Abstract. In these mini-courses, I will provide 3-4 short lectures on how to apply an adaptive spectral method to tackle both forward-type problems to numerically solving spatiotemporal partial integral differential equations and to tackle inverse-type problems of reconstructing spatiotemporal equations, especially those with a spatial variable defined in unbounded domains.
Specifically, how traditional spectral methods can be integrated into the design of parameterized neural networks will be explored. The prerequisite of this course will be numerical analysis, numerical PDEs (preferably spectral methods), and neural network models.

Abstract. High-order accurate finite-difference (FD) methods allow for versatile numerical solutions to various partial differential equation (PDE) problems in science and engineering. However, boundary closures, efficiency, and algebraic complexity limit the usage of such methods. In this mini-course, we provide practical tools to improve the order of accuracy of FD PDE schemes without losing efficiency, simplicity, and scalability. Our model problem will be a scalar wave equation initial-boundary-value problem on complicated geometry. Using the model problem, we introduce Local Compatibility Boundary Conditions (LCBC) to support high-order accurate boundary treatment. We also discuss the FActored Modified Equation (FAME) scheme, a hierarchical algorithm to solve wave equation problems on complex geometry to arbitrary accuracy with high efficiency. Both methods utilize computational ideas that can help accelerate the design and testing of automated high-order accurate FD methods for PDEs.

Abstract. Randomization has proven to be an effective tool for accelerating a number of linear-algebra tasks, both in terms of asymptotic complexity and real-world runtime. In this mincourse we survey some of the key algorithms and analysis techniques commonly used in randomized linear algebra.

Abstract. TBA.

Abstract. In this minicourse, I will give a quick and informal introduction to a number of numerical techniques for rare event probability estimation in stochastic differential equations (SDEs), an important problem in many scientific disciplines. First, I will provide an overview of importance sampling and splitting methods for SDEs. Afterwards, I will describe sampling-free computational methods for asymptotic rare event estimates based on large deviation theory (LDT). Both leading-order LDT asymptotics as well as recent progress on pre-exponential refinements will be addressed. Prerequisites for the minicourse would be a basic familiarity with SDEs and optimization/control, but I will try to briefly introduce all necessary concepts.