Revolutionizing Rational Approximation: Nick Trefethen Presents Distinguished Lecture

Nick Trefethen, a renowned professor of applied mathematics at Harvard University and expert in numerical analysis, captivated a packed auditorium with his lecture on October 31, 2024 at the Oden Institute of Computational Engineering and Sciences. Trefethen’s talk was the third in the Oden Institute's Distinguished Lectures series, which was launched in the spring of 2024.

Many students know Trefethen as the author of Numerical Linear Algebra and Approximation Theory and Approximation Practice, two foundational textbooks in the field. However, his recent work has centered on the development, implications, and applications of the AAA, pronounced “triple A” and derived from “adaptive Antoulas-Anderson.” This innovative algorithm acts as an ‘energizer bunny’ for rational approximation, transforming what was once a nearly impossible task into a relatively simple one—assuming you have access to MATLAB and about 0.0359 seconds of free time.

Rational approximation is crucial in numerical analysis because many complex functions, particularly those found in fields like signal processing, control theory, and computational physics, are difficult to represent. “Polynomials never truly fail,” said Trefethen. “However, they may fail to be efficient. That’s where rational functions come in.” 

These complex functions may exhibit singularities, sharp peaks, or oscillatory behaviors that complicate direct computation or analysis. Rational functions, which are the ratio of two polynomials, offer a flexible and powerful tool for approximating such complex functions because they can better model these irregularities and exhibit more efficient convergence properties.

The AAA algorithm introduces a key innovation in computing rational approximations by streamlining the process of finding efficient rational functions to represent complex data. Unlike traditional methods, which often require manual selection of interpolation points, the AAA algorithm automatically adapts to the function's characteristics, ensuring the best points for approximation are chosen.

“The function essentially acts like magic,” Trefethen noted. This adaptability reduces both computational complexity and the risk of instability, even when working with functions that exhibit singularities or other challenging behaviors. By making the process faster, more stable, and easier to implement, the AAA algorithm has significantly enhanced the practical application of rational approximation in numerical analysis.

After summarizing the utility and efficiency of the AAA algorithm, Trefethen’s lecture transitioned into a practical, mathematical show-and-tell, showcasing its broad applicability across various fields, featuring spirited discussions with attendees. Trefethen also demonstrated how AAA can handle real-world datasets with gaps, filling in missing data with high accuracy. In fluid dynamics, the algorithm proved valuable for simulating Stokes flow, modeling low-Reynolds-number fluid motion. Lastly, he discussed its role in Helmholtz scattering, where rational approximations are key for modeling wave propagation in complex media. 

As the AAA algorithm and Trefethen’s broader body of work continue to shape the future of numerical analysis, his visit left a lasting impact on both students and faculty alike.