陽明交通大學應用數學系

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
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
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.

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.