Research Fields: Mathematical Modeling and Scientific Computing

We are a research group on “Numerical Algebra and Geometry” in the ST Yau Center and Applied Math Department , nycu.

Our primary research interests include the following two parts:

(a) Eigendecomposition and Fast Eigensolver for 3D Maxwell’s equations：
We use finite difference Yee’s scheme to discretize the Maxwell’s equations with simple cube (SC) or face-centered cube (FCC) periodic domain into generalized or rational eigenvalue problems. We explicitly derive the eigendecomposition of the double-curl operator and SVD of the single curl operator. Then we develop a FFT-based algorithm for compact representations of ∇ × ∇ × and ∇ × operators and propose null-space free methods with Newton-type and nonequivalence deflation technique for solving Maxwell’s equations. We compute the band structures of (I) Dielectric materials, where ε>0, μ>0, ξ=ζ=0; (II) Dispersive metallic photonic crystals with Drude or Drude-Lorentz models, where the permittivity ε=ε(ω) depends on the high-frequency ω with some poles; (III) Left-handed materials or negative-index material (complex media), where ε<0, μ<0, ξ,ζ≠0 are chiral or pseudo-chiral parameters. All trivial zero eigenvalues caused by the degenerate curl operator have been successfully deflated in our fast eigensolvers. FFT-based fast eigensolvers in Matlab code are developed to compute various band structures of I, II and III efficiently, reliably and robustly.

(b) 3D Computational conformal geometry with applications：
We mainly develop efficient numerical methods to compute conformal mappings of 3D surfaces such as a genus zero closed surface, a simply connected surface with a single boundary and a closed surface of genus ≥ 1 conformally to a sphere S², a unit disk D₁, a parallelogram on R² (g=1) and a polygon domain on H² (g ≥ 2). We propose a Quasi-Implicit Euler Method to a nonlinear heat equation, and the discrete Ricci flow with circle packing technique on the closed surface of g ≥ 1 to compute the associated conformal maps ( Figure 1 and 2 ). A high-resolution 3D dynamical scanner equipped in the Lab of ST Yau Center can be used to capture various 3D human faces and the corresponding conformal disks ( Figure 3 ). We actively cooperate with industry and develop the related applications in 3D animation, retargeting/driving with texture mapping ( Figure 4 and 5 ). Furthermore, facial expression analysis, face recognition, antiquities presentation, industrial inspection, and medical image are under investigation.

The study of the incompressible flows with interfaces is of major interests among applied mathematics community. It plays an important role in numerous natural phenomena and industrial applications, especially, for the soft matter physics and hydrodynamics of micro-fluidic systems. We have successfully developed simple and efficient numerical schemes based on immersed boundary method for simulating fluid-surfactant and fluid-vesicle dynamics. Brief descriptions are as follows：

(I) Interfacial flows with insoluble/soluble surfactant:
Surfactant are surface active agents that adhere to the fluid interface and affect the interface surface tension. Surfactant play an important role in many applications in the industries of food, cosmetics, oil, water purification , etc. In microsystems with the presence of interfaces, it is extremely important to consider the effect of surfactant since in such cases the capillary effect dominates the inertia of the fluids. We demonstrate the equilibrium for a hydrophobic drop with clean (dashed line) and contaminated interface (solid line) in Figure 1; the concentration distribution of a soluble droplet imposed to a flow is shown in Figure 2.

