Numerical Analysis seminars

Michael Borinsky introduces tropical quantum field theory as a simpler deformation limit that preserves useful information about an interacting theory. Its numerical algorithm computes massive-scalar perturbation theory with runtime polynomial in loop order and external-leg count, avoiding the factorial growth of a Feynman-diagram expansion and making very high loop orders feasible. Geometrically, amplitudes become volumes of graph-moduli spaces, whose recursions resemble those for Weil–Petersson volumes. The talk connects the limiting construction, geometry and algorithm.

This introduction connects Kazhdan's property T with algebraic spectral gaps for a group Laplacian. Numerical searches for sum-of-squares expressions can suggest proofs, while certification converts approximate solutions into exact mathematical conclusions. Examples include new proofs for integer special linear groups and results for automorphism groups of free groups.

Unbiased stochastic queries can support sampling whose cost grows only polylogarithmically with inverse target error. This talk applies the framework to diffusion generation with sufficiently accurate score estimates and to log-concave sampling with subexponential gradient noise. The latter result contrasts with stochastic convex optimization, where even simple additive noise can require polynomially many queries.

Katarzyna Swirydowicz uses an asynchronous solver on an AMD system to examine the practical implementation of computational linear algebra across CPUs and GPUs. The talk introduces the relevant computational ideas, programming models, and software tools, then considers how algorithmic structure interacts with hardware capabilities. This case study illustrates both the opportunities and implementation challenges of asynchronous methods for large-scale scientific computing on GPU-accelerated systems.

Yousef Saad examines relaxed gradient descent for large-scale optimization. Relaxing the optimal step length in Cauchy's steepest descent avoids its characteristic zigzag behavior and can bring the search direction close to an eigenvector of the Hessian. Once that alignment is sufficiently accurate, properties of the Lanczos method can accelerate convergence. The talk analyzes several such strategies and illustrates them in global minimization of strictly convex functions, retaining the simplicity and low memory requirements that make gradient methods attractive for machine learning.

Matthias Bolten discusses the scalability of multigrid solvers for linear systems arising from discretized partial differential equations. On modern supercomputers, heterogeneous CPUs and GPUs and the widening gap between computation, network, and memory speeds complicate parallelization. Classical multigrid analysis relies on tightly coupled multiplicative components, whereas additive and asynchronous variants relax this coupling. The talk compares approaches to improving high-performance multigrid scalability, including asynchronous execution.