Mathematicians at the Center for Advanced Mathematics in Energy Research and Application (CAMERA) in Lawrence Berkeley Laboratory have developed a mathematical algorithm to decipher the rotational dynamics photon correlation spectroscopy (XPCS) experiments of twisted particles in large complex systems from the X-ray scattering patterns observed in highly complex X-rays.
These experiments are aimed at studying the properties of suspensions and solutions of colloids, macromolecules and polymers, and have become the key scientific driving factors for the upgrading of many coherent light sources being carried out by the US Department of Energy. The new mathematical method developed by the camera team of Zixi Lake, Jeffrey Donatelli and James Setian may reveal more information about the functions and characteristics of complex materials, which was previously impossible.
Suspended particles move in Brownian motion, and will shake when they move (translate) and rotate (rotate). The size of these random fluctuations depends on the shape and structure of materials, and contains information about dynamics, which is applied to molecular biology, drug discovery and material science.
The working principle of XPCS is to focus a coherent X-ray and capture the light scattered by suspended particles. The detector receives the generated speckle pattern, which contains several small fluctuations in the signal and encodes detailed information about the observation system dynamics. In order to make full use of this ability, Advanced Light Source (ALS) in Berkeley Lab, Advanced Photon Source (APS) in Argonne and SLAC coherent light source to be upgraded soon are all planning some of the most advanced XPCS experiments in the world, using unprecedented coherence and brightness.
But once you collect data from all these images, how can you get useful information from them? The main technology to extract dynamic information from XPCS is to calculate the so-called time autocorrelation, which measures the changes of pixels in speckle pattern after a period of time. Autocorrelation stitching static images together, just like an old movie comes alive, and closely related postcard images fly by.
At present, the algorithm is mainly limited to the extraction of translation motion; It's like jumping from place to place on stilts. However, the previous algorithms can not extract the "spin diffusion" information of how the structure spins and rotates, which is very important for understanding the functional and dynamic characteristics of physical systems. Obtaining these hidden information is a great challenge.
Twisted light
From 2065438 to February 2009, experts gathered together to attend the camera seminar on XPCS and discussed the emerging key requirements in this field. Extracting rotational diffusion is a key goal. Hu is a graduate student majoring in mathematics at the University of California, Berkeley. Donatelli, the camera leader of mathematics; In cooperation with Sethian, a professor of mathematics at the University of California, Berkeley and director of CAMERA, this problem was solved.
The result of their work is a powerful new mathematical and algorithmic method to extract rotation information, which is now effective in 2D and can be easily extended to 3D. Because there are few images (less than 4000), this method can easily predict the simulated rotational diffusion coefficient, and the error is within a few percentage points. The details of this algorithm are published in the August 18 issue of Proceedings of the National Academy of Sciences.
Its core idea is to go beyond the standard autocorrelation function, but to find the extra rotation information contained in the angle-time correlation function, which compares the changes of pixels in time and space. This is a great leap in the complexity of mathematics: a simple data matrix has become a four-way data tensor, and the theory of associating rotation information with these tensors involves advanced harmonic analysis, linear algebra and tensor analysis. In order to connect the required rotation information with the data, Hu developed a highly complex mathematical model and described how to express the angle-time correlation as a function of rotation dynamics according to this new complex equation set.
He said: "In order to establish a good mathematical and algorithmic framework to solve this problem, there are many multi-level mysteries to be solved." "There is information about static structures and dynamic attributes, which need to be used systematically to build a consistent framework. Generally speaking, they provide an excellent opportunity to weave many mathematical ideas together. It is very interesting to get useful information from an environment that seems very noisy at first glance in this way. "
However, it is challenging to solve this set of equations to restore rotational dynamics, because it contains several different types of mathematical problems and it is difficult to solve them at one time. In order to meet this challenge, the team worked on Donatelli's early Multi-layer Iterative Projection (M-TIP), aiming at solving complex inverse problems, with the goal of finding inputs that produce observable outputs. The idea of M-TIP is to decompose a complex problem into sub-problems, use optimal inversion/pseudo-inversion for each sub-problem, and iterate these sub-solutions until they converge to a solution that solves all parts of the problem.
Hu and his colleagues adopted these ideas and established a sister method, "Multi-layer Estimated Correlation Spectra (M-TECS)", to solve the complex layered equation through the sub-steps of the system.
"The advantage of M-TECS method is that the problem can be decomposed into high-dimensional linear parts and low-dimensional nonlinear nonconvex parts, and each part has its own effective solution, but if these problems are solved once, it will become an extremely difficult optimization problem," Donatelli said.
This enables M-TECS to effectively determine the rotational dynamics from such a complex set of equations, but the standard optimization method has encountered trouble in convergence and calculation cost. "
Opened the door for new experiments.
"XPCS is a powerful technology and will play an important role in ALS upgrade. This work opens a new dimension for XPCS, which will allow us to explore the dynamics of complex materials, such as rotating molecules in water channels, "said Alexander himer, head of als computing project.
For this, Mr. Hu won the Bernard Friedman Award from the University of California, Berkeley. He joined camera in the Computing Research Department of Berkeley Lab as its newest member. Cecian said: "This * * * design of mathematics and algorithms is a sign of excellent applied mathematics, and new mathematics plays a key role in solving practical problems at the forefront of scientific exploration."
The camera team is currently working with the beam scientists of ALS and APS to design a new XPCS experiment, which can make full use of the team's mathematics and algorithm to study the new rotational dynamics of important materials. The team is also expanding their mathematical and algorithmic framework to restore the more general dynamic characteristics of XPCS and apply these methods to other related 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 Science under the Science Office of the US Department of Energy.