Many applied mathematicians spend their careers working closely with engineers – writing code to find approximate solutions to partial differential equations for specific applications. But Lewis and Zhang go deeper: They’re hoping to change the methods applied mathematicians use.
Analysts and engineers often use something called monotone methods to handle a particularly challenging class of partial differential equations. These methods require a lot of heavy lifting to find solutions, with complicated code required for each specific problem. Lewis is working to replace monotone methods with “narrow-stencil methods,” using an idea that began with Lewis’ PhD advisor, who was later also Zhang’s postdoctoral supervisor.
While the 1991 proof that monotone methods would work was relatively straightforward, Lewis and his advisor’s 2021 paper for their new narrow-stencil method took 39 pages in a top journal where papers are usually capped at 20 pages. Basically, they trade onerous coding and an ‘easy’ proof for an onerous proof with ‘easy’ coding.
This approach could completely change the field. Lewis says just about “anyone can code” narrow-stencil methods, including undergraduate students.
“We basically broke analytic barriers that’ve been around since the ’90s,” Lewis says. “There’s top research that implies what we’re doing is not feasible, and we proved that it was.”
Lewis and Zhang are also creating new versions of methods that analysts have used since the 1970s to address so-called optimization problems. Think: minimizing cost or energy or maximizing a certain outcome in a complex system.
With their new funding, Lewis and Zhang will further develop these “penalty-free” methods and work to demonstrate their reliability and efficiency in solving industry problems. They’ll also apply both narrow-stencil and penalty-free methods to expanding classes of partial differential equations – with the potential to impact areas ranging from optics to economics.
The grants are another feather in the cap of UNCG’s unique computational math program, which now offers applied-math doctoral students rigorous training in both partial differential equations theory and numerical computing, preparing them for both academic and industrial work.
“Yi and I are basically the numerical group in the computational math program, which means we’re honored to be pioneering and growing that reputation,” Lewis says.
Adds Zhang, “It helps our graduates compete and our department continue to make a name for itself.”