Georgia Institute of Technology College of Sciences

Classical solvers for large-scale scientific and data-driven problems often face limitations when uncertainty, multiscale effects, or ill-conditioning become dominant. In this talk, I will present hybrid algorithmic frameworks that unify ideas from numerical analysis, stochastic computation, and machine learning to address these challenges. In the first part, I will introduce Preconditioned Truncated Single-Sample (PTSS) estimators, a new class of stochastic Krylov methods that integrate preconditioning with truncated Lanczos iterations. PTSS provides unbiased, low-variance estimators for linear system solutions, log-determinants, and their derivatives, enabling scalable algorithms for inference and optimization. In the second part, I will discuss a data-driven approach to constructing approximate inverse preconditioners for partial differential equations (PDEs). By learning the Green’s function of the underlying operator through neural representations, this framework captures multiscale behavior and preserves essential spectral structure. The resulting solvers achieve near-linear complexity in both setup and application. Together, these developments illustrate how stochastic and learning-based mechanisms can be embedded into classical numerical frameworks to create adaptive and efficient computational methods for complex systems.