陽明交通大學應用數學系

Abstract. Transformer-based machine learning models have achieved state-of-the-art performance in many areas. However, the quadratic complexity of the self-attention mechanism in Transformer models with respect to the input length hinders the applicability of Transformer-based models to long sequences or large images. To address this, we present Fast Multipole Attention (FMA), a new attention mechanism that uses a divide-and-conquer strategy to reduce the time and memory complexity of attention from $O(n^2)$ to $O(n \log n)$ or $O(n)$, while retaining a global receptive field. The hierarchical approach groups queries, keys, and values into $O(\log n)$ levels of resolution, where groups at greater distances are increasingly larger in size and the weights to compute group quantities are learned. As such, the interaction between tokens far from each other is considered in lower resolution in an efficient hierarchical manner. This multi-level divide-and-conquer strategy is inspired by fast summation methods from n-body physics and the Fast Multipole Method. We perform evaluation on language modeling and image processing tasks and compare our FMA model with other efficient attention variants on medium-size datasets. We find empirically that the Fast Multipole Transformer outperforms other efficient transformers in terms of memory size and accuracy. For large language models, the FMA mechanism has the potential to enable greater sequence lengths, taking the full context into account in an efficient, naturally hierarchical manner during training and when generating long sequences.

Abstract

A domino (resp., lozenge) tiling is a covering of a region on the square (resp., triangular) lattice using dominoes (resp., lozenges) without gaps or overlaps. In this talk, I will introduce the physical background of these objects and several classical techniques for enumerating tilings of specific regions. Finally, I will present recent results regarding the symmetry classes of domino tilings of the Aztec diamonds. This talk does not assume any prior background; undergraduate students are welcome.

Abstract. The heat equation with a random potential driven by space-time white noise serves as a basic model for continuum directed polymers in random media, a topic of longstanding interest in the physics literature. While this model generates systems exhibiting scale invariance and universality, establishing their fundamental properties has posed deep mathematical challenges. In particular, although classical stochastic analysis can construct the corresponding Gibbs measures for polymers as two-dimensional curves, extending these results to three dimensions introduces substantial complexity due to the critical, non-Gaussian behaviour.

This talk will present the basic ideas of continuum directed polymers in random media, focusing on their behaviour as three-dimensional curves.

Abstract. Newton-type methods are among the most effective tools for solving large-scale nonlinear systems arising in scientific computing. Despite their fast local convergence, their global behavior can be unpredictable, with common issues such as overshooting, stagnation, and sensitivity to problem scaling—especially in stiff or highly unbalanced systems.

In this talk, we present a dynamical systems perspective for understanding these behaviors by interpreting Newton iterations as discrete approximations of an underlying continuous-time flow. This viewpoint provides an intuitive characterization of nonlinear imbalance in terms of stiffness, offering insight into why classical globalization strategies, particularly line search, may become ineffective or overly restrictive.

Motivated by this perspective, we revisit line search methods and introduce improved strategies, including curve search techniques, that better align with the intrinsic dynamics of the nonlinear system. We further show that nonlinear preconditioning can be naturally interpreted as a transformation that reduces stiffness and restores balance, leading to improved robustness and convergence.

Numerical examples from nonlinear PDEs illustrate how this framework not only enhances performance, but also provides a unified viewpoint for understanding globalization, acceleration, and stabilization in Newton-type methods.

Abstract

The Kadomtsev-Petviashvili II (KPII) equation is one of the few physically relevant integrable systems in more than one spatial dimension. In this talk, we present an overview of the inverse scattering theory and the stationary phase method, and explain how these tools are used to derive the long-time asymptotic behavior of solutions.

Abstract
Artificial intelligence (AI) has become increasingly integrated into modern radiology, offering tools that support image interpretation, quantitative analysis, workflow prioritization, and clinical decision-making. Recent advances in machine learning and deep learning have enabled the detection of subtle imaging patterns and the efficient analysis of large-scale imaging data, helping to enhance diagnostic accuracy and improve workflow efficiency. In current radiology practice, AI is already being applied in areas such as lesion detection, organ segmentation, triage, and risk prediction.
Pancreatic cancer remains one of the most lethal malignancies, in part because of its subtle imaging features and the difficulty of early diagnosis on routine CT. In this talk, we will review the emerging role of AI in pancreatic cancer detection using CT imaging, with a focus on recent advances in AI-based detection models and their potential to facilitate earlier diagnosis. We will also briefly discuss the broader real- world applications of AI in radiology and consider the challenges and opportunities for translating these technologies into clinical practice.

