Providing a Scientific Backbone: Francis Aznaran and his numerical solutions of differential equations
The pouring of milk into a coffee cup, the collapse of a bridge, and the way air flows around an airplane are all completely different scenarios, yet are modeled using the same differential equations.
This is the focus of Francis Aznaran, postdoctoral researcher at the University of Notre Dame. His work in the Department of Applied and Computational Mathematics and Statistics (ACMS) revolves around the numerical analysis of such equations.
In undergraduate and high school mathematics, students often work to find the exact solutions to problems. Yet, the equations Aznaran deals with are far too complex for that, so he works to create numerical algorithms on computers that allow him to solve them approximately.
“What I do is take these mathematical statements of physical laws and study them in their own right, as pure mathematicians do, but then also move on and design schemes to come up with approximate solutions to them,” said Aznaran. He subsequently proves properties about the resulting schemes and implements them in publicly available code.
Numerical analysis provides the backbone for many other areas of science: “Entire areas of physics revolve around the solutions of their underlying differential equations,” Aznaran said. “It’s particularly important in modeling and predictive contexts.”
These equations have strong pragmatic impacts, helping to predict things like how a disease moves through a population, or how a new roadway might deform under the weight of cars.
While in the abstract they might be approximate, Aznaran said that he can usually prove that his approximate solutions obtained on a computer will actually converge to the true one.
Notre Dame has offered Aznaran a strong community of other people in his field. Many of these researchers are also trained in the same methods as he is, particularly the finite element method, which was the focus of his doctoral degree.
“I was attracted to the excellent research group there is here at Notre Dame, not only in the field of numerical analysis, but even more specifically, the numerical solutions of differential equations, which is my area of focus,” Aznaran said. He was also deeply impressed with the thoroughness of the University’s Catholic mission and identity.
Alongside his supervisor, Martina Bukač, he is now working on more multi-physics applications of such numerical solutions.
As Aznaran pursues his research, he has found an outlet in distance running, which has helped train him for the perseverance that is necessary to have when his work gets challenging.
Aznaran recently won first place in the College of Science Postdocs Lightning Talk Competition, which allowed him to showcase the research he has been conducting and its practical relevance.
“You prove things and see them play out in your computer screens and simulations, but then, you hopefully get to see that mirrored in real-life physics,” Aznaran said. “I love my work because of that deep connection between quite abstract mathematical concepts and very concrete applications.”
Originally published by science.nd.edu on October 18, 2024.
atLatest Research
- From card catalogs to AI? Technology and workflows at the Notre Dame ArchivesWhen I started working in the Notre Dame Archives in 1983, we relied on technology developed in the 19th century or earlier: the typewriter, telephone, photography, snail mail. We kept typewritten inventories in file cabinets. We responded to reference requests by mail or by telephone. We had card…
- Notre Dame students travel to Croatia for a summer school program on resilience and recoveryFrom June 27 through July 7, 2024, the Catholic University of Croatia hosted the third summer school program focused on resilience and recovery. This year’s program was titled “Practicing Resilience – Preparing…
- Notre Dame’s Religious Liberty Clinic and Education Law Project file amicus brief in support of religious school’s freedom to integrate faith and learningThis week Notre Dame Law School’s Lindsay and Matt Moroun Religious Liberty Clinic filed an amicus brief on behalf of the National…
- The Significance of PrecedenceIt is the responsibility of architects to understand the multiple traditions that have preceded them, according to Associate Dean of Faculty Affairs and Curriculum and Professor Samir Younés said. Precedent is imperative. “One of the first things you do is to study the written, designed, and…
- Dissecting the Roman Forum Digitally — From Pen to Pixel StudiesThe 2015 Gorkha Nepal earthquake devastated some of the world’s architectural wonders, but what if those buildings could be restored? What if history could be brought back to life exactly as it was? Since 2010, Associate Professor and Director of the Digital Historic Architecture and Material…
- Bryan Stevenson delivers powerful message about ‘proximity’ at Institute for Social Concerns lecture in downtown South BendA maximum-security prison in rural Georgia was the place where everything changed for Bryan Stevenson. He was still a law student, interning with a human rights organization at the time, and felt unqualified to help the man on death row that he was about to meet. When the shackled man entered…