Computational Mathematics & Scientific Computing for Young Researchers

A Semi-Analytical Method for the Poisson Equations on Unbounded Domains with Boundaries

Chien-Chang Yen, Fu Jen Catholic University

The Poisson equations on a finite region have been well studied in historical literature. In this talk, we consider domains with boundaries that are unbounded, such as a half plane or the quarter plane. The proposed method, based on the integral form, is free of artificial boundaries under the assumption that the density has compact supports. Moreover, the order of accuracy is induced from the order of Taylor expansion of the density, and fast nearly linear computational complexity can be obtained if uniform grid discretization is used. Finally, the numerical comparison study shows that this method outperforms the finite difference approach.

Co-authors: Tzu-Ching Liu, Zhi-Yi Liu, Feng-Nan Hwang.

Slicing Support Functions, Recovery Formulas, and Monge-Ampère Structures in Convex Geometry

Yen-Chang Huang, National Yang Ming Chiao Tung University

The support function is a fundamental tool in convex geometry, encoding the shape of a convex body through its supporting hyperplanes. In this talk, we introduce the slicing support function, a variant arising from extremal affine slices of a convex body. This function provides an alternative representation of convex geometric data and offers a lower-dimensional perspective on the classical support function.

We show that the slicing support function can be obtained from the classical support function through an infimal-convolution-type formula. This recovery structure reveals how slicing data are encoded in the original support function while reducing the dimensional complexity of the problem.

Under suitable convexity assumptions, the associated minimizer is unique almost everywhere, which leads to differentiability properties of the slicing support function. We further derive a nonlinear partial differential equation of Monge-Ampère type satisfied by the slicing support function. This equation highlights a connection between slicing geometry, curvature reconstruction, and nonlinear geometric analysis.

SDRE Control in IAG, IBVS and Bipedal Robot

Chin-Tien Wu, National Yang Ming Chiao Tung University

This talk presents an accelerated nonlinear optimal control framework based on the State-Dependent Riccati Equation (SDRE) for three critical robotic applications: Impact Angle Guidance (IAG), Image-Based Visual Servoing (IBVS) for quadrotors, and bipedal leg control. For IAG and IBVS, achieving finite-time control under noisy perturbations is practically essential. We propose the Mi-SDRE-DKF framework, which integrates SDRE control with a Discrete Kalman Filter by employing the Structure-Preserving Doubling Algorithm (SDA) naturally combined with binary power iteration. This synergistic approach significantly accelerates the computation of Riccati and Lyapunov equations, enabling its use in real-time control. The SDA algorithm adapts high hardware efficiency, maintaining a 641 Hz control frequency on FPGA.

In parallel, a suspended bipedal robot is utilized as a test bench for exoskeleton devices, where high reliability and repeatability of human lower limb gaits are mandatory. To address challenges such as motor under-actuation and model dynamics uncertainty, we implement a three-stage control strategy featuring an offline PID-LQR compensation step. Since the first and third stages of this strategy are formulated under the SDRE framework, the computational acceleration provided by SDA is also successfully extended to the bipedal system. Experimental results demonstrate high-fidelity motion reproduction with an average RMSE below 3°.

Generalized Debye-Hückel Theory Based on the Poisson-Fermi Model for Mixed-Salt and Mixed-Solvent Electrolyte Systems

Chin-Lung Li, National Tsing Hua University

The concept of ion activity coefficients plays a fundamental role in describing the non-ideal behavior of electrolyte solutions, where interactions among ions and solvent molecules cause deviations from ideal solution assumptions. Accurate modeling of ion activities is therefore crucial for predicting and understanding the thermodynamic behavior of electrolyte systems. In this talk, a generalized Debye-Hückel theory is proposed for the thermodynamic modeling of ion activities in mixed-salt and mixed-solvent electrolyte solutions.

The theory is based on a molecular mean-field Poisson-Fermi model that treats ions and solvent molecules of any volume and shape with interstitial voids and takes account of ion-ion, ion-solvent, and solvent-solvent correlations, polarizability of solvent molecules and ions, and permittivity variations with ionic strength, temperature, polarizability, and location. The unique analytic solution of the electric potential can be obtained by the linearization of the fourth-order Poisson-Fermi model with some additional boundary and interface conditions in the spherically symmetric domain. The mean ionic activity coefficient is explicitly derived from the excess chemical potential of the ions in electrolyte solutions. Some numerical examples are presented to illustrate the novelty of this theory in capturing the physical properties and interactions.

Age of Information for Periodic Status Updates Under Sequence-Based Scheduling

Yuan-Hsun Lo, National Chengchi University

In many applications of Internet of Things, such as temperature and air pollution monitoring or traffic condition detection for autonomous driving, received information usually has a higher value when it is fresher. Age-of-information (AoI) is a newly defined performance metric to quantify the information freshness over such a wireless access networks. In this talk, we focus on AoI performance in the scenario where multiple users transmit periodically generated packets to an access point without feedback. The adopted scheme is based on protocol sequences, where each user is assigned a periodic binary sequence to schedule their transmissions. We will provide some progress on the average AoI under sequence-based scheme, including low-complexity closed-form expressions for some special cases and some properties of the sequence structure to optimize the AoI performance.

