Professor  Chunmei Su will talk about: 'Temporal high-order structure-preserving parametric finite element methods for curvature flows'

The quality of the mesh is crucial for simulating curvature flows, as standard approaches may fail due to mesh distortion. We first present a series of high-order parametric finite element methods based on the Barrett-Garcke--Nurnberg formulation for solving various types of flows involving curves and surfaces. Extensive numerical experiments demonstrate the anticipated high-order accuracy while maintaining favorable mesh quality throughout the evolution process. Secondly, for flows involving multiple geometric structures, such as surface diffusion—which reduces area while preserving volume—we propose a type of structure-preserving method that incorporates two scalar Lagrange multipliers along with two evolution equations related to area and volume, respectively. These schemes effectively preserve the geometric structure at a fully discrete level. Comprehensive numerical experiments illustrate that our methods achieve the desired temporal accuracy, while simultaneously preserving the geometric structure of the surface diffusion.
 

Professor Sara Shashaani is going to talk about: 'Adaptive Sampling and Regularization for Stochastic Trust-region Methods'

Trust-region methods have proven highly effective for unconstrained nonconvex stochastic optimization problems where objective and gradient information are available only through noisy stochastic oracles. ASTRO is a class of adaptive sampling trust-region methods that dynamically determine sampling effort while constructing local quadratic models from noisy function and gradient observations. By exploiting dependence among samples and the stochastic structure of the problem, ASTRO achieves strong convergence and complexity guarantees. Its derivative-free variant, ASTRO-DF, relies solely on noisy function evaluations and also enjoys almost-sure convergence guarantees.