(II) Vesicle hydrodynamics:
Vesicles dynamics in fluid flow has become a quite active research area recently in the communities of soft matter physics and computational fluid mechanics. Indeed, the understanding of vesicle behaviors in fluids might lead to a better knowledge of red blood cells in bloods since they both share some similar mechanical behaviors. It is well-known that the phospholipid membrane exhibits a resistance against area dilation and bending; therefore, it is natural to regard this membrane as an inextensible surface with mechanical properties defined by some energy functional. We have proposed a series of immersed boundary methods to simulate the dynamics of two- and three-dimensional inextensible vesicles in Navier–Stokes flows; for demonstration, the snapshots of a freely suspended vesicle in quiescent flow are shown in Figure 3.

http://www.math.nycu.edu.tw/~mclai

We are interested in developing mathematical methodologies to investigate nonlinear dynamical systems. The systems of particular interest are some mathematical models about biological processes or phenomena, including the ones on cell differentiation, somitogenesis, gene regulation, neuronal systems, and competitive species under dispersal. Mathematical analysis on the collective behaviors of these coupled cells or coupled systems enhances the understanding of these models. The other research interest is mathematical theory in neural networks.

Our research interests are centered around the real world problems that can be modeled by coupled networks. Among the problems under considerations are as follows: (i) Coupled chaotic oscillators; Coupled maps lattices. Our group has been studying synchronization phenomena in coupled systems for the last few years. We develop some analytical theorems to ensure the emerge of synchronization. These results can be applied to the realistic design of electronic circuit system and secure communication (Figure 1). (ii) Biological neural networks; Neural pathways: In the last few decades, the study of the neuronal dynamical behaviors has shifted to a network of neurons from a single neuron. Our group focuses on the study of synchronization phenomena in the neural network and probes the distinct neuronal dynamical behaviors under coupled neural networks (Figure 2). (iii) Flocking behavior; Collective animal behavior exhibited by many living beings such as birds, fish, bacteria and insects: Recently, we consider the flocking behaviors of living beings and discuss the mechanism of avoiding collision between them (Figure 3). (iv) Epidemic models; Disease transmission models : We are also interested in considering the impact of the individual contact networks, awareness and vectors on the disease spreading. (v) Microscopic and macroscopic traffic flow models : The traffic flow problem is one of our research interests. We wish to study some well-known traffic models to explain the traffic flow and the occur of traffic jams. In the future, we aim to use our model to predict and control the traffic flow in freeways of Taiwan.

We are interested in transport and diffusion problems in heterogeneous media. These problems arise from contaminant flows in the subsurface, heat transfer in two-phase media, stress in composite materials, etc. Our attention is mostly on the mathematical modeling, convergence analysis, and numerical computations.

Consider an elliptic equation in a periodic domain with period size ?. It is known that if ? tends to 0, the solution of the elliptic equation approaches a solution of a simple macroscopic equation. Moreover, a convergence estimate can be derived. Therefore, we would like to ask the same questions for realistic problems. If a mathematical model for a realistic problem is available, is it possible to find a simple macroscopic model so that the solution of the simple model still keeps properties close to those of the original realistic problem when measured on long space-time scales? If yes, can we derive a convergence estimate for the realistic solution? Also is it possible to find a simple way to compute the solution of the original realistic problem?

To answer these questions, some basic tools are necessary, for example, knowledge of partial differential equations and ergodic theory, understanding of functional spaces and homogenization methods, and numerical methods.

I am interesting in scientific computing and its applications. In particular, we use finite element method to solve problems from fluid dynamics, elastic mechanics, and geometric optics. In fluid dynamic, we simulate incompressible flow, shallow water and thin film. Robust multigrid methods combined with Krylov iteration are employed to speed up our simulator. In elastic mechanics, we solved the non-linear geometric deformation of thin shell with composite or piezoelectric material. We also apply large diffeomorphic metric map to compute the deformation of 3d images. In geometric optics, we constructed non-image freeforms by solving the Monge-Ampere equation using high order finite element and optimized the design using quantum particle swarm method. Recently, we like to further study on topics including 3d modeling, video tracking and machine learning etc.

In Scientific Computations, we focus on Numerical Computations of Nonlinear Schrodinger's Equations that there are two researching topics in this field: (1) The Ground State and Excited States for the Bose-Einstein Condensates (BECs) (see Fig. 1), (2) Stability Analysis in the Soliton Waves (see Fig. 2). Then in Dynamical Systems, we focus on Chaotic System and its Applications that there are also two researching topics in this field: (3) Investigation on Chaotic Behavior, (4) Digital Chaotic Generators in Secure Communication (see Fig. 4).

