arXiv subjects

Po-Yi Wu

Publications and source records attributed to Po-Yi Wu.

5 recordsLinked to original sources

arXiv2025

On the Pre-Asymptotic Stability and Inverse Structure of Extended-Domain Spectral Methods

The extended-domain method is a strategy for applying spectral methods to complex geometries. Its stability is complicated by the ill-conditioning of the Fourier extension frame. This paper provides an analysis of the method's pre-asymptotic behavior. We confirm that the spectral collocation system is asymptotically ill-conditioned for both the Poisson and convection-diffusion operators, driven by the redundancy of the underlying frame. However, we prove a fundamental structural dichotomy in their discrete Green's functions. We show that the inverse of the convection-diffusion operator is numerically highly asymmetric, exhibiting exponential upstream decay, in contrast to the numerically dense inverse of the Poisson operator. This intrinsic asymmetry explains why the convection-diffusion operator is significantly more robust to the underlying frame instability in practical computations.

math.NA↗

arXiv2025

MVA 2025 Small Multi-Object Tracking for Spotting Birds Challenge: Dataset, Methods, and Results

Small Multi-Object Tracking (SMOT) is particularly challenging when targets occupy only a few dozen pixels, rendering detection and appearance-based association unreliable. Building on the success of the MVA2023 SOD4SB challenge, this paper introduces the SMOT4SB challenge, which leverages temporal information to address limitations of single-frame detection. Our three main contributions are: (1) the SMOT4SB dataset, consisting of 211 UAV video sequences with 108,192 annotated frames under diverse real-world conditions, designed to capture motion entanglement where both camera and targets move freely in 3D; (2) SO-HOTA, a novel metric combining Dot Distance with HOTA to mitigate the sensitivity of IoU-based metrics to small displacements; and (3) a competitive MVA2025 challenge with 78 participants and 308 submissions, where the winning method achieved a 5.1x improvement over the baseline. This work lays a foundation for advancing SMOT in UAV scenarios with applications in bird strike avoidance, agriculture, fisheries, and ecological monitoring.

cs.CV↗

arXiv2025

Convergent Operator-Splitting Scheme for Viscosity Solutions: A Foundation for Learning Domain-to-Solution Maps

This work introduces and rigorously analyzes a novel operator-splitting finite element scheme for approximating viscosity solutions of a broad class of constrained second-order partial differential equations. By decoupling the primary PDE evolution from the enforcement of constraints, the proposed method combines a stabilized finite element method for spatial discretization with an efficient semi-implicit time-stepping strategy. The cornerstone of our analysis is a proof that the scheme satisfies a discrete comparison principle. We demonstrate that under a mild time-step restriction and with appropriate stabilization, the discrete operator yields an M-matrix, which is sufficient to guarantee the scheme's monotonicity and consequent $L^\infty$-stability. These properties -- consistency, stability, and monotonicity -- are shown to be sufficient to prove convergence of the numerical approximation to the unique viscosity solution within the celebrated Barles--Souganidis framework. For solutions with enhanced regularity, we further establish an optimal-order error estimate of $O(Δt + h^2)$. The rigorously established stability of the scheme provides a blueprint for a novel Physics-Constrained Neural Operator (PCNO) architecture. We prove that by emulating the scheme's structure, the PCNO can provably break the curse of dimensionality for the challenging class of domain-to-solution mapping problems with complex topological variations, a problem for which standard learning approaches often fail. Numerical experiments for both a Hamilton-Jacobi equation with state constraints and a controlled reaction-diffusion system are presented to validate the theoretical findings and demonstrate the scheme's effectiveness.

math.NA↗

arXiv2025

A Projection Method for Compressible Generic Two-Fluid Model

A new projection method for a generic two-fluid model is presented in this work. Specifically, we extend the projection method, originally designed for single-phase variable density incompressible and compressible flows, to viscous compressible two-fluid flows. The key idea is that the single pressure $p$ can be uniquely determined by the products of volume fractions and densities $ϕ_k ρ_k$, of the two fluids. Additionally, the stability of the method is ensured by appropriately assigning intermediate step variables at the time-discrete level and incorporating a stabilizing term at the fully discrete level. We prove the energy stability of the proposed numerical scheme, and its validity is demonstrated through three numerical tests.

math.NA↗

arXiv2018

Development of a New Spectral Collocation Method Using Laplacian Eigenbasis for Elliptic Partial Differential Equations in an Extended Domain

The recent development of spectral method has been praised for its high-order convergence in simulating complex physical problems. The combination of embedded boundary method and spectral method becomes a mainstream way to tackle geometrically complicated problems. However, the convergence is deteriorated when embedded boundary strategies are employed. Owing to the loss of regularity, in this paper we propose a new spectral collocation method which retains the regularity of solutions to solve differential equations in the case of complex geometries. The idea is rooted in the basis functions defined in an extended domain, which leads to a useful upper bound of the Lebesgue constant with respect to the Fourier best approximation. In particular, how the stretching of the domain defining basis functions affects the convergence rate directly is detailed. Error estimates chosen in our proposed method show that the exponential decay convergence for problems with analytical solutions can be retained. Moreover, two-dimensional Poisson equations and convection-diffusion equations with simple and complex geometrical domains will be simulated. The predicted results justify the advantages of applying our method to tackle geometrically complicated problems.

math.NA↗