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One will present a two expansions of the spectral information to yield the pointwise structure of the Green;s function.

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Consider the set of positive integers representing the number of spanning trees in simple graphs with n vertices. How quickly can this set grow as a function of n? In this talk, we discuss a proof of the exponential growth of this set, which resolves an open problem of Sedlacek from 1966. The proof uses a connection with continued fractions and advances towards Zaremba’s conjecture in number theory. This is joint work with Alex Kontorovich and Igor Pak. This talk is intended for general audience.

Abstract
Group testing (GT) is a mathematical trick to identify a small number of targets from a large population using pooled tests, and it has become increasingly relevant in modern communications. For instance, for uplink, the challenge is to arrange devices who want to talk into frequency–time slots; for downlink, on the other hand, the challenge is to send messages to devices without Alice mistaking Bob's message for her own. 6G will benefit from group testing tricks to support a massive number of devices with highly irregular activity.

This talk applies GT to both downlink and uplink through a single design idea that we call cutting the plum pudding (CTPP). The analogy is simple: plums are randomly distributed in a pudding, and it is difficult to cut out exactly one plum. In our setting, the base station performs randomized cuts over device sets so that exactly one device is highlighted. Once the singleton is found, the task reduces to one-to-one communication, which is significantly easier to handle reliably.

Abstract
Unitary quantum channels play a central role in modeling coherent quantum dynamics and quantum circuits. Reconstructing such channels from finite input–output data is inherently challenging, particularly in the presence of noise, and is closely related to nonconvex optimization on matrix manifolds.
In this work, we develop a unified geometry-aware framework for unitary quantum channel reconstruction by formulating the problem as a constrained optimization task on the Stiefel manifold. For noisy measurement data, we propose an iterative algorithm based on polar decomposition that embeds the unitary constraint directly into the update rule. We prove that the resulting sequence monotonically decreases the objective function and converges to a critical point on the manifold.
In the noise-free setting, we further introduce a direct reconstruction methodology. We show that the global minimizers of the objective function form an equivalence class of unitary matrices, differing only by a global phase factor, and establish conditions under which the underlying quantum channel can be recovered exactly. Leveraging spectral properties of non-degenerate quantum states, we derive a reconstruction procedure that significantly reduces the effective search dimension and requires only a minimal number of quantum observables.
Together, the proposed iterative and direct methods provide a theoretically rigorous and computationally efficient approach for approximating or exactly recovering unitary quantum channels from limited data.

Abstract
We continue the analysis of the defocusing nonlinear Schrödinger equation in the semiclassical limit. With the scattering data approximated using the WKB method, we turn to the inverse-scattering problem which is formulated as a matrix Riemann-Hilbert problem, here also involving the small parameter in a singular fashion. We develop the key ideas of the Deift-Zhou steepest descent method, and show how it leads to phenomena such as wave breaking.

Abstract. Boundary integral equations (BIEs) efficiently reduce elliptic and wave problems to the boundary, but standard implementations require explicit surface parametrizations and produce fully dense matrices. The Implicit Boundary Integral Method (IBIM) avoids parametrization by using a level-set representation and evaluating layer potentials in a tubular neighborhood of a Cartesian grid, at the cost of dense extended operators and high computational expense.

I will present a complementary approach based on spectral-bias-aided multilevel training of neural-network surrogates for IBIM operators. Exploiting the tendency of neural networks to learn low frequencies first, we design a coarse-to-fine training strategy aligned with the IBIM grid hierarchy. This allows information from coarse levels to accelerate training and inference on finer grids, yielding speedups of about 40–600×. I will show results for Laplace and Poisson problems, and briefly discuss extensions to Helmholtz equations and “numerically consistent” machine learning for scientific computing.

