Research

Computational and data-driven modeling has been widely applied in science and engineering communities, but the marketing and management communities have been slower in their acceptance. Marketers already know that they can use data analytics to detect preferences in books, friends, movies, music, and travel and predict future consumer choices. My research in this area is to develop innovative mathematical tools, and machine learning approaches, which will allow for more accurate and informative models of consumer motivation and reactions.

One of the new tools for consumer study is the eye tracker. An eye tracker records the eye position, movement, and pupil size of a subject who is viewing text, pictures, or an animation, so as to detect areas in which the user has an interest. Collaborating with colleagues in Business, we have analyzed consumer Eye tracking data to determine their attention and preference from gaze information, such as gaze sequence and gaze duration; see Figure 2. In experiments, eye tracking data is recorded as fixation points, fixation duration, and saccades. The result is a classic example of Big Data, having big velocity, big volume, and big variety. For example, eye trackers save 60 data points per second, and a single session may last as long as half an hour. Finding and analyzing patterns in the data is thus a challenge, but machine learning algorithms can discover complex patterns without priori specification. We compare and improve machine learning algorithms for analyzing eye-tracking data and build an innovative discovery-driven (e.g. inductive) approach. Inspired by this research, we have also co-developed a multidisciplinary course – Marketing Revolution with Machine Learning – in the Department of Mathematics and Statistics and the Department of Business and Information Technology at the Kummer College of Innovation, Entrepreneurship, and Economic Development.

Markov transition map of Eye tracking gaze sequence (Fukawa, Zhang, Steward, and Burkardt, 2018)

Student:  Yumeng Wang (Ph.D. student in Department of Mathematics and Statistics, co-advised with Prof. Nobuyuki Fukawa in Marketing)

Bose-Einstein condensation (BEC), predicted by S. N. Bose and A. Einstein in the early 1920s, is a state of matter near absolute zero temperature for which all atoms lose their individual properties and condense into a macroscopic coherent “super-wave”. Since its first experimental realization in 1995 (2001 Nobel Prize in Physics awarded to E. Cornell, C. Wieman, and W. Ketterle), Bose-Einstein condensation has been the focus of active research, both experimentally and theoretically. It provides a new tool to investigate quantum properties of matter and also opens new perspectives for understanding the phenomena of superconductivity and superfluidity. The recent launch of the Cold Atom Laboratory to the space station on May 21, 2018 has once again drawn a spotlight on Bose-Einstein superfluidity. So far, tremendous progress has been made in this area, but the comprehensive understanding and control of BEC still remains an elusive goal.

The objectives of my research are to develop mathematical and computational techniques to study the properties of BEC, so as to advance experimental approaches and applications. It is well-known that BEC is realized at extremely low temperatures, which greatly hinders its extensive laboratory studies but makes mathematical studies of BEC very important. On the computational side, we have developed highly efficient and highly accurate numerical methods for simulating the stationary states and dynamics of BECs. With the advance of new technologies, BEC has been achieved in various types of atoms/molecules and tailored to study different properties; consequently new mathematical models and computational methods are required to describe the new phenomena. Motivated by these new experiments, we have been continuing our modeling and computational studies on different aspects of BECs. On the application side, we have performed mathematical analysis and numerical simulations to understand the properties of BECs. We are especially interested in dynamics of quantized vortices. A quantized vortex is a signature of superfluidity and plays a significant role in the understanding and application of BEC and superconductors. The study of quantized vortices is challenging as they represent multiscale material defects; see illustration in Figure 3. We have explored the structure of the vortex states under different external fields, interactions between vortices, stability of vortices under noise, as well as their dynamical laws. To control quantum vortices, we introduced an optimal control approach to study the pinning of vortices in superconductors.

Giant vortex lattice (left) and vortex ring (right) in rotating BEC.
(Zeng and Zhang, 2009)

Student:  Shiping Zhou (Ph.D. student in Department of Mathematics and Statistics)

Seismic waveform inversion is a widely used technique in the oil and gas industry, and geophysical research. Over recent decades, full-waveform inversion has proved to be the most powerful technique to reconstruct high-resolution and high-fidelity subsurface structures from seismic data. The success of full-waveform inversion relies on accurate forward modeling of seismic wave propagation and efficient data-driven waveform inversion for processing seismic data; see illustration of full-waveform inversion in Figure 1. Existing seismic imaging methods are implemented in the classical framework of elasticity, which mostly ignores the viscous (attenuation) property of the Earth’s subsurface. The recently proposed fractional viscoacoustic wave model is capable of accounting for both seismic attenuation and dispersion in wave propagation, but its effective incorporation into full-waveform inversion of seismic data is still missing.

Collaborating with geophysicists, the objective of my research is to create a new paradigm for data-driven seismic modeling and inversion, so as to promote the subsurface seismic modeling and imaging techniques. This interdisciplinary research combines the search for accurate modeling, efficient algorithms, and practical geophysical advances in data inversion, validation, and applications. On one hand, we are developing accurate forward models for describing seismic wave propagation. So far, we mainly focus on models compatible with existing well used software in oil and gas industry, but which add a proper treatment of the attenuation of the Earth’s subsurface. On the other hand, we are developing data-driven inversion approaches, to learn the properties of the Earthp’s subsurface, such as quality factors.

Schematic illustration of full waveform inversion for seismic data.
(Wu, Zhang, and Zhu, 2021)

Student:   Yixuan Wu (Ph.D. student in Department of Mathematics and Statistics)