|Colloquium, 台灣大學大氣科學系 郭鴻基 講座教授|
Thursday, October 8, 15:10—16:00 數學系3174
Title: Two-Dimensional Turbulence and Typhoon Vorticity Dynamics (兩度空間亂流與颱風渦度動力)
Abstract:Earth’s atmosphere and ocean, under the influence of rotation and density stratification, exhibit complex patterns of fluid motion over a wide range of space and time scales. The two-dimensional turbulence, which may be stemmed either from the fluid rotation or the fluid internal stratification, has been a paradigm for atmospheric oceanic fluid dynamics (AOFD). It is a remarkable fact that for any type of random initial state or external forcing, a two-dimensional fluid will rapidly organize itself into a system of coherent, interacting vortices swimming through a sea of passive filamentary structure produced from earlier vortex interactions. This discipline has also been instrumental in the development of single charge plasma physics, galaxy spiral, the Great Red spot, the ozone-hole problem and the typhoon dynamics. A brief review of 2D and 3D turbulence will be given. The mathematical nature of the 2D turbulence will be addressed. The nonlinear typhoon vorticity dynamics, such as merger dynamics, the eyewall rotation and mixing, the successive formation of tropical vortices and the concentric eyewall formation dynamics will be presented. These nonlinear vorticity dynamics are shown to bear great resemblance to the observations. 演講討論兩度空間亂流與颱風渦度動力。地球大氣海洋流體受地球旋轉與密度層化影響，其動力特性是兩度空間亂流學的典範；兩度空間亂流非線性尺度交互是往大尺度移動，渦漩合併成大渦漩以及大量環繞周邊細絲渦度帶，是兩度空間亂流特性。介紹以兩度空間亂流相關的物理現象，並以此觀點探討的颱風渦度動力，也會討論數學與此動力的相關性。
|Colloquium, 淡江大學數學系 楊定揮教授|
Thursday, October 8, 16:10—17:00 數學系3174
Title: Some Recent Works on the Dynamics and Traveling Wave Solutions of Non-Monotone Three Species Food Web Models
Abstract: Three species food web models are fundamental building blocks of large scale ecosystems. To clarify the local or global and short-term or long-term behavior of ecosystems, it is essential to understand the interacting dynamics of three species food web models. A monotone ecosystem whose interactions among n-species are all cooperative or competitive (n = 2) have been well studied in the past three decades by the theory of monotone dynamical systems. However, for a non-monotone system whose interactions are blended at least with one consumption (i.e. herbivory, predation or parasitism), most known results are constrained on two species cases since the classical Poincare-Bendixson Theorem can be applied. Hence recent attention has been attracted to the dynamics of a non-monotone ecosystem with at least three species. In this talk, we will first survey some recent progress on the non-diffusive predator-prey food web models. Then, with diffusion, the existence of traveling wave solutions by upper lower solutions/shooting method are investigated. Finally, some open problems of these topics are proposed.
|Colloquium, 交通大學通識中心 陳明璋教授|
Thursday, October 15, 15:10—17:00 數學系3174
|Colloquium, 國立交通大學AI學院 魏澤人教授|
Thursday, October 22, 15:10—16:00 數學系3174
Abstract: 本次演講介紹一些基礎數學手法在深度學習中的應用。主要的主題如下: * 用數學上的平面、立體幾何來幫助設計神經網路模型 * 用不變量的概念，設計及訓練神經網路結構 * Neural ODE, 用微分方程配合神經網路 * 數學定義模式與神經網路結構的相似性
|Colloquium, 交通大學數學系 林奕亘助理教授|
Thursday, October 29, 15:10—16:00 數學系3174
Title: Monotonicity-Based Inversion of the Fractional Schrodinger Equation
Abstract: We consider an inverse problem for the fractional Schrodinger equation by using monotonicity formulas. We provide if-and-only-if monotonicity relations between positive bounded potentials and their associated nonlocal Dirichlet-to-Neumann maps. Based on the monotonicity relation, we can prove uniqueness for the nonlocal Calderon problem in a constructive manner. Secondly, we offer a reconstruction method for an unknown obstacles in a given domain. Our method is independent of the dimension and only requires the background solution of the fractional Schrodinger equation.
|Colloquium, 國立中央大學數學系 黃楓南教授|
Thursday, October 29, 16:10—17:00 數學系3174
Title: Data Assimilation Technique with Long Short Term Memory Networks for Highway Traffic Flow Prediction
Abstract: The development of accurate and reliable computational traffic flow prediction tools has always been an active research topic in transportation engineering and planning. Generally, the available predictive tools are divided into three categories, namely parametric methods, nonparametric methods, and PDE-based simulations. In particular, the machine learning methods (such as the k-nearest neighbor (K-NN) method and the long short term memory networks (LSTM)) are the nonparametric techniques, and the autoregressive integrated moving average (ARIMA) and its variants are one of the most representative parametric methods. In this work, we propose the data assimilation technique with the long short term memory networks (LSTM) for predicting the highway traffic flows. The proposed method is developed based on the framework of the Karman filtering (KF) algorithm, which consists of two key components: the prediction step and the correction step. The predicted value is obtained by performing numerical simulation and then corrected by Karman filtering with real data. As the numerical simulator, which is a kernel component of the predictive tool, we use an explicit Godunov’s method to discretize the Lighthill-Whitham-Richards model, where the MacNicholas formulation is used as the fundamental relation between the velocity and density. Since the data at the upstream boundary point in the future period is not available. The pseudo-predicted values obtained by using LSTM are used for setting boundary conditions. In this study, we use Seasonal ARIMA (SARIMA), LSTM methods as baseline methods and compare them with our proposed method. The numerical results based on the real traffic for the Hsuehshan Tunnel extracted from the Freeway Bureau Ministry of Transportation and Communications (MOTC) database in Taiwan show that our method outperforms SARIMA and LSTM as well as the classical KF method.
|PDE Seminar, 國立台灣大學數學系 王振男教授|
Friday, October 30, 11:10—12:00 數學系3樓會議室
Title: Non-Radiating Sources for the Elastic Waves in Anisotropic Inhomogeneous Media
Abstract: In this talk, I would like to discuss the characterization of non-radiating volume and surface (faulting) sources for the elastic waves in anisotropic inhomogeneous media. Each type of the source can be decomposed into a radiating part and a non-radiating part. The radiating part can be unique determined by an explicit formula containing the near-field measurements. On the other hand, the non-radiating part does not induce scattered waves at a certain frequency. In other words, such non-radiating source can not be detected by measuring field at one single frequency in a region outside of the domain where the source is located. This is a recent joint work with Pu-Zhao Kow.