Key research themes
1. How can projection methods be adapted and visualized to better understand distortions and topological preservation in multidimensional data projections?
This research area focuses on enhancing continuous projection techniques, such as Principal Component Analysis and Curvilinear Component Analysis, by developing tools to visualize and quantify the distortions and topological changes that occur when projecting high-dimensional data onto lower-dimensional spaces. Understanding these distortions is crucial for accurate data interpretation, especially in exploratory data analysis and manifold learning.
2. What advances in projection methods facilitate efficient and convergent solutions for large-scale linear and nonlinear systems, including inverse problems and optimal control?
This theme addresses recent methodological developments that leverage projection algorithms, including alternating projection, relaxed and successive projection, and Krylov subspace methods, to solve large-scale linear systems, inverse tomography problems, and nonlinear equations arising in optimal control and imaging. Research encompasses theoretical convergence guarantees, relaxation schemes, and iterative solvers designed to enhance computational efficiency and accuracy in high-dimensional or constrained settings.
3. How do projection-based numerical methods, including meshless and immersed boundary projection algorithms, enhance simulations of complex physical and fluid-structure interaction problems?
This theme explores computational techniques employing projection methods to handle complex boundary conditions and coupling in fluid dynamics and fluid-structure interaction (FSI) simulations. These methods improve robustness and efficiency in 2D and 3D simulations involving incompressible flows, multiphase media, and deformable structures by mathematically decoupling constraints, enforcing boundary conditions weakly, or developing monolithic solvers that preserve accuracy and stability.