歷史演講 PAST TALKS

Abstract：

      This thesis investigates the unconstrained optimization problem and its special cases, such as nonlinear least-squares and nonlinear equations, which hold substantial importance in numerical optimization and have numerous practical uses.

      Various methods are available to address these optimization challenges. This thesis investigates a family of quasi-Newton methods, a class of iterative techniques for solving these problems, and demonstrates their advantages over traditional optimization methods such as the steepest descent and Newton’s method. Quasi-Newton methods approximate the Hessian matrix, which encodes second-order information about the objective function, using only gradient evaluations. This approximation significantly reduces computational cost and complexity, making quasi-Newton methods particularly appealing for large-sized problems where calculating the exact Hessian is impractical or impossible.

      By using secant-like diagonal matrix approximations, quasi-Newton methods provide efficient solutions for a wide range of optimization problems, demonstrating their effectiveness across diverse scenarios. These methods also exhibit global convergence properties under suitable conditions.

      This thesis advances the understanding and application of quasi-Newton methods in solving nonlinear least squares and symmetric nonlinear equations. Additionally, by exploring their use in addressing the nearest correlation matrix problems through numerical experiments, this study establishes a foundation for further research and practical implementations in real-world applications.

20240618演講_ 黃德國先生 (國立中央大學數學系)

Abstract：

As the first focus of this work, we aim to develop an efficient and robust solution algorithm for solving hyperbolic partial differential equations exhibiting solution discontinuity and highly local nonlinearity behavior. Such problems arise in numerous applications in computational science and engineering. Traditionally, parallel solution algorithms using time-marching methods focused on parallelizing computations within each time step while maintaining a sequential approach between time steps. Such a sequential step is the potential bottleneck of parallel algorithm performance. On the other hand, with the increasing computational power of modern parallel computer systems, there has been a growing interest in fully coupled space-time solution algorithms for time-dependent partial differential equations (PDEs), enabling parallelism in the temporal domain. To make the solution algorithm practical, it is crucial to have a robust and efficient nonlinear solver for large, sparse systems of equations in space-time algorithms. The inexact Newton with backtracking (INB) method is a well-known approach for solving nonlinear equations, but it exhibits slow convergence in the presence of shocks of our interests. In this work, we propose a new variant of nonlinear elimination preconditioning techniques to enhance the robustness and improve the efficiency of INB tailored to the solution characteristics of hyperbolic PDEs. By employing nonlinear elimination as a right or left preconditioner in conjunction with INB, we introduce two methods: INB with adaptive nonlinear elimination (INB-ANE) and nonlinear elimination preconditioned inexact Newton (NEPIN). These methods aim to reduce the impact of highly local nonlinearity and provide a better initial guess for the next Newton iteration. Numerical results show that the modification enhances the robustness of INB for Burgers' and Buckley-Leverett equations and reveal that NEPIN outperforms INB-ANE in identifying the correct shock location and introducing less interface pollution after the subspace correction before the global update. Additionally, the number of Newton iterations required to converge for NEPIN is almost independent of the time step and mesh size. 

Although nonlinear preconditioning has been shown numerically to be a valuable technique to enhance the robustness and efficiency of Newton-type methods, a comprehensive theoretical or heuristic understanding of this technique still needs to be completed. Therefore, the second focus of this thesis is to introduce a new metric known as the stiffness ratio, which is borrowed from stiff ordinary differential equations (ODEs) to quantify the unbalanced nature of nonlinear systems. This metric is analogous to the condition number of the coefficient matrix for linear systems, which is used to assess matrix quality. Such insights offer valuable guidance and analytical tools for developing preconditioners tailored to specific applications. An investigation of a highly nonlinear algebraic system reveals that NEPIN reduces stiffness compared to INB. To illustrate this concept, we present numerical examples demonstrating the reduction of stiffness ratios of the corresponding Newton ODE system through nonlinear elimination.

