陽明交通大學應用數學系

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.

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Abstract
Semiclassical analysis involves the study of the solutions of singularly-perturbed partial differential equations with given initial data or boundary conditions. The singular perturbation enters via small multipliers on derivatives, and the aim is to describe how the given data taken independent of the small parameter evolves over space and time in the asymptotic limit that the parameter tends to zero. This lecture will introduce the topic, present illustrations from the theory of linear and integrable nonlinear equations, and then embark on a detailed analysis of the Cauchy problem for the defocusing nonlinear Schrödinger equation in the semiclassical limit. The latter problem is solved via an inverse-scattering transform and here we focus on the computation of scattering data in the semiclassical limit, which employs the WKB method.

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
Ice sheet flow laws currently use coefficients that depend on their respective power-law exponents, complicating systematic parameter exploration in models. This study proposes dimensionless formulations for both basal sliding and internal deformation laws, which decouple coefficients from exponents, simplifying sensitivity studies and making independent variation of these parameters feasible in ice-sheet models.

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.