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Hamiltonian boundary value methods applied to KdV-KdV systems

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applied Numerical mathematics 2025年 214卷 1-27页

作者： Luo, Qian Xiao, Aiguo Luo, Xuqiong Yan, Xiaoqiang School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan 411105 China Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Hunan 411105 China School of Mathematics and Statistics Changsha University of Science and Technology Changsha410114 China National Center for Applied Mathematics in Hunan Xiangtan University Hunan 411105 China

In this paper, we propose a highly accurate scheme for two KdV systems of the Boussinesq type under periodic boundary conditions. The proposed scheme combines the Fourier-Galerkin method for spatial discretization with Hamiltonian boundary value methods for time integration, ensuring the conservation of discrete mass and energy. By expanding the system in Fourier series, the equations are firstly transformed into Hamiltonian form, preserving the original Hamiltonian structure. Applying the Fourier-Galerkin method for semi-discretization in space, we obtain a large-scale system of Hamiltonian ordinary differential equations, which is then solved using a class of energy-conserving Runge-Kutta methods, known as Hamiltonian boundary value methods. The efficiency of this approach is assessed, and several numerical examples are provided to demonstrate the effectiveness of the method. © 2025 IMACS

关键词： Ordinary differential equations

Splitting algorithms for paraxial and Itô-Schrödinger models of wave propagation in random media

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arXiv 2025年

作者： Bal, Guillaume Nair, Anjali Departments of Statistics and Mathematics Committee on Computational and Applied Mathematics University of Chicago ChicagoIL60637 United States Department of Statistics Committee on Computational and Applied Mathematics University of Chicago ChicagoIL60637 United States

This paper introduces a full discretization procedure to solve wave beam propagation in random media modeled by a paraxial wave equation or an Itô-Schrödinger stochastic partial differential equation. This method bears similarities with the phase screen method used routinely to solve such problems. The main axis of propagation is discretized by a centered splitting scheme with step ∆z while the transverse variables are treated by a spectral method after appropriate spatial truncation. The originality of our approach is its theoretical validity even when the typical wavelength θ of the propagating signal satisfies θ ∆z. More precisely, we obtain a convergence of order ∆z in mean-square sense while the errors on statistical moments are of order (∆z)2 as expected for standard centered splitting schemes. This is a surprising result as splitting schemes typically do not converge when ∆z is not the smallest scale of the problem. The analysis is based on equations satisfied by statistical moments in the Itô-Schrödinger case and on integral (Duhamel) expansions for the paraxial model. Several numerical simulations illustrate and confirm the theoretical findings. Copyright © 2025, The Authors. All rights reserved.

关键词： Schrodinger equation

Stability and Lorentzian geometry for an inverse problem of a semilinear wave equation

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Analysis & PDE 2025年第5期18卷 1065-1118页

作者： Matti Lassas Tony Liimatainen Leyter Potenciano-Machado Teemu Tyni Department of Mathematics and Statistics University of Helsinki Helsinki Finland Department of Mathematics and Statistics University of Jyväskylä Jyväskylä Finland Research Unit of Applied and Computational Mathematics University of Oulu Oulu Finland

This paper concerns an inverse boundary value problem for a semilinear wave equation on a globally hyperbolic Lorentzian manifold. We prove a Hölder stability result for recovering an unknown potential ☆q☆ of the nonlinear wave equation ☆□☆g☆u☆+☆q☆u☆m☆=☆0☆, ☆m☆≥☆4☆, from the Dirichlet-to-Neumann map. Our proof is based on the recent higher-order linearization method and use of Gaussian beams. We also extend earlier uniqueness results by removing the assumptions of convex boundary and that pairs of light-like geodesics can intersect only once. For this, we construct special light-like geodesics and other general constructions in Lorentzian geometry. We expect these constructions to be applicable in studies of related problems as well.