20240618演講_梁長雯小姐 (國立中央大學數學系)

Abstract：

The well known Maxwell’s equations in classical electro-magnetirsm give rise to gauge theory with profound mathematical values, especially in the theory of vector bundles. An obvious property in Maxwell’s equations is the dual correspondence of electric and magnetic field by a sign. Such a correspondence can be generalized in the form if “gauge groups” and “dual gauge groups”. We give a brief overview of this correspondence, and describe how they gradually evolve into the theory of Hitchin moduli spaces.

20240516演講_劉之中教授 (成功大學數學系)

Abstract：

Simulating the time evolution of a Hamiltonian system on a classical computer is hard —The computational power required to even describe a quantum system scales exponentially with the number of its constituents, let alone integrate its equations of motion. Hamiltonian simulation on a quantum machine is a possible solution to this challenge – Assuming that a quantum machine composed of spin-1⁄2 particles that we could manipulate at will, then it is tenable to engineer the interaction between those particles according to the one that is to be simulated, and thus predict the value of physical quantities by simply performing the appropriate measurements on the system. Establishing a linkage between the unitary operators described mathematically as a logic solution and the unitary operators recognizable as quantum circuits for execution is therefore essential for algorithm design and circuit implementation. Most current techniques are fallible because of truncation errors or the stagnation at local solutions. This work offers an innovative avenue by tackling the Cartan decomposition with the notion of Lax dynamics. The approach employs an eclectic mix of techniques from Lie theory, dynamical systems, linear algebra, combinatorics, and numerical ODEs. Not only that the process is numerically feasible, but also it produces a genuine unitary synthesis that is optimal in both the precision with controllable integration errors and the usage of only minimally required synthesis components. This talk aims at providing a glimpse into the theoretic and algorithmic foundations by exploiting the geometric properties of Hamiltonian subalgebras and describing a common mechanism for deriving the Lax dynamics. (Lots of examples will be given. This talk does not assume a priori knowledge on quantum computing.)

20240509演講_Prof. Moody T. Chu (Department of Mathematics, North Carolina State University, USA)

Abstract：

布林可滿足性問題（Boolean Satisfiability Problem, SAT）是第一個被證明為 NP-Complete 的問題。如果能夠找到更有效率地求解 SAT 問題的計算方法，對於計算機科學及數位電路的驗證，在求解效率上將非常有利。近年來，由於量子電腦的問世，也愈來愈多研究著手於量子 SAT 求解器的設計，試圖解決在傳統電腦中求解 SAT 問題時所遇到的狀態空間爆炸問題。其中，基於格羅弗演算法 (Grover's algorithm) 的 SAT 求解器也成為了熱門的研究對象。雖然現有的量子 SAT 求解器已經可以一次考慮所有可能的輸入，但它們必須照順序求解函式中的每個子句，使得時間複雜度為 O(m)，與子句的數量 m 呈線性的關係。 為此，我們開發了並行式的量子 SAT 求解器，利用額外的量子位元將每次格羅弗迭代的時間複雜度減少到 O(1)。並且為了更進一步解決並行式 SAT 求解器使用過多量子位元，以及待驗證設計資訊過於集中的問題，我們利用量子遙傳（Quantum Teleportation）提出了分散式量子 SAT 求解器，使得所需的量子位元能夠被分配在各台量子電腦中，並透過這些不同的量子電腦共同完成 SAT 問題。

20240502演講_陳盈如教授 (成功大學電機工程學系)

Abstract：

Multi-agent patrolling generally indicates the scenario where single/multi-mobile agents (i.e., robots, drones, autonomous cars...) persistently cover and scout a physical environment. Efficient assigning resources to ensure security is crucial and unlike typical resource-constraint problems, the challenges have the following characteristics. First, the agents are required to visit/re-visit targets and the feedback may not be obtained in the short run. Secondly, the realistic model naturally combines adversarial settings and the problems are much more complicated and also varied a lot based on different adversary settings.

In this talk, I will briefly introduce the recent trend of Multi-agent patrolling and then show our recent theoretical contributions to one of the close-related objectives: latency. Later, I will show how this objective connects to the realist game-theoretic models and their applications.

