 | Dr Zhiwen ZHANG Associate Professor of Department of Mathematics
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Q: What do you think are your most significant research accomplishments, and what has been the impact of your research?
A: Since I joined HKU, I have built my own research program and made significant progress in several new areas, including structure-preserving schemes for computing effective diffusivity and computational methods for Schrodinger equations in the semiclassical regime.
Computing effective diffusivity for particles moving in chaotic and stochastic flows is a fundamental problem in studying the diffusion enhancement phenomenon in fluid advection, which is of great theoretical and practical importance. Many existing works use numerical methods (e.g. finite element methods and spectral methods) to solve a convection-diffusion type corrector problem to compute effective diffusivity, which becomes extremely expensive when the diffusion coefficient is small and/or flows are in 3D space. I developed robust structure-preserving schemes (which are Lagrangian particle methods) to compute effective diffusivity for chaotic flows (including 3D ABC flow and Kolmogorov flow) and provided a sharp and uniform-in-time error estimate for the numerical schemes. My work is the first one in the literature to develop Lagrangian particle methods to compute effective diffusivity in 3D chaotic flows. I also developed stochastic structure-preserving schemes to compute effective diffusivity for stochastic flows, which is more challenging and interesting.
In recent years, people use Schrodinger equations to model quantum hetero-structures with tailored functionalities, such as heterojunctions and quantum metamaterials. The potentials in these models can be aperiodic and/or discontinuous, bringing challenges to existing methods such as asymptotic-based methods (e.g. WKB method), Bloch decomposition methods, and spectral methods. To address this challenge, I proposed a multiscale finite element method (MsFEM) to solve this problem in the semi-classical regime. In addition, I developed an enriched MsFEM to solve Schrodinger equations with time-dependent potentials for studying electron dynamics under external controls and a multiscale basis method to compute eigenvalues and eigenfunctions of Schrodinger equations. Later, I combined the MsFEM with the quasi-Monte Carlo method to solve Schrodinger equations with random potentials, which allows us to simulate the famous Anderson localization phenomenon. My work appears to be the first one in the computational mathematics community to compute the Anderson localization phenomena for Schrodinger equations.
Q: Please give a brief description of 1 - 2 ongoing research projects that best reflect your visions in the scientific field.
A: Uncertainty quantification (UQ) is an emerging research area in scientific computing. The main goal of UQ is to develop fast and accurate numerical methods to solve partial differential equations (PDEs) with random coefficients. Over the past several decades, many efficient computational methods have been developed in the literature and these methods are very accurate when the problems have random inputs of low or moderate dimensions and their solutions admit good regularities with respect to the random parameter space. In practice, however, many problems can be parametrized by high-dimensional random variables and their solutions do not admit good regularities with respect to the random variables. This brings essential challenges to the existing methods. Therefore, designing fast and accurate computational methods to solve these types of problems is crucial.
I intend to further explore the specific structures of the high-dimensional parametric PDEs and design model reduction methods via adaptive basis functions to solve them efficiently. In my recent GRF project, I aim to develop adaptive computational methods for solving high-dimensional parametric PDEs arising from reduced-order modeling (ROM) and UQ. I will work on two model problems: (1) Helmholtz equations in random media; and (2) convection-diffusion equations with parametric velocities and small diffusivities. In my recent NSFC project, I aim to develop numerical methods to solve random PDEs with low-regularity solutions. I will work on two model problems: (1) reaction-diffusion-advection equations with random velocities and small diffusivities; and (2) Schrodinger equations with random potentials in 3D space. The results of the proposed project are expected to make significant advances in solving high-dimensional parametric PDEs arising from ROM and UQ. With these new methods, we can simulate many real-world problems, such as seismic inversion, pollutant transport in the atmosphere or groundwater, and turbulent combustion.
Q: What is the most important question you want to address?
A: Many complex phenomena in science and engineering are modeled by parametric PDEs that depend on a range of (potentially high-dimensional) parameters. These parameters may describe the conductivity or refractive index of a material, wave frequencies, uncertainties in complicated fluid flows, among others. For instance, Helmholtz equations with random refractive indices are used to model wave propagation or scattering through a media with spatially heterogeneous or random properties. Designing efficient computational methods to solve this problem has an important application in seismic inversion. To understand the influences of these parameters on the physical phenomena, one needs to solve these PDEs with different parameters, which requires significant computational time.