关键词： Lorentzian geometry nonlinear wave equations inverse problems

PRECISE: PRivacy-loss-Efficient and Consistent Inference based on poSterior quantilEs

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arXiv 2025年

作者： Zhou, Ruyu Liu, Fang Department of Applied and Computational Mathematics and Statistics University of Notre Dame IN46556 United States

Differential privacy (DP) is a mathematical framework for releasing information with formal privacy guarantees. Despite the existence of various DP procedures for performing a wide range of statistical analysis and machine learning tasks, methods of good utility are still lacking for obtaining valid statistical inference with DP guarantees. We address this gap by introducing the notion of valid Privacy-Preserving Interval Estimation (PPIE) and proposing PRivacy-loss-Efficient and Consistent Inference based on poSterior quantilEs (PRECISE). PRECISE is a general-purpose Bayesian approach for constructing privacy-preserving posterior intervals. We establish the Mean-Squared-Error (MSE) consistency for our proposed private posterior quantiles converging to the population posterior quantile as sample size or privacy loss increases. We conduct extensive experiments to compare the utilities of PRECISE with common existing privacy-preserving inferential approaches in various inferential tasks, data types and sizes, privacy loss levels. The results demonstrated a significant advantage of PRECISE with its nominal coverage and substantially narrower intervals than the existing methods, which are prone to either under-coverage or impractically wide intervals. © 2025, CC BY.

关键词： Differential privacy

Topological edge states of continuous Hamiltonians

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arXiv 2025年

作者： Frazier, Matthew Bal, Guillaume Committee on Computational and Applied Mathematics University of Chicago ChicagoIL60637 United States Departments of Mathematics and Statistics Committee on Computational and Applied Mathematics University of Chicago ChicagoIL60637 United States

This paper concerns the topological classification of continuous Hamiltonians that find applications in biased cold plasmas and photonics. Besides a magnetic bias, the Hamiltonians are parametrized by a plasma frequency and a fixed vertical wavenumber. Eight distinct phases of matter are identified as these parameters vary. When insulating gaps are shared by two such phases, asymmetric edge modes propagate along interfaces separating the two phases. Here we apply the notion of a bulk difference invariant (BDI) to this Hamiltonian, and show by numerical diagonalizations of interface Hamiltonians that after an appropriate regularization our BDI correctly predicts edge transport as described by a bulk edge correspondence. We also derive theoretical tools to compute the BDI and show the limitations of the bulk edge correspondence (BEC) when the phase transition is too singular. Copyright © 2025, The Authors. All rights reserved.

关键词： Topology

GLIVENKO–CANTELLI FOR f-DIVERGENCE

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arXiv 2025年

作者： Wang, Haoming Lim, Lek-Heng Computational and Applied Mathematics Initiative Department of Statistics University of Chicago ChicagoIL60637 United States

We extend the celebrated Glivenko–Cantelli theorem, sometimes called the fundamental theorem of statistics, from its standard setting of total variation distance to all f-divergences. A key obstacle in this endeavor is to define f-divergence on a subcollection of a σ-algebra that forms a π-system but not a σ-subalgebra. This is a side contribution of our work. We will show that this notion of f-divergence on the π-system of rays preserves nearly all known properties of standard f-divergence, yields a novel integral representation of the Kolmogorov–Smirnov distance, and has a Glivenko–Cantelli theorem. We will also discuss the prospects of a Vapnik–Chervonenkis theory for *** Codes 60B10, 60F15, 60F25 © 2025, CC BY.

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GEOMETRIC PROGRAMMING FOR 3D CIRCUITS

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arXiv 2025年

作者： Wang, Rongbiao Lim, Lek-Heng Committee on Computational and Applied Mathematics University of Chicago ChicagoIL60637 United States Computational and Applied Mathematics Initiative Department of Statistics University of Chicago ChicagoIL60637-1514 United States

With the soaring demand for high-performing integrated circuits, 3D integrated circuits (ICs) have