Abstract. The global navigation satellite system (GNSS) has been widely used in positioning, navigation, and timing. The GNSS satellite orbit serves as a reference datum in connection to the International GNSS Service (IGS)-defined reference frame, not only for GNSS ranging measurements but also for the so-called precise point positioning (PPP) technique. Therefore, the accuracy of the reference orbit is crucial for precise geodetic applications. On the other hand, in recent years, with the rapid development of the space satellite industry, countries around the world have been actively promoting the research, development, and application of Low Earth Orbit (LEO) satellites. These satellites have been widely utilized in scientific research and commercial sectors and are quickly advancing toward commercialization. LEO satellites refer to satellites operating at altitudes of approximately 200 to 2,000 kilometers above the Earth’s surface. Compared to medium- and high-orbit satellites, LEO satellites offer advantages such as lower orbital altitude, reduced latency, and higher coverage accuracy. As technology progresses and demand increases, the application scope of LEO satellites continues to expand, particularly playing a critical role in Positioning, Navigation, and Timing (PNT) as well as in communication services.

Abstract. Exploring the mathematical principles behind remote sensing, starting with how to define the location using Earth Coordinate Systems and Geodetic Coordinate Systems, and demonstrate how to map satellite images to a Ground Coordinate System. The presentation will also introduce image processing techniques for both optical and Synthetic-aperture radar (SAR) images, including radiometric calibration and geometric correction. Finally, I'll showcase specific applications such as Total Variation (TV) models, image fusion using neural networks and SAR image denoising.

Abstract. Computational galaxy formation has been remarkably successful in recreating realistic galaxies on a supercomputer using the so-called “cosmological simulations”, where a smooth mixture of gas and dark matter in the early Universe gradually evolves into thousands of galaxies similar to those observed today. The key to this success lies in the physical processes (collectively referred to as "feedback") that drive the cycling of gas in and around galaxies. However, all existing cosmological simulations face a fundamental limitation due to their empirical "sub-resolution" models. In this talk, I will introduce the successes and challenges in this field and discuss exciting recent progress on high-resolution, small-scale simulations that aim to tackle the problem by directly modeling the physics in the interstellar medium. I will also discuss the critical role of innovative numerical algorithms in advancing our field.

Abstract.
Mean field games (MFGs) are formulated as nonlinear coupled systems of partial differential equations, consisting of the Fokker–Planck equation, which governs the density distribution of agents, and the Hamilton–Jacobi–Bellman equation, which describes the temporal evolution of their control inputs. Such systems arise in a broad range of applications, including crowd dynamics, control of autonomous vehicle fleets, mathematical biology, engineering, and economics. Since MFGs are typically posed as space–time boundary value problems, numerical schemes designed for standard initial value problems cannot be directly applied. This motivates the development of new computational methods together with a rigorous mathematical foundation. In this talk, I present an implementation-friendly approach based on a generalized conditional gradient (GCG) method and discuss its convergence properties. In particular, I report recent results, obtained in collaboration with H. Nakamura, for MFGs with local coupling terms.

Abstract
Given a graph and a function on its vertices, how do we partition the vertices into clusters so that (1) vertices with similar function values are in the same cluster and (2) the induced subgraph on each cluster is connected as much as possible? Such a problem has applications in detecting the sources of air pollution, image segmentation, and so on. We will go through the theoretical background of this algorithm and demonstrate some of its applications.

Abstract
The Quantic Tensor Train (QTT) is a tensor network framework that provides highly compact representations of high-dimensional functions and operators. This makes it a promising tool for tackling nonlinear partial differential equations that are otherwise computationally demanding. In this work, we explore the use of QTT for solving the Gross–Pitaevskii equation (GPE), a fundamental nonlinear model describing Bose–Einstein condensates and related quantum systems. By using QTT, we significantly reduce the computational complexity compared to conventional discretization methods, achieving efficient and scalable solutions. Our results demonstrate the potential of QTT to extend tensor network techniques beyond linear problems, opening the door to new applications in nonlinear quantum dynamics and many-body physics.