歷史演講 PAST TALKS

Abstract：

The Boltzmann equation, a fundamental type of kinetic equation, serves as a high-fidelity model for simulating fluid dynamic. By capturing dynamics in both space and velocity dimensions (also known as phase space), it offers valuable insights into plasma instabilities, one of the most critical challenges in fusion devices and other plasma applications. However, directly discretizing the six-dimensional phase space is computationally prohibitive. Low-rank approximation provides an efficient model reduction technique for handling large system matrices across various applications, such as image compression, linear dynamical systems, and regularization methods for ill-posed problems. Dynamical low-rank approximation specifically targets time-dependent data matrices and solutions to matrix differential equations. By projecting the time derivative onto the tangent space of the manifold of rank-r matrices, this approach ensures that the solution matrix maintains a fixed rank throughout the simulation. This talk will first introduce the plasma kinetic equations and then focus on application of dynamical low-rank approximation to the Vlasov-Poisson equation, a simplified form of the plasma Boltzmann equation that excludes the collision operator and magnetic force. Numerical experiments demonstrate strong alignment with the theoretical predictions, showcasing the advantages of this method over direct phase-space discretization. Dynamical low-rank approximation thus emerges as a promising strategy to mitigate the computational cost of solving plasma kinetic equations.

20241226演講_鄭宇皓先生 (Department of Applied Mathematics, University of Washington)

Abstract：

In this talk, we will see the fundamental ideas and tools needed to perform numerical optimization on matrix manifolds. We will introduce the main tools from Riemannian geometry that allow us to extend the classical steepest descent algorithm to matrix manifolds, with particular emphasis on retraction mapping, and present some numerical examples. To keep the presentation more concrete, we will use the Stiefel manifold as a running example, but the concepts discussed herein remain valid for any embedded submanifold. We will discuss applications on the Stifel manifold, the power manifold of unit spheres, and the manifold of fixed-rank matrices.

20241212演講_Dr. Marco Sutti (國家理論科學研究中心)

Abstract：

In the talk, I would like to introduce some histories of calculus of variation and apply a direct method to find the global minimizer of a functional. In addition, some partial differential equations with different boundary conditions will be studied from the viewpoint of variational method. Finally, I will construct the existence of traveling waves of reaction-diffusion equations by a variational approach.

20241205演講_黃志強教授 (國立中正大學數學系)

Abstract：

Modern economics research tries to obtain causal effects, not just correlations or associations with an outcome, and instrumental variables (IV) are one method of doing so. In this seminar, I will use linear algebra to show the proof of the Gauss-Markov theorem and its derivation of ordinary least squares (OLS) regression as the best linear unbiased estimator. That result holds when errors are uncorrelated between observations, but that is often not the case in real data. Next, I will show the extension to IV estimators and describe when they can be used to obtain unbiased results. The paper I will present uses the effects of wind direction on daily air pollution in New York City (NYC) to show that PM2.5 and Ozone pollution both cause children to get sick and be absent from school. The direction that the wind is blowing on any given day is quasi-random, satisfying the condition that errors are uncorrelated between observations, and pollution gets carried hundreds of miles by wind such that the wind direction in NYC can nearly double PM2.5 pollution that day. Using daily attendance data on all NYC public schools from 2006 to 2019, I show that both Ozone and PM2.5 pollution cause children to be sick and miss school. I show that improvements in air quality have prevented approximately 380,000 absences per year which increased NYC school budgets by at least $19 million per year.

20241205演講_Dr. Daniel Guth (University of Rochester Medical Center)

【原定10/31演講，延期至11/28】

Abstract：

This study examines clusterability testing for a signed graph in the bounded-degree model. Our contribu6ons are two-fold. First, we provide a quantum algorithm with query complexity Õ(N^{1/3}) for testing clusterability, which yields a polynomial speedup over the best classical clusterability tester known [Adriaens and Apers, 2023]. Second, we prove an ΩV(N^{1/2}) classical query lower bound for testing clusterability, which nearly matches the upper bound from [Adriaens and Apers, 2023]. This seWles the classical query complexity of clusterability testing, and it shows that our quantum algorithm has an advantage over any classical algorithm.

20241128演講_陳國進博士 (鴻海研究院)

Abstract：

Program verification is a foundational discipline in computer science that ensures software correctness through formal methods, often contrasted with software testing. While testing explores specific inputs and observes outputs to detect errors, verification provides mathematical guarantees about a program’s behavior for all possible inputs. On the other hand, with the rise of quantum computing, a new frontier in program verification has emerged: quantum program verification. Quantum programs, unlike classical ones, exploit the principles of superposition, entanglement, and probabilistic outcomes, presenting unique challenges.

In this talk, I will give an introduction to program verification and several recent approaches to the challenges of quantum program verification.

20241114演講_林濬璈教授 (臺北科技大學創新前瞻科技研究學院)

Abstract：

In the era of big data, real-world data like digital photographs, MRI scans, or speech signals usually has a high dimensionality, we need to reduce the dimensionality adequated in order to save data in the limited storage space and to transmit efficiently. Some common methods of dimensionality reduction such as singular value decomposition, principal component analysis, independent component analysis and autoencoders. However, for high dimensionality data, these methods suffer from heavily computational complexity and memory requirement in practice. Therefore, we provide an efficient and accurate method, which is the numerical solution of the projected nonsymmetric algebaric Riccati equations (pNAREs) arisen in the refinement of estimates of invariant subspaces (REIS). The theoretical contribution of our paper is threefold. Firstly, we introduce how to apply SVD to do image compression and provide some mathematics measurement tools. Secondly, the core of this paper is to adapt REIS and find the first singular values to do image compression, especially for high resolution images. The main idea of REIS is applied to large-scale and real matrices via pNAREs or their associated Sylvester equations through Newton’s method. Thirdly, we describe some measurement tools like compression ratio, mean square error, peak signal to noise ratio and structural similarity index to evaluate the performance of the compression factors and the quality of the compressed images. Furthermore, the operation counts and computational complexity are also provided. Finally, we present some numerical experiments about some real world image dataset to show the feasibility of our proposed algorithm.

20241024演講_翁章譯教授 (國立嘉義大學應用數學系)

Abstract：

在ML領域，人們心中一直藏著幾個不敢對外說出口的秘密。其中最大的一個秘密是，實際上我們並不確切知道為什麼DNN如此有效。缺乏對背後理論的共識，使我們在機器學習工作中常常將AI視為黑盒，無腦地試手氣方式調整參數。本場次中，Birdman將以輕鬆易懂的科普方式，從機器學習的發展，一路漫談到當前的機器學習理論，與類神經網路的動力系統，並分享個人的學習心得和獨到見解，希望能啟發更多人一同探索機器學習算法背後的真相。

20241017演講_邱銘彰 (Birdman)先生（奧義智慧科技 創辦人）