From Differential Operators to Matrices: Computational Perspectives on Enumerative Polynomials

Chun-Cheng Yeh, National Kaohsiung Normal University

Many important polynomials in enumerative combinatorics — such as the Eulerian polynomials, the second-order Eulerian polynomials, and the alternating run polynomials — arise naturally from the repeated action of a differential operator. Starting from this operator viewpoint, the talk follows a path from continuous operations to discrete and matrix-theoretic structures, describing several complementary approaches that speak to one another. On the symbolic side, context-free grammars offer a systematic way to generate these polynomials. On the combinatorial side, simple algorithms translate the expansion of operator powers into discrete structures such as increasing trees, set partitions, and Young tableaux. On the algebraic side, these polynomials admit determinantal representations — in particular of lower Hessenberg type — that recast them in the language of numerical linear algebra, with the zeros and positivity of the polynomials reflecting the underlying combinatorial structure. The talk is intended to be expository and accessible to a general audience.

A Characterization of Positive Entropy of Markov Tree-Shifts

Nai-Chu Huang, National University of Kaohsiung

Topological entropy is often considered as an indicator of complexity. However, there is no finitely checkable algorithm to determine the positive entropy of multidimensional shifts of finite type. Indeed, it is right recursively enumerable. In this presentation, we will present the characterization of positive entropy in Markov tree-shifts by adjacency matrices.

Inverse Fiedler Vector Problem of a Graph

Jephian C.-H. Lin, National Yang Ming Chiao Tung University

Given a graph and one of its weighted Laplacian matrices, a Fiedler vector is an eigenvector with respect to the second smallest eigenvalue. The Fiedler vectors have been used widely for graph partitioning, graph drawing, spectral clustering, and finding the characteristic set. For a given tree, we characterize all possible Fiedler vectors among its weighted Laplacian matrices. As an application, the characteristic set can be anywhere on a tree, except for the set containing a single leaf.

This is a joint work with Mahsa N Shirazi.

White-Box Physics-AI for Reaction–Diffusion Systems: EnVarA-Guided Constitutive Learning and Stage-Consistent Time Integration

Min-Jhe Lu, Institute of Computational and Modeling Science, National Tsing Hua University

We present a white-box physics-AI framework for learning reaction–diffusion dynamics while retaining the mathematical and numerical structure of the governing equations. Guided by the energetic variational approach (EnVarA), we separate prescribed transport kinematics from model-dependent constitutive physics.

For systems of the form ∂_tρ + ∇·(ρu) = ρr, the density is factorized as ρ = MI, where I describes local compression and M follows material trajectories while accumulating reaction. This decomposition provides a transparent interface between learned velocity and reaction laws and a conventional numerical solver: a finite-volume method updates the compression factor, while trajectory-based transport and reaction splitting update the mass factor.

The resulting neural solver is tested across seven PDE families, including diffusion, phase separation, porous-medium dynamics, reaction variants, and a coupled two-species system. The numerical experiments demonstrate transfer across spatial grids without retraining, coherent evolution beyond the training interval, and stable long-time pattern formation.

We then address a temporal-accuracy issue that becomes important when the constitutive law depends on the evolving density. At every Runge–Kutta stage, the density must first be reconstructed and the constitutive law evaluated again at that stage state. A matched comparison between an updated law and a reused law, together with a fixed-grid Fourier calculation, explains the observed split between second- and first-order time accuracy. The rigorous analysis covers prescribed smooth fields; extensions involving nonlinear learned feedback require additional regularity and stage-wise control.

This work is carried out jointly with Professor Shih-Hsuan Hung and master’s student Shang-Ke Chen from the Department of Computer Science, master’s student Kuan-Yu Chen from the Institute of Computational and Modeling Science, all at National Tsing Hua University, and Chao-Shun Zhan and Johnson Sun from the NVIDIA AI Technology Center, NVIDIA Corporation, Taipei, Taiwan.

On Synchronization Analysis of Kuramoto Oscillators Incorporating Amplitude Dynamics

Shih-Hsin Chen, National Chung Hsing University

Synchronization is regarded as a universal feature. It has also been observed and applied in various disciplines, including circadian rhythms, firefly flashes, biological oscillators, etc. To demonstrate such phenomena, the Kuramoto model has been investigated and widely applied to Josephson junction arrays, power systems, network science, etc. In this talk, I shall briefly give an introduction and introduce the 1st-order and 2nd-order Kuramoto models with the amplitude dynamics. Then I shall present the main results and key lemmas. Finally, I will show the numerical simulations and give a summary.

Sphere Retraction Normalizations for Stable Deep Transformers

Cheng-Fang Su, National Yang Ming Chiao Tung University

Stable training of deep neural networks remains a central issue in modern deep learning. Although residual connections and normalization methods have been widely used in architectures such as ResNets and Transformers, increasing model depth still raises challenges related to hidden-state scale control, gradient propagation, and numerical stability under low-precision computation.

