Huayi Wei

Yangyang Zheng, Huayi Wei, Shuhao Cao et al.

In this work, we propose a joint operator learning method for reconstructing images of conductivity coefficients from boundary data. Inspired by the idea of employing partial differential equation (PDE) solvers as preconditioners for this inverse problem, we investigate a ``solver-in-the-loop'' training mechanism. It allows the interaction of learnable parameters integrated in a PDE solver module and those in neural networks for reconstructing images. Specifically, we employ a fractional Laplace-Beltrami operator with a learnable fractional order, which transforms boundary data into high-dimensional features. These features then serve as input to a neural network, significantly improving reconstruction accuracy. For this purpose, a Learning-Automated FEM (LA-FEM) package, facilitating this ``solver-in-the-loop'' property, is developed with PyTorch as a backend. The new LA-FEM module conveniently allows the auto-differentiation regarding an objective function to freely propagate through the PDE solver from the forward problem and the coupled neural networks for the inverse problem.

Yunqing Huang, Huayi Wei, Wei Yang et al.

We design and numerically validate a recovery based linear finite element method for solving the biharmonic equation. The main idea is to replace the gradient operator $\nabla$ on linear finite element space by $G(\nabla)$ in the weak formulation of the biharmonic equation, where $G$ is the recovery operator which recovers the piecewise constant function into the linear finite element space. By operator $G$, Laplace operator $Δ$ is replaced by $\nabla\cdot G(\nabla)$. Furthermore the boundary condition on normal derivative $\nabla u\cdot \pmb{n}$ is treated by the boundary penalty method. The explicit matrix expression of the proposed method is also introduced. Numerical examples on uniform and adaptive meshes are presented to illustrate the correctness and effectiveness of the proposed method.

Dong Wang, Chunyu Chen, Huayi Wei

This paper constructs an energy function for simplex mesh based on the radius ratio and develops a corresponding mesh optimization method. The method combines vertex relocation and connectivity improvement, and can effectively remove slivers and improve the overall mesh quality. Based on the structure of the gradient of the energy function, we design a preconditioner, which reduces the number of iterations and improves the efficiency of the optimization algorithm. Numerical experiments show that the proposed method is effective in both sliver removal and mesh quality improvement.