Fall 2019

The waveform morphology in the cardiovascular signal data reflects physiological dynamic interaction inside the human body. By organizing the subtle change of waveform morphology from large amount of data, unsupervised manifold learning help delineate a high dimensional structure. Using diffusion map, we may gain more insight for signal processing, data analysis and classification. Development in engineering warrants the reliability of clinical application. Therefore, I will present several challenging problems for further discussion.

In this talk we will introduce several inverse problems, such as the famous Calder\'on problem. Roughly speaking, inverse problems investigate the question of determining the interior physical information of a medium from its boundary measurements. Furthermore, we also introduce linear, non-linear and non-local inverse problems in this talk.

We present a strongly-coupled immersed-boundary method for flow–structure interaction problems involving rigid bodies. The method is stable for arbitrary choices of solid-to-fluid mass ratios and for large rigid-body motions. Dynamics of rigid bodies is characterized by the velocity of the center of mass and the angular velocity about the center of mass, which are governed by the translational and rotational equations of motion involving boundary forces. By introducing the summation and distribution operators, equations of motion are coupled with the incompressible Navier-Stokes equations through the no-slip constraint and boundary forces. Boundary forces are regarded as Lagrange multipliers that enable the no-slip constraint to be implicitly determined to arbitrary precision without associated time-step restrictions. Through projections, the current method removes not only slip component of the velocity field but also components of rigid-body dynamics that do not satisfy equations of motion. We verify our method by simulating the free fall of a circular cylinder and further apply our method for investigating the starting of a flow-driven vertical-axis wind turbine.

In the early 1980 Bannai asked whether the Q-polynomial distance-regular graphs with large diameter can be classified. In this talk I will first give an overview what is known. If there is time I will discuss some open problems.

Global optimization of expensive functions has important applications in physical and computer experiments. It is a challenging problem to develop efficient optimization scheme, because each function evaluation can be costly and the derivative information of the function is often not available. We propose a novel global optimization framework using adaptive Radial Basis Functions (RBF) based surrogate model via uncertainty quantification. The framework consists of two iteration steps. It first employs an RBF-based Bayesian surrogate model to approximate the true function, where the parameters of the RBFs can be adaptively estimated and updated each time a new point is explored. Then it utilizes a model-guided selection criterion to identify a new point from a candidate set for function evaluation. The selection criterion adopted here is a sample version of the expected improvement (EI) criterion. We conduct simulation studies with standard test functions, which show that the proposed method is more efficient and stable in searching the global optimizer than two existing methods.

Two-dimensional free-surface flow over a localised bottom topography is examined with an emphasis on calculating steady, forced solitary-wave solutions. In particular we focus on the case of a Gaussian dip topography. Most of the focus is on the weakly-nonlinear limit where a forced KdV equation is applicable, and the problem essentially boils down to solving a forced nonlinear ODE with a single parameter that quantifies the amplitude of the topography. This equation has a rich solution space with a large (probably infinite) number of solution branches. Asymptotic analysis for small topography amplitude reveal some interesting features, for example an internal boundary layer which mediates a change from exponential to algebraic decay of the free-surface in the far-field. Traditional boundary-layer theory fails beyond the first two solution branches, where the surface profiles feature multiple waves trapped over the topography. The stability of the steady solutions will also be briefly discussed.

這個演講分成兩個部分，前半段分享美國工業與應用數學會（Society for Industrial and Applied Mathematics，簡稱 SIAM）在2012年發表了一篇報告: 產業中的數學(SIAM Report on Mathematics in Industry)。他們發現不論是在傳統或嶄新產業，數學與科學計算皆有愈來愈多的應用。這些應用會對公司的盈虧造成極大幅的影響。我將跟大家分享我個人的經驗裡數學在哪些產業與應用提供了什麼貢獻，並且稍微介紹計算數學的一些核心思想。 後半段我介紹近期在可壓縮歐拉方程問題中，解接近真空時的一個簡化模型。在這研究中我們提中一個新的角度來思考當密度接近0時，如何逼近歐拉方程的解。我們提出的簡化模型可以解決Glimm方法中，因為解接近真空而造成逼近解的總變化量過大的問題。進而證明了當初使密度夠小時，歐拉方程的解都會在O(1)時間存在。這份工作是與中央大學洪盟凱教授與李信儀博士一起完成的。

Minimizing the logarithmic loss is an essential task to many applications, such as positron emission tomography, optimal portfolio selection, and quantum state tomography. The optimization problems are indeed convex. However, the logarithmic loss violates standard smoothness assumptions in optimization literature, so most existing optimization algorithms and/or their convergence guarantees do not directly apply. In this talk, I will introduce some of our recent developments in addressing the logarithmic loss.

