots of attention in image processing. However, developing fast and stable algorithms for optimizing the EE energy is a great challenge due to its nonconvexity, strong nonlinearity, and singularity. This talk will present a novel, fast, globally convergent hybrid alternating minimization method (HALM) algorithm for the Euler elastica model based on a bilinear decomposition. The HALM algorithm comprises three sub-minimization problems, and each is either solved in the closed form or approximated by fast solvers, making the new algorithm highly accurate and efficient. Numerical experiments show that the new algorithm produces good results with much-improved efficiency compared to other state-of-the-art algorithms for the EE model. This work is joint with Baochen Sun, Xue-Cheng Tai, Qi Wang, and Huibin Chang.

Continuum Limit of Hypergraph p-Laplacians on Point Clouds

Kehan Shi China Jiliang University Peoples Rep of China Co-Author(s):

Abstract: As a generalization of graphs, hypergraphs are widely used to model higher-order relations in data. In this talk, we discuss the benefit of the hypergraph structure for handling point cloud data that contain no explicit structural information. To this end, we define both oriented and unoriented hypergraphs from point clouds and study the continuum limit of the associated hypergraph $p$-Laplacians when the number of data points goes to infinity. The application of our hypergraph models for semi-supervised learning is also discussed.

The second fundamental form: an effective regularizer for multiplicative noise removal

Shengzhu Shi Harbin Institute of Technology Peoples Rep of China Co-Author(s):

Abstract: Incorporating appropriate geometric priors into a variational denoising model has shown superiority in noise elimination while preserving important geometric features of the image, such as contrasts, corners, and edges. In this paper, we propose an effective variational model that utilizes the second fundamental form as the regularizer for multiplicative noise removal.

A class of positive-preserving, energy stable and high order numerical schemes for the Poission-Nernst-Planck system

Minqiang Xu Zhejiang University of Technology Peoples Rep of China Co-Author(s):    Waixiang Cao,Yuzhe Qing

Abstract: In this paper, we present a class of efficient, positive-preserving, energy stable and high order numerical schemes are presented and studied for solving the time-dependent Poisson-Nernst-Planck (PNP) system. The numerical scheme is based on the energy variational formulation and the PNP system is reformulated as a non-constant mobility $H^1$ gradient flow, with singular logarithmic energy potentials involved. The fully discrete numerical scheme is constructed by using the first/second order semi-implicit time discretization coupled with the $k$-th order direct discontinuous Galerkin (DDG) method or the finite element (FE) method for space discretization. The scheme is shown to be positivity preserving and energy stable. Furthermore, optimal error estimates and some superconvergence results are established for the fully-discrete numerical solution. Numerical experiments are provided to demonstrate the accuracy, efficiency, and robustness of the proposed scheme.

Spectral method based fractional physics-informed neural networks for solving tempered fractional partial differential equations

Tianxin Zhang Harbin Institute of Technology Peoples Rep of China Co-Author(s):    Dazhi Zhang

Abstract: Physics-informed Neural Networks (PINNs) have emerged as a popular method for solving both forward and inverse differential equations. However, the automatic differentiation techniques employed by PINNs face challenges when solving fractional-order equations. To address this issue, we propose the spectral method based fractional PINNs, termed spectral-fPINNs. This method adopts a more efficient global discretization approach based on Jacobi polynomials, which reduces the need for auxiliary points. Meanwhile, the transformation between physical value and expansion coefficients is computed efficiently by a standard matrix-vector multiplication, thereby increasing the efficiency of the algorithm. The performance of spectral-fPINNs is validated via several examples. We first consider the accuracy, stability and efficiency of our method for solving steady-state fractional partial differential equations. We also analyze the errors under different parameters. Subsequently, experiments are conducted on more complex time-dependent equations. Additionally, an application of the tempered equations on finance and the inverse problems are presented. These results demonstrate the advantages of spectral-fPINNs in terms of efficiency in solving tempered fractional partial differential equations.