Victor Minden thought he would study international relations or music at Tufts University in Boston for his bachelor’s degree, but the contrasts between his calculus and economics courses were too great to ignore.
Minden found he was more comfortable in math classes. “I just felt it was something I understood better – the meaning of it or the significance,” says Minden, a Department of Energy Computational Science Graduate Fellowship (DOE CSGF) recipient. For him, mathematics and what he could accomplish with it was more concrete than the concepts he learned in economics.
“Which is funny, because a lot of people don’t think of mathematics as a concrete discipline,” Minden says. “It’s very abstract, but at the same time, it’s very fundamental in the sense that you’re saying things that are true. They are right or they are wrong.”
That certainty also is one of the things Minden likes about his research at Stanford University. Working with Lexing Ying, he creates and refines techniques that exploit simplified physics and the underlying spatial relationships in a problem to develop efficient algorithms for extremely large problems.
The linear algebra methods Minden creates could be used in a range of high-performance computing (HPC) applications. For example, researchers may have scattered sea-surface temperature measurements and want to estimate temperatures throughout the rest of the ocean. Minden’s approach provides a statistically sound method to interpolate temperatures between the physical observations.
Other possible applications include recreating oil field distribution and subsurface flows, such as tracking a contaminant moving through groundwater.
Usually these problems center on big matrices – something like giant spreadsheets filled with numbers that describe the relationships between data at different points in space. These data could be temperature measurements gathered over a large area at particular moments, perhaps, or something more abstract, like the results of calculations over a discretized computational grid – points spread through a simulated physical space. Computers calculate the physical properties at each point. Taken together, they portray what’s happening throughout the space, similar to how millions of pixels taken together form a single digital image.
Minden’s algorithms seek a matrix’s underlying structure. Mathematically, it may just be an array of numbers, but “on the back end it’s related to a bunch of either physical observations” or points on a computational grid. “There’s a lot of different operations, a lot of different computations, you need to do with these matrices,” Minden says, but performing them in a standard, naïve fashion can consume excessive computer time and power. He develops faster ways to do the math.
“There’s this underlying sense of locality” in the data that his methods tap into, Minden says. Data from spots or points near each other are more likely to be similar while data from more distant locations are less likely to be correlated. Minden’s algorithms use that underlying data structure to make approximations that accelerate the matrix calculations. “You can do it a lot faster to whatever accuracy you need. It’s not exact, but it’s good enough.”
Despite their many uses, Minden rarely focuses on applications for his algorithms. “I’m not always working on the whole picture. I get a piece of the problem and I want to solve that piece as best as possible. And that piece may be hard to relate back” to a particular piece of software.
Minden appreciates his research’s behind-the-scenes nature. While at Tufts he worked for a summer at Argonne National Laboratory on PETSc, the Portable, Extensible Toolkit for Scientific Computation. Computational science researchers around the world use the library of data structures and routines. That piqued Minden’s interest in developing tools and ideas the entire HPC community could use for a variety of purposes.
Since moving into more detailed numerical linear algebra work, Minden also has come to appreciate his research for other reasons.
“There’s something about the mathematical level, the various structures that you get to work on, the various ideas that you see used in these algorithms.” The concepts couple nicely to physical concepts, “but also lead to pretty mathematics.”
Besides his Argonne internship, Minden has worked at Lawrence Livermore National Laboratory and did his DOE CSGF practicum at Lawrence Berkeley National Laboratory. With that experience under his belt, he hopes to get a postdoctoral fellowship at a DOE lab after his expected 2017 graduation.
Image caption: Atlantic Ocean surface temperatures interpolated from scattered measurements taken from the International Comprehensive Ocean-Atmosphere Data Set. The data, colored by temperature, are projected onto a two-dimensional plane through a Mercator projection and scattered on a white background for visualization. Image courtesy of Victor Minden.