To understand and manipulate the numerical error is an important issue in scientific computing. The behavior of errors is various and interesting. Errors are not always a bad things in applications. In this lecture, I will introduce some interesting and unexpected examples. Moreover, some useful applications designed by the numerical error will be shown.

透過大量的資料取得與電腦計算能力的增強，資料科學的研究領域近年來在學界與產業界裡均受到高度的重視，許多在機器學習與資料探勘中被提出來的預測模型也確實滿足了不少在資料分析上的需求，而機器學習如何落地也是大家很關注的議題。這次的演講主要是分享過去與業界產學合作的經驗，包含金融核貸、資料安全與物聯網相關的議題。

In this talk, we will introduce nonlinear eigenvalue problems, including tensor eigenvalue problems and nonlinear Schrödinger equations. We will discuss its numerical methods and some numerical results. A great advantage of this method is that it converges quadratically and is positivity preserving in the sense that the vectors approximating the Perron vector (or the ground state vector) are strictly positive in each iteration.

We construct singular solutions to a semilinear elliptic equation with exponential nonlinearity on a bounded domain in $\mathbb{R}^2$. To show the existence of singular solutions, we introduce an inverse problem and utilize a Born-Infeld approximation scheme. Ruling out a possible occurrence of bubbling phenomenon, we show that as the Born-Infeld parameter tends to infinity, solutions of the inverse problem on a subdomain with finitely many holes can be used to approximate singular solutions to the original problem. Our work rigorously justifies the Born-Infeld-Higgs approximation to the abelian Maxwell-Higgs theory. This is a joint work with Yong Yu and Ho Man Tai (CUHK).

In this talk, I will first describe briefly my backgrounds about how to encounter such a fascinating topic of solving polynomial systems. Three different tools, Sylvester resultants, Wu’s method, and Groebner bases will be introduced as different viewpoints in solving polynomial systems. Some applications to point vortex problems and N-body problems will be prevented.

Brain source imaging is an important method for noninvasively characterizing brain activity using Electroencephalogram (EEG) or Magnetoencephalography (MEG) recordings. Traditional EEG/MEG Source Imaging (ESI) methods usually assume that either source activities at different time points are unrelated, or that similar spatiotemporal patterns exist across an entire study period. The former assumption makes ESI analyses sensitive to noise, while the latter renders ESI analyses unable to account for time-varying patterns of activity. To effectively deal with noise while maintaining flexibility and continuity among brain activation patterns, we propose a novel probabilistic ESI model based on a hierarchical graph prior. In our method, a spanning tree constraint is imposed to ensure that activity patterns have spatiotemporal continuity. An efficient algorithm based on alternating convex search is presented to solve the proposed model and is provably convergent. Comprehensive numerical studies using synthetic data on a real brain model are conducted under different levels of signal-to-noise ratio (SNR) from both sensor and source spaces. We also examine the EEG/MEG data from two real applications, in which our ESI reconstructions are neurologically plausible. All results demonstrate significant improvements of the proposed algorithm over the benchmark methods in terms of source localization performance, especially at high noise levels.

In this talk, we aim to have a glance of the spark between dynamics and geometry. I will start from ``what are dynamical systems'' and try to answer ``how could dynamics tell us about the geometry." Along the way, I will introduce works of Sinai (2014 Abel Prize winner), Margulis (1978 Fields medalist), Thurston (1982 Fields medalist), McMullen (1998 Fields medalist), Mirzakhani (2014 Fields medalist), and some of my recent works on entropy rigidity and pressure metrics.