Considerable amounts of effort have been devoted to study the parametric PDEs with many efficient methods being successfully developed. These include adaptive methods based on mesh refinement or sparsity, model reduction, compressed sensing, sparse grid, and low-rank tensor methods. However, there are still many difficult problems that deserve investigation, such as high-dimensional parametric PDEs, especially when these PDEs admit low-regularity solutions. High-dimensional parameter spaces bring essential difficulty in numerical approximation, due to the so-called curse of dimensionality. Moreover, the low regularity in the solutions may further deteriorate the convergence rate for numerical approximation. Based on my research progress over the past several years, I want to develop new and efficient computational methods to solve these challenging problems.
Q: Can you give an example of your translational work?
A: I also made progress in some interdisciplinary research problems. In a recent paper, which has been published in the Journal of the Royal Society Interface, we proposed a mathematical model for shape evolution and locomotion of fish epidermal keratocytes on elastic substrates. The model is based on mechanosensing concepts: cells apply contractile forces onto the elastic substrate, while cell shape evolution depends locally on the substrate stress generated by themselves or external mechanical stimuli acting on the substrate. We use the level set method to study the behavior of the model numerically. Despite its simplicity, the model predicts multiple different modes of locomotion behaviour, owing to its rich bifurcation response. These include symmetry breaking from the stationary centrosymmetric to the well-known steadily propagating crescent shape as observed. We show how mechanosensitive coupling between cell shape evolution and substrate stress acts as a feedback loop to bring about the symmetry breaking necessary for locomotion. Asymmetric bipedal oscillations seen in experiments and traveling waves in the lamellipodium leading-edge occur in model simulations. These results suggest that the mechanism here is further symmetry breaking caused by actin flow polarization. In addition, simulated cells exhibit tensotaxis, or motion towards mechanical tension externally applied to the substrate (seen in human keratinocytes), and away from compression, as observed in lamellipodium fragments without a nucleus. The model also exhibits durotaxis or turning towards an interface with a rigid substrate as observed in various locomoting cells.
Q: Where do you see yourself in five years/ ten years? What do you want to accomplish the most?
A: I will continue applying for grants, publishing high-quality papers, and mentoring students. In addition to the several research projects mentioned above, I will study the deep learning methods for solving high-dimensional PDEs and stochastic dynamical systems in the next five years. In recent years, deep learning methods have achieved unprecedented successes in various application fields, including computer vision, speech recognition, natural language processing, audio recognition, social network filtering, and bioinformatics, where they have produced results comparable to and in some cases superior to human experts. Motivated by this exciting progress, there are increased new research interests in the scientific computation community where researchers apply deep neural networks (DNNs) based methods for scientific computation, including approximating multivariate functions and solving differential equations using the DNNs. This is a fascinating research area where new and exciting research results come out every day. However, there are several issues that remain open. For instance, we do not get the convergence rate for the DNN method and we have little understanding of the parameter space of the DNN. In addition, the issue of local minima and saddle points in the optimization problem is highly nontrivial. I am interested in studying these issues in the future.
Q: Who has influenced you the most?
A: I would like to thank my Ph.D. advisors, Professor Houde HAN and Professor Shi JIN, for bringing me into the scientific computing research area. I think my academic career was mostly influenced by my postdoc advisor, Professor Thomas HOU, who is the Charles Lee Powell Professor of Applied and Computational Mathematics in the Department of Computing and Mathematical Sciences at the California Institute of Technology. Professor HOU always encourages us to have our own research styles and tastes. If people have more solid mathematic foundation and research training than you, never give up and just continue to work harder. Appreciate and learn from people’s good points, but do not lose yourself or demean yourself. You just need to be yourself. Moreover, Professor HOU used his own story to encourage us. If a person has the courage to challenge difficult problems, works hard with passion, is good at learning from others, and at the same time insists on himself, it is possible to succeed through unremitting efforts through exerting one’s strengths and creativity. People should take a long-term perspective, not care too much about short-term gains and losses. Only in this way can you become a leader rather than a follower. Professor HOU is always a role model for my students and me to follow.
Q: Can you tell us more about your research group? What are the roles and the missions?
As a junior staff, I have received enormous help and support from colleagues in my department, which I very much appreciate. I have had the great fortune to work with four talented students over the past five years. They all have gone to join well-known universities or corporations over the last years. For example, Dr Zhongjian WANG joined the faculty of the University of Chicago as a William H. Kruskal Instructor in September 2020. Dr Dingjiong MA and Dr Junlong LYU joined Huawei Theoretical Research Lab in Hong Kong in the summer of 2020 and 2021, respectively. I look forward to working with future students to address challenging problems in the deep learning methods for computational science.