Mathematicians build an algorithm to “do the twist”

Mathematicians at the Center for Advanced Mathematics for Energy Research Applications (CAMERA) at the Lawrence Berkeley National Laboratory (Berkeley Lab) have developed a mathematical algorithm to decipher the rotational dynamics of torsion particles in large complex systems from the diffusion models of X-rays observed in sophisticated X-ray Photon Correlation Spectroscopy (XPCS) experiments.

These experiments, designed to study the properties of suspensions and solutions of colloids, macromolecules, and polymers, have been established as key scientific drivers for many ongoing coherent light source upgrades within the U.S. Department of Energy (DOE). The new mathematical methods, developed by the CAMERA team of Zixi Hu, Jeffrey Donatelli and James Sethian, have the potential to reveal much more information about the function and properties of complex materials than was previously possible.

Suspended particles undergo Brownian motion, wriggling as they move (translate) and spin (spin). The sizes of these random fluctuations depend on the shape and structure of the materials and contain information on dynamics, with applications in molecular biology, drug discovery, and materials science.

XPCS works by focusing a coherent beam of X-rays to capture light scattered by suspended particles. A detector picks up the resulting speckle patterns, which contain several tiny fluctuations in the signal that encode detailed information about the dynamics of the observed system. To take advantage of this capability, future upgrades from Coherent Light Source to Berkeley Lab’s Advanced Light Source (ALS), Argonne Advanced Photon Source (APS), and Coherent Light Source SLAC’s Linac all provide some of the world’s most advanced XPCS experiences, leveraging unprecedented consistency and brightness.

But once you’ve collected the data for all of these images, how do you get some useful information from them? An advanced technique for extracting dynamic information from XPCS is to calculate something called temporal autocorrelation, which measures how pixels in speckle patterns change after a certain amount of time. The autocorrelation function puts the still images together, just like a movie of yesteryear comes to life as closely related postcard images scroll.

Current algorithms are mainly limited to extracting translational movements; think of a Pogo stick that jumps from place to place. However, no previous algorithm was able to extract “rotational scattering” information about how structures turn and turn, information essential to understanding the function and dynamic properties of a physical system. Obtaining this hidden information is a major challenge.

By deflecting the light

A breakthrough came when experts gathered for a CAMERA workshop on XPCS in February 2019 to discuss critical emerging needs in the field. Extracting the rotational scattering was a key focus, and Hu, a graduate math student at UC Berkeley; Donatelli, CAMERA manager for mathematics; and Sethian, a math professor at UC Berkeley and director of CAMERA, have teamed up to tackle the problem head-on.

The result of their work is a powerful new mathematical and algorithmic approach to extract rotational information, now operating in 2D and easily scalable to 3D. With remarkably few images (less than 4000), the method can easily predict the simulated rotational scattering coefficients to within a few percent. Details of the algorithm were released on August 18 in the Proceedings of the National Academy of Sciences.

The key idea is to go beyond the standard autocorrelation function, instead looking for the additional information about rotation contained in the angular-time cross-correlation functions, which compare how pixels change to both in time and in space. This is a major leap in mathematical complexity: simple data matrices transform into 4-way data tensors, and the theory relating rotational information to these tensors involves advanced harmonic analysis, linear algebra, and a tensor analysis. To relate the desired rotational information to the data, Hu developed a very sophisticated mathematical model that describes how the angular-temporal correlations behave depending on the rotational dynamics of this new complex set of equations.

“There were a lot of superimposed mysteries to be unraveled in order to build a good mathematical and algorithmic framework to solve the problem,” said Hu. “There was information related to both static structures and dynamic properties, and these properties had to be systematically exploited to build a cohesive framework. Taken together, they present a wonderful opportunity to weave many mathematical ideas. this approach be useful information out of what at first glance appears to be terribly noisy was a lot of fun. “

However, solving this set of equations to recover the rotational dynamics is difficult, because it consists of several layers of different types of math problems that are difficult to solve at the same time. To address this challenge, the team built on Donatelli’s previous work on iterative multilevel projections (M-TIP), designed to solve complex inverse problems where the goal is to find the input that produces an observed output. The idea of ​​M-TIP is to break a complex problem into sub-parts, using the best possible inversion / pseudo-inversion for each sub-part, and iterate through these sub-solutions until they converge to a solution that solves all parts of the problem.

Hu and his colleagues took these ideas and built a sister method, “Multilevel Estimation for Correlation Spectroscopy (M-TECS)”, solving the complex set of layered equations through systematic substeps. .

“The powerful thing about the M-TECS approach is that it exploits the fact that the problem can be separated into high dimensional linear parts and low dimensional non-linear and non-convex parts, each having its own efficient solutions, but they would turn into an extremely difficult optimization problem if they were to be solved all at once, ”Donatelli said.

“This is what allows M-TECS to efficiently determine the rotational dynamics from such a complex system of equations, whereas standard optimization approaches would encounter problems both in terms of convergence and computational cost. “

Open the door to new experiences

“XPCS is a powerful technique that will feature prominently in upgrading the ALS. This work opens up a new dimension to XPCS and will allow us to explore the dynamics of complex materials such as rotating molecules within water channels, ”said Alexander Hexemer, IT Manager program at the ALS.

Hu, who won the Bernard Friedman Award from UC Berkeley for this work, joined CAMERA, which is part of the computer research division of Berkeley Lab, as a newest member. “This type of mathematical and algorithmic co-design is the hallmark of good applied mathematics, in which new mathematics plays a central role in solving practical problems at the forefront of scientific research,” Sethian said.

The CAMERA team is currently working with beamline scientists from the ALS and APS to design new XPCS experiments that can take full advantage of the team’s mathematical and algorithmic approach to study new properties of rotational dynamics from important materials. The team is also working to extend its mathematical and algorithmic framework work to recover more general types of dynamic properties from XPCS, as well as to apply these methods to other correlation imaging technologies.

This work is supported by CAMERA, which is jointly funded by the Office of Advanced Scientific Computing Research and the Office of Basic Energy Sciences, both within the Office of Science of the US Department of Energy.