Shi Shu

Hui Zhu, Shi Shu, Jianping Zhang

Solving variational image segmentation problems with hidden physics is often expensive and requires different algorithms and manually tunes model parameter. The deep learning methods based on the U-Net structure have obtained outstanding performances in many different medical image segmentation tasks, but designing such networks requires a lot of parameters and training data, not always available for practical problems. In this paper, inspired by traditional multi-phase convexity Mumford-Shah variational model and full approximation scheme (FAS) solving the nonlinear systems, we propose a novel variational-model-informed network (denoted as FAS-Unet) that exploits the model and algorithm priors to extract the multi-scale features. The proposed model-informed network integrates image data and mathematical models, and implements them through learning a few convolution kernels. Based on the variational theory and FAS algorithm, we first design a feature extraction sub-network (FAS-Solution module) to solve the model-driven nonlinear systems, where a skip-connection is employed to fuse the multi-scale features. Secondly, we further design a convolution block to fuse the extracted features from the previous stage, resulting in the final segmentation possibility. Experimental results on three different medical image segmentation tasks show that the proposed FAS-Unet is very competitive with other state-of-the-art methods in qualitative, quantitative and model complexity evaluations. Moreover, it may also be possible to train specialized network architectures that automatically satisfy some of the mathematical and physical laws in other image problems for better accuracy, faster training and improved generalization.The code is available at \url{https://github.com/zhuhui100/FASUNet}.

Ruigang Shen, Shi Shu, Ying Yang et al.

We study a two-grid strategy for decoupling the time-dependent Poisson-Nernst-Planck equations describing the mass concentration of ions and the electrostatic potential. The computational system is decoupled to smaller systems by using coarse space solutions at each time level, which can speed up the solution process compared with the finite element method combined with the Gummel iteration. We derive the optimal error estimates in $L^2$ norm for both semi- and fully discrete finite element approximations. Based on the a priori error estimates, the error estimates in $H^1$ norm are presented for the two-grid algorithm. The theoretical results indicate this decoupling method can retain the same accuracy as the finite element method. Numerical experiments including the Poisson-Nernst-Planck equations for an ion channel show the efficiency and effectiveness of the decoupling two-grid method.

Jie Peng, Junxian Wang, Shi Shu

Balancing domain decomposition by constraints (BDDC) algorithms with adaptive primal constraints are developed in a concise variational framework for the weighted plane wave least-squares (PWLS) discritization of Helmholtz equations with high and various wave numbers. The unknowns to be solved in this preconditioned system are defined on elements rather than vertices or edges, which are different from the well-known discritizations such as the classical finite element method. Through choosing suitable "interface" and appropriate primal constraints with complex coefficients and introducing some local techniques, we developed a two-level adaptive BDDC algorithm for the PWLS discretization, and the condition number of the preconditioned system is proved to be bounded above by a user-defined tolerance and a constant which is only dependent on the maximum number of interfaces per subdomain. A multilevel algorithm is also attempted to resolve the bottleneck in large scale coarse problem. Numerical results are carried out to confirm the theoretical results and illustrate the efficiency of the proposed algorithms.

Xiaoqiang Yue, Shi Shu, Xiaowen Xu et al.

The paper investigates a non-intrusive parallel time integration with multigrid for space-fractional diffusion equations in two spatial dimensions. We firstly obtain a fully discrete scheme via using the linear finite element method to discretize spatial and temporal derivatives to propagate solutions. Next, we present a non-intrusive time-parallelization and its two-level convergence analysis, where we algorithmically and theoretically generalize the MGRIT to time-dependent fine time-grid propagators. Finally, numerical illustrations show that the obtained numerical scheme possesses the saturation error order, theoretical results of the two-level variant deliver good predictions, and significant speedups can be achieved when compared to parareal and the sequential time-stepping approach.

Jie Peng, Shi Shu, Junxian Wang

A balancing domain decomposition by constraints (BDDC) algorithm with adaptive primal constraints in variational form is introduced and analyzed for high-order mortar discretization of two-dimensional elliptic problems with high varying and random coefficients. Some vector-valued auxiliary spaces and operators with essential properties are defined to describe the variational algorithm, and the coarse space is formed by using a transformation operator on each interface. Compared with the adaptive BDDC algorithms for conforming Galerkin approximations, our algorithm is more simple, because there is not any continuity constraints at subdomain vertices in the mortar method involved in this paper. The condition number of the preconditioned system is proved to be bounded above by a user-defined tolerance and a constant which is dependent on the maximum number of interfaces per subdomain, and independent of the mesh size and the contrast of the given coefficients. Numerical results show the robustness and efficiency of the algorithm for various model problems.

