How do scientists and engineers solve numerical problems that would require years upon years to solve on today’s fastest supercomputers?

They don’t.

Instead, with the assistance of mathematics, scientists and engineers can rely on approximations that provide answers that are sufficiently accurate for many applications and that can be computed in seconds on laptop computers.

Reducing the complexity of numerical simulations with mathematical tools is what keeps Benjamin Peherstorfer busy at his desk all day. Peherstorfer, hired through the college’s Grainger Institute for Engineering, is the newest faculty member in the Department of Mechanical Engineering. Peherstorfer takes complex computer models from engineers and scientists, develops mathematical tools to reduce them to only the very essential components, and performs the mathematical analysis to ensure the error introduced by this approximation is within an acceptable tolerance.

While approximations may seem not ideal, Peherstorfer says the computer models in engineering and science have become so complex that simply buying a larger computer is not an option. “Using reduced models is really not about making it fast, but about making it possible at all,” Peherstorfer says.

So, what are these incredibly complex problems Peherstorfer is interested in?

One is related to numerical simulations of liquid rocket engines (LREs). Peherstorfer uses his computer modeling methods to account for the uncertainty inherent in these problems: “In virtually any engineering system, uncertainties are introduced because of incomplete data, measurement errors, or tiny variations in the manufacturing process,” Peherstorfer explains. “So how do these small variation affect the system? To estimate the effects, we perform millions of numerical simulations to make statistical statements on how likely it is for the rocket engine to fail, for example. Reduced models are essential to make these computations tractable.”

To complicate Peherstorfer’s research, he works in the context of inverse problems, as opposed to forward problems.

“In forward problems, you have the inputs and you feed them into your numerical simulation and to get your output. In inverse problems, you have the output and you would like to know what inputs gave rise to that output.” Peherstorfer says.

A common real-world example of an inverse problem is x-ray computed tomography, where an object is imagined based on how it scatters incoming x-rays. A computationally very demanding inverse problem is imaging subsurface Earth based on seismic waves. Roughly speaking, inverse problems require more computing power and are more difficult to answer than forward problems, Peherstorfer says.

“When you now think that you additionally want to quantify uncertainties in inverse problems, then you can again see the need for developing cheap and certified reduced models,” he says.

Peherstorfer has long been interested in computational problems in science and engineering and harnessing the power of computers to help find solutions—or at least approximations—to them. He received his bachelor’s, master’s and PhD in computer science at the Technical University of Munich (TUM) in Germany. There, he was a member of the scientific computing group and worked on machine learning to detect patterns in data streams in real-time.

Peherstorfer comes to UW-Madison from a postdoctoral position at Massachusetts Institute of Technology (MIT) in its aerospace computational design laboratory. While at UW-Madison, he plans to work on multi-fidelity modeling to combine multiple computational models for uncertainty quantification in inverse problems. In addition, he’s establishing research collaborations that go beyond the department and college.

“I am absolutely excited about the incredible collaboration opportunities at UW-Madison,” Peherstorfer says. “UW-Madison provides a very collaborative environment, which is a great opportunity for me and my interdisciplinary research that cuts across math, computer science, and engineering.”

Author: Will Cushman