Yujie Gong, a Ph.D. student from the Department of Mathematics of the Faculty of Science and Technology, University of Macau, won the Student Paper Competition at the 18th Copper Mountain Conference on Iterative Methods (CMCIM 2024), it further highlights the achievements of the University of Macau in cultivating high-quality scientific research talents and advancing cutting-edge research in the field. The paper focuses on solving highly nonlinear hyperelasticity problems on three-dimensional unstructured grids. Experimental results show improved algorithm efficiency, which is of significant importance in various scientific and engineering applications such as biomedicine and aerospace.

Since its establishment over 40 years ago, CMCIM has been a prestigious academic event in the field of numerical computation and iterative methods. It is jointly organized every two years by the University of Colorado and the Society for Industrial and Applied Mathematics (SIAM), an internationally renowned academic organization and it is supported by the the US Department of Energy. It has developed into an important platform for scholars in the global mathematics community to exchange research results and promote academic progress. The conference took place from April 14th to 19th, 2024, in the United States. The award-winning paper by Ph.D. student Yujie Gong is titled “Stagnation-shortening inexact Newton method based on unsupervised learning for highly nonlinear hyperelasticity problems on three-dimensional unstructured meshes.” The paper was co-supervised by Prof Cai Xiao-Chuan and Prof Luo Li from the University of Macau.

Although the traditional inexact Newton method is widely used, it often encounters long stagnation periods when dealing with highly nonlinear problems, where the nonlinear residual does not decrease significantly, resulting in a significant waste of computational time. However, the reasons for this stagnation are often difficult to quantify. To address this, Yujie Gong, with the assistance of the supervising professors, proposed an innovative “stagnation-shortening inexact Newton method.” The method introduces a unique “stagnation analysis phase” where the norms of the residuals are used as a guiding curve, and the residual vector is decomposed into a slow subspace and a regular subspace. The research team applied unsupervised learning techniques based on principal component analysis (PCA) to perform subspace Newton calculations in the slow subspace, and the obtained solution is then projected back to the global space. Experimental results demonstrate that this embedded learning phase significantly improves algorithm efficiency.

As a practical application example, the paper selected the modeling of human arterial stenosis using the hyperelasticity equation with multiple material parameters. The model presents extremely high nonlinear complexity due to the significant differences in material coefficients between the plaque and the surrounding healthy blood vessel sections. Through numerical experiments, the proposed method successfully reduces the number of nonlinear iterations and exhibits excellent robustness in handling such challenging hyperelasticity problems, highlighting its great potential in practical applications.

This award will inspire more scholars to pay attention to the intersection of iterative methods and unsupervised learning and provide new insights for solving complex real-world nonlinear problems.