Yun Liu, Chen Cui, Shi Shu et al.

The numerical solution of partial differential equations (PDEs) is fundamental to scientific and engineering computing. In the presence of strong anisotropy, material heterogeneity, and complex geometries, however, classical iterative solvers often suffer from reduced efficiency and require substantial problem-dependent tuning. The Fourier neural solver (FNS) is a learning-based hybrid iterative solver for such problems without extensive manual parameter tuning, but its original design is primarily effective for scalar PDEs on structured meshes and is difficult to extend directly to unstructured meshes or strongly coupled PDE systems. Building on the FNS framework, we introduce block smoothing operators and graph neural networks to construct a solver for unstructured systems, termed the graph Fourier neural solver (G-FNS). We further incorporate a coordinate transformation network to develop the adaptive graph Fourier neural solver (AG-FNS), and then extend this formulation to a frequency-domain multilevel variant, ML-AG-FNS. Rigorous analysis shows that, under suitable mathematical assumptions, the proposed method achieves mesh-independent convergence rate. Error-spectrum visualizations further indicate that AG-FNS can capture complex multiscale error modes. Extensive experiments on two-dimensional anisotropic diffusion and on two- and three-dimensional isotropic/anisotropic linear elasticity problems over unstructured meshes demonstrate strong robustness and efficiency. The proposed framework can be used either as a solver or as a preconditioner for Krylov subspace methods. Overall, it substantially extends the original FNS methodology and broadens the applicability of this class of neural solvers.

Zheng Li, Shi Shu

Inspired by dynamic programming, we propose Stochastic Virtual Gradient Descent (SVGD) algorithm where the Virtual Gradient is defined by computational graph and automatic differentiation. The method is computationally efficient and has little memory requirements. We also analyze the theoretical convergence properties and implementation of the algorithm. Experimental results on multiple datasets and network models show that SVGD has advantages over other stochastic optimization methods.

Xiaoqiang Yue, Yehong Xu, Shi Shu et al.

In this study we construct a time-space finite element (FE) scheme and furnish cost-efficient approximations for one-dimensional multi-term time fractional advection diffusion equations on a bounded domain $Ω$. Firstly, a fully discrete scheme is obtained by the linear FE method in both temporal and spatial directions, and many characterizations on the resulting matrix are established. Secondly, the condition number estimation is proved, an adaptive algebraic multigrid (AMG) method is further developed to lessen computational cost and analyzed in the classical framework. Finally, some numerical experiments are implemented to reach the saturation error order in the $L^2(Ω)$ norm sense, and present theoretical confirmations and predictable behaviors of the proposed algorithm.

Xiaoqiang Yue, Weiping Bu, Shi Shu et al.

The paper aims to establish a fully discrete finite element (FE) scheme and provide cost-effective solutions for one-dimensional time-space Caputo-Riesz fractional diffusion equations on a bounded domain $Ω$. Firstly, we construct a fully discrete scheme of the linear FE method in both temporal and spatial directions, derive many characterizations on the coefficient matrix and numerically verify that the fully FE approximation possesses the saturation error order under $L^2(Ω)$ norm. Secondly, we theoretically prove the estimation $1+\mathcal{O}(τ^αh^{-2β})$ on the condition number of the coefficient matrix, in which $τ$ and $h$ respectively denote time and space step sizes. Finally, on the grounds of the estimation and fast Fourier transform, we develop and analyze an adaptive algebraic multigrid (AMG) method with low algorithmic complexity, reveal a reference formula to measure the strength-of-connection tolerance which severely affect the robustness of AMG methods in handling fractional diffusion equations, and illustrate the well robustness and high efficiency of the proposed algorithm compared with the classical AMG, conjugate gradient and Jacobi iterative methods.

Jie Peng, Shi Shu, Haiyuan Yu et al.

The finite volume methods are frequently employed in the discretization of diffusion problems with interface. In this paper, we firstly present a vertex-centered MACH-like finite volume method for solving stationary diffusion problems with strong discontinuity and multiple material cells on the Eulerian grids. This method is motivated by Frese [No. AMRC-R-874, Mission Research Corp., Albuquerque, NM, 1987]. Then, the local truncation error and global error estimates of the degenerate five-point MACH-like scheme are derived by introducing some new techniques. Especially under some assumptions, we prove that this scheme can reach the asymptotic optimal error estimate $O(h^2 |\ln h|)$ in the maximum norm. Finally, numerical experiments verify theoretical results.