In this talk, I will revisit deep connection and normalization mechanisms from the viewpoint of spherical geometry. We propose p-SpheretNorm, a class of normalization methods based on retraction maps on the hypersphere. The main idea is to constrain hidden states to a fixed-scale spherical manifold and interpret layer-wise transformations as geometric residual updates along tangent directions. From this perspective, p-SpheretNorm preserves the scale of hidden states during forward propagation and provides a mathematical framework for studying backward propagation and Jacobian stability.

The talk will introduce the motivation, geometric construction, and stability viewpoint behind p-SpheretNorm, followed by a brief discussion of its potential application to Transformer-style architectures and empirical observations. This is joint work with Professor Min-Te Sun and Ph.D. student Jie Zhang from the Department of Computer Science and Information Engineering, National Central University.

Multi-Objective Bayesian Optimization of CPU Cooling Design with Mixed Variables using Category Tree Gaussian Process

Ray-Bing Chen, National Tsing Hua University

As computing power advances, optimizing CPU cooling systems has become increasingly important for maintaining reliability and thermal performance. Designing such systems often requires computationally expensive high-fidelity simulations involving both qualitative and quantitative inputs while producing multiple performance outputs, leading to a challenging mixed-input multi-objective optimization (MOO) problem. To address this challenge, we investigate surrogate modeling for multi-response computer experiments with mixed-type inputs.

Taking advantage of the Category Tree Gaussian Process (ctGP), originally developed for efficient modeling with large categorical input spaces, we propose a multi-response extension termed ctmGP. The proposed approach adaptively captures dependence structures among objectives while remaining effective across both highly correlated and weakly correlated settings. Using ctmGP, we develop surrogate-assisted multi-objective Bayesian optimization frameworks based on two acquisition criteria and sequential design strategies. Numerical experiments on synthetic benchmarks and a comprehensive CPU cooling-system case study demonstrate that the proposed methods provide effective and robust optimization performance across diverse correlation structures.

An SVD-Based Algorithm for Maxwell’s Equations with PEC and Quasi-Periodic Boundary Conditions

Jia-Wei Lin, Tunghai University

This study presents FAME, a fast algorithm for computing the band structures of three-dimensional photonic crystals with supercell geometries under mixed perfect electric conductor and quasi-periodic boundary conditions. Maxwell’s source-free equations are discretized using an oblique Yee finite-difference scheme applicable to various Bravais lattices. By exploiting the FFT-based singular value decompositions of the directional discrete differential operators, we derive an explicit SVD of the discrete curl operator. Its range-space basis is then used to reformulate the Maxwell eigenvalue problem as a nullspace-free generalized eigenvalue problem solvable by the conjugate gradient method.

The resulting discrete cosine, sine, and quasi-periodic transforms are accelerated by FFTs, yielding O(NlogN) complexity per matrix–vector multiplication. Numerical experiments demonstrate the efficiency and effectiveness of FAME and reveal surface states in three-dimensional photonic crystals with gyroid supercell structures.

Scalable Spectral Methods for Sensor Fusion via Landmark Alternating Diffusion

Sing-Yuan Yeh, National Central University

Manifold learning aims to recover the intrinsic geometry of high-dimensional data through spectral analysis of graph operators. In this talk, I focus on scalable spectral methods for multi-sensor data and large datasets. In particular, I introduce Landmark Alternating Diffusion (LAD), a scalable spectral method for sensor fusion, which builds on the Alternating Diffusion (AD) framework to extract the common manifold structure shared by multiple observations.

By approximating the alternating diffusion process using a small set of landmark points, LAD significantly reduces computational cost while preserving the geometry captured by AD. Under standard common manifold assumptions, we establish theoretical guarantees including consistency, convergence, and finite-sample error bounds. We also demonstrate the practical effectiveness of LAD on an EEG sleep stage annotation task using multi-channel signals. Finally, I will briefly discuss how the landmark framework can be extended to Vector Diffusion Maps (VDM) for scalable analysis of directional and connection-valued data.

Computational Mathematics for Public-Health Modeling: Methods, Applications, and Collaborative Perspectives

Sayooj Aby Jose, Seoul National University

Mathematical modelling and scientific computing provide powerful frameworks for understanding complex public-health systems, particularly the transmission and control of infectious diseases. This talk will present selected research in mathematical epidemiology, nonlinear dynamical systems, data-driven modelling, machine learning, and epidemic forecasting. The presentation will illustrate how mathematical theory, computational techniques, and real-world health data can be integrated to analyse disease dynamics, evaluate intervention strategies, and improve public-health decision-making.

The talk will also highlight the role of interdisciplinary research networks in connecting mathematicians, computational scientists and public-health researchers. Selected ongoing research activities and collaborative experiences will be discussed to demonstrate how international partnerships can support methodological innovation, knowledge exchange, and the development of practical computational solutions. The presentation aims to encourage young researchers to explore emerging opportunities at the intersection of computational mathematics, scientific computing, and public-health research.