講者簡介：

Hao-Tsung Yang is an assistant professor at National Central University. Before that, he was a research associate at the School of Informatics in University of Edinburgh, U.K., supervised by Prof. Rik Sarkar. He receives his Ph.D. degree in Computer Science, Stony Brook University, in 2020, advised by Prof. Jie Gao and Prof. Shan Lin.

Hao-Tsung Yang's research theme lies between autonomous systems, algorithms, and privacy/ fairness/ transparency machine learning. He focuses on new problems and challenges when A.I. comes into human life, including serving humans, interacting & cooperating with humans, or defense from a human-like adversary. For example, an autonomous system such as multi-robot path planning involves multiple works; the control-feedback loop, the algorithm design, privacy, and data misuse. The solutions of these works influence one another, especially considering the human factor in the environment. One can use machine learning techniques to learn and generate good path planning solutions but also may invade people's privacy such as revealing their routine schedule, misusing sensitive data,...etc. On the other hand, the solution may also reveal to the adversary who wants to damage the system and take advantage of it. In a patrol mission, the adversary can predict the arrival time of patrolling robots and launch attacks in vulnerable time slots. These bring new challenges and solutions could be found from algorithmic or machine learning perspectives, and sometimes, combining both together.

20240425演講_楊晧琮教授 (國立中央大學資工系)

Abstract：

In this talk, we propose a policy improvement algorithm for an optimal investment problem to maximize an expected exponential utility of terminal wealth. We employ a general stochastic factor model where the returns and volatilities of assets are random and affected by some economic factors, modeled as diffusion processes. We establish an iteration procedure that converges to the value function and the optimal strategies.

20240321演講_ Prof. Hiroaki Hata (Hitotsubashi University)

Abstract：

In this paper, we introduce multivariate Hawkes processes with jump-diffusion CIR intensity and provide their exact simulation algorithm. The algorithm is an extension of the exact simulation algorithms given in Dassios and Zhao (2017) for generalised self-exciting point processes with CIR-type intensities and their extensions. Namely, we adopt a distributional decomposition for the processes. In addition, we also consider two time discretization schemes in which we approximately generate sample paths of not only counting processes but also intensity processes via time discretization. Through some numerical experiments, we discuss the validity and efficiency of each method.

20240321演講_ Prof. Kazuhiro Yasuda (Hosei Univerisity)

Abstract：

This talk is structured into two parts.

Part I: Introduction to Molecular Communication and Channel Modeling

        To begin, I will give a brief overview of the emerging field of Molecular Communication (MC). Subsequently, we will delve into the mathematical modeling of communication channels used in MC systems. After developing a suitable mathematical model, we will explore the analysis of the Channel Impulse Response (CIR) within diffusive MC systems. In our recent research, we introduce a novel approach where the transmission position, rather than the transmission time, serves as an information carrier. This new channel is termed the First-Arrival-Position (FAP) channel, in which the information is recorded in the first hitting position on the receiver surface. By meticulously examining the relationship between the convection-diffusion equation (a PDE model) and the Itô diffusion process (an SDE model), we aim to find a closed-form FAP density formula for characterizing the channel response of such FAP systems. For this purpose, tools from stochastic analysis are employed, including the generator of a diffusion semigroup and Dynkin′s formula.

Part II: Introduction to Information Theory and Channel Capacity Analysis

        The second part begins with an introduction to basic concepts from information theory. A central concept is the "channel capacity", which encapsulates the fundamental information transmission rate of the channel under consideration. Our recent research tackles the problem of estimating the channel capacity of FAP channels from two perspectives. First, we focus on capacity estimation within a subclass of FAP channels termed VDFAP channels (where VD denotes vertical drift). Second, we explore scenarios under zero-drift assumption. Our results demonstrate that FAP channels exhibit behavior akin to heavy-tailed Cauchy distributions as the background drift velocity approaches zero. Because Cauchy-like distributions lack first and second moments, traditional constraints (e.g., the mean-square constraint) on input distributions are rendered ineffective. To address this challenge, we introduce a relative power measure, the $\alpha$-power, into the field of MC. We explicitly depict the signal space required to characterize the zero-drift FAP channel capacity. Additionally, we derive the Shannon capacity of FAP channels in 2D and 3D spaces in closed form, under a newly proposed logarithmic constraint.

