Dimensions on a diet

As a young musician, Paul Constantine enjoyed exploring variations on Caribbean drum rhythms and new arrangements of jazz chord progressions. Now that same passion for pattern drives Constantine’s research as an assistant professor at the Colorado School of Mines, where he finds hidden structures within data computational models generate.

By applying a mathematical test to different physics models, Constantine has shown it’s possible to greatly reduce the computational cost of running the models without sacrificing accuracy. The promise of this method netted him both Advanced Scientific Computing Research program support and a book contract with the Society for Industrial and Applied Mathematics (SIAM).

With an education that includes courses in philosophy and music theory besides advanced math and engineering, Constantine has thrived in multidisciplinary environments. As a graduate student in Stanford University’s Institute for Computational and Mathematical Engineering, he studied uncertainty quantification and enjoyed being on teams addressing big problems. A John von Neumann Fellowship at Sandia National Laboratories in New Mexico from 2009–2011 focused his efforts on DOE-relevant problems in computational modeling.

“Physical models are becoming increasingly complex,” Constantine says. “They include more and more components and they are becoming more expensive to run on the computers. My work has been to look at the models to see if we can make these problems tractable.”

To do that, Constantine has focused on what is known as the dimension-reduction problem. For many physical models, increasing the number of variables or inputs exponentially increases the computing time. As Constantine explains it, reducing the number of dimensions can make it possible to solve an otherwise impossibly large problem.

For example, a model may have five parameters – not a large number on its face. But to test 10 possible values for each parameter would require 100,000 computer runs. By mathematically reducing five parameters to one, the same problem requires only 10 runs. “If each run costs $1, then the dimension reduction bought you a $100,000 study for $10,” he says.

As computational models become more and more complex, mathematical treatments will be necessary to keep a rein on data.

Savings like these make Constantine’s active subspace methods a promising tool to save both computational time and money for complex modeling problems. But he cautions that the technique is not a cure-all and can only be applied to models that have a particular structure. “It just turns out that lots of models we care about in engineering have this exploitable structure,” he says.

Mathematically, active subspace methods work by finding the combinations of inputs that explain the majority of a model’s variability. Then the method allows users to ignore or average the less important combinations, reducing the number of parameters in any given calculation.

Constantine and collaborator Qiqi Wang, an assistant professor at the Massachusetts Institute of Technology near Boston, began developing the method at Stanford, where they were graduate students together. The two joined collaborators at Stanford’s DOE-sponsored NNSA Predictive Science Academic Alliance Program, to show that active subspace methods could be applied to a model seeking to determine the safe operating range of a hypersonic scramjet in design.

“Without the dimension reduction from the active subspaces, we would not have been able to determine the sets of parameters that maintained thrust from the scramjet engine,” he says.

Since this early foray three years ago, Constantine says, the active subspace method keeps finding uses in models as diverse as chemical kinetics and designs for an airfoil and a photovoltaic solar cell.

He plans to use his current ASCR support (from a Mathematical and Statistical Methodologies for DOE Data-Centric Science at Scale project) to study the use of active subspace in a class of problems known as inverse calibration problems. Here, a large amount of data collected from an array of measurements is used to calibrate a model’s parameters. In these kinds of problems there’s often noise in the data that affects the ability to calibrate the model. Active subspace methods search the data to find which values are important and which can be safely ignored.

The question, he says, is how to compress the data for use in the calibration problem. For example, active subspace methods could allow analysis of engine emissions data to monitor for pollutants.

Many of these examples will become chapters in the SIAM book Constantine is writing on active subspace methods. It also will have an explanation of what he hopes will become a useful tool for physical scientists hoping to use the active subspace method to reduce dimensionality in their data sets.

He has developed what he calls a “quick-and-dirty check” that will quickly analyze data output from series of a simulation runs and determine whether the data are amenable to the active subspace method. The quick-check method uses a statistical approach adapted to create a visualization tool that easily shows whether relationships exist between data of interest.

Constantine has posted a short description of the method on arXiv, the open-access preprint archive for research articles. It gives examples in which he used the quick check to identify the active subspace in a design optimization problem involving a transonic wing. He was able to combine 60 design variables into a single variable that sufficiently described the wing’s drag coefficient behavior. This transformed a 50-dimensional optimization into a one-dimensional optimization.

As computational models become more and more complex, mathematical treatments will be necessary to keep a rein on data and preserve the usefulness of computational methods, Constantine says. But to do that, physicists and mathematicians must work together and learn each other’s language.

“The big problems we have in science are going to take interdisciplinary teams,” he says.

When he’s not solving complex physics problems, Constantine can be found on social media tweeting as @DrPaulynomial and giving informal public talks at events like Nerd Nite Denver. “It’s important to promote my work beyond my small circle of researchers to demonstrate its broad impact outside of academia.”