The motion of an incompressible Newtonian fluid is described by the incompressible Navier- Stokes equation in the Euclidean space setting. Though the standard derivation of the incompress- ible Navier-Stokes equation based on the Newton’s second law was well-understood already in the 19 century, serious mathematical study of viscous incompressible fluid flows from the view point of Partial Differential Equations only began with the existence theory of Leray-Hopf weak solutions to the incompressible Navier-Stokes equation as established by Leray (1930’s) and Hopf (1950’s). Since then, P.D.E. specialists in this area had devoted great efforts in obtaining the classical smoothness property of Leray-Hopf weak solutions under suitable space-time integrability condi- tions imposed on the weak solutions themselves. However, it is still a long standing open problem to decide whether the breakdown of classical smoothness of Leray-Hopf weak solutions could occur.

My research focuses on the study of analytical, geometrical, and topological properties of viscous fluid flows taking place in a curved space setting. In the past few years, I and professor Magdalena Czubak made a simple yet striking observation about the non-uniqueness phenomena of finite energy viscous fluid flows on a negatively curved surface of constant sectional curvature. and subsequentally established the proper framework of weak solutions which restores uniqueness in the same setting. In the near future, we will try to find a deeper connection between mathematical properties of fluid flows taking place in a curved space setting and those of fluid flows taking place in the standard flat space setting.

The focus of my research is concerned with the development of analytical and computational tools for the problems that arises in fluid dynamics, currently in thin liquid films, and further to communicate with scientists from other disciplines to solve engineering problems in practice.

My current research topics includes (1) Fingering instabilities in Newtonian films; (2) Modeling and analysis of nematic droplets and films; (3) Coherent structures in non-local dispersive active-dissipative systems; (4) Pulse interaction and bound state formation in electrified falling films. For example, Figures. 1 and 2 present 2D and 3D numerical simulations of fingering instabilities arising in falling liquid films, respectively. Figure 3 presents the pattern formation on a thin film of nematic liquid crystal. Figure 4 shows the simulation that represents a liquid film sheared by a turbulent gas.

Currently, my research interests focus on analysis and computation of the Partial Differential Equations (PDEs) arising from geophysical fluid dynamics. For example, equations related to weather prediction and oceanography are the inviscid Primitive Equations (PEs) and the Shallow Water Equations (SWEs).

In Figure 1, the basic level of physical complexity in geophysical fluid dynamics is presented. The Navier-Stokes equations govern the motion of the fluid. In the ocean or atmosphere, due to the thin layer structure (the ratio of the vertical scale of the domain to the horizontal scale is relatively small), Navier-Stokes equation can be simplified in the form of Primitive equations. Shallow water equations are derived from the first mode of the Primitive equations.

The following issues are focused:
1.Nonreflecting boundary conditions
2.Time periodic flows
3.Long time stability

For more details, please see my homepage (Figure 2).

A general principle in many areas of mathematics and science is that the dynamics of a system tend to evolve in a direction which reduces the available energy. This is the least action principle, which postulates the possibility of defining an energy - a functional whose input is taken from a set of possible states - and supposing that the output or “action” of that state prefers local minima. From this perspective, the calculus of variations is very important, since it is precisely the study of the extrema of functionals.

Our research focuses on the regularity of minimizers and how the qualitative properties depend on the differential order of the energy. Here we utilize an object whose study in the literature is surprisingly quite sparse – Riesz fractional gradients – and early research suggests that although there are significant differences in the properties of energies of integer differential order, there is no difference in the regularity or qualitative properties.

Thus far we have obtained several new existence and convergence results for the Partial Differential Equations we consider, as well as new inequalities that provide a nice complement to more classical results. Future research will look to new regularity results for non-linear problems as well as the application to physical problems modeled by these energies.