20240314演講_ 李彥奇博士 (鴻海研究院)

Abstract：

Let M be a tensor product of unitarizable irreducible highest weight modules over the Lie (super)algebra g, where g is gl(m|n), osp(2m|2n) or spo(2m|2n). We show that the singular eigenvectors of the quadratic Gaudin Hamiltonians for g on M can be obtained from the singular eigenvectors of the quadratic Gaudin Hamiltonians for the corresponding Lie algebras on some tensor products of finite-dimensional irreducible modules. Consequently, the simultaneous diagonalization of the quadratic Gaudin Hamiltonians for g on the singular space of M is established under a generic condition. This is joint work with Ngau Lam.

20240307演講_ 章源慶教授 (成功大學數學系)

Abstract：

Nowadays, artificial intelligence (AI) has evolved into an indispensable tool driving advancements in modern sciences and technology. Its applications span various domains, including bioinformatics, molecular design, computer science, image analysis and processing, and multimedia applications. Notably, foundational mathematical techniques such as probability theory, statistics, differential equations, convex optimization, and optimal transport form the cornerstone of AI and machine learning (ML) models. Recently, besides these mathematical tools, geometry and topology have revealed their prowess in analyzing intricate structures within data and ML architectures, leading to a notable enhancement in the performance of ML models. Specifically, tools derived from topological data analysis (TDA) and geometric deep learning, such as persistent homology and graph neural networks, have captured the attention of the AI community. These tools exhibit promising results, particularly in molecular design and social network analysis. This presentation offers a comprehensive introduction to the developmental roadmap showcasing the contribution of geometry and topology in modern AI. Additionally, it includes an overview of our current studies on the applications of TDA in material designs, introducing the power of topological and geometric methods in analyzing solar cell materials by showing our promising results in predicting their physical properties.

20240229演講_ 胡全燊博士 (新加坡南洋理工大學)

Abstract：

Diffusion models, pioneers in Generative AI, have significantly propelled the creation of synthetic images, audio, 3D objects/scenes, and proteins. Beyond their role in generation, these models have found practical applications in tasks like media content editing/restoration, as well as in diverse domains such as robotics learning.

In this talk, we'll explore the origins of diffusion models, gaining insights into their mechanisms as differential equations (DE) solving (Song et al. ICLR 2020). With this, we introduced, FP-Diffusion (Lai et al. ICML 2023), improves the diffusion model by aligning it with its underlying mathematical structure, the Fokker-Planck (FP) Equation. Additionally, the link between diffusion models and DE solving reveals limitations associated with the slow sampling speed of thousand-step generation. Motivated by this, we'll introduce the Consistency Trajectory Model (CTM) (Kim & Lai et al. ICLR 2024), an innovative method enabling one-step diffusion model generation while preserving high fidelity and diversity. If time permits, we'll delve into controllable generation using pre-trained diffusion models, showcasing their utility in tasks such as media restoration and user-specified applications, and exploring their practical applications in industrial business.

20240222演講_ 賴杰昕博士 (Sony AI, Tokyo, Japan)

Abstract：

This talk will provide an in-depth exploration of the Tits Alternative, a crucial dichotomous property in group theory. It will trace its origins from the Banach-Tarski paradox to its interpretation through group theory by Van Neumann. The presentation will also highlight the role of the Ping-Pong Lemma, a key theorem in set theory, and the concept of growth rate in the understanding of the Tits Alternative. Further, the speaker will introduce the concept of dynamical canonical height and use it to prove the main result. The talk will conclude with a demonstration of how rational functions can exhibit either polynomial or exponential growth, and an examination of the practical applications of the Tits Alternative in the search for nontrivial solutions in functional equations.

20240222演講_ 彭俊文博士 (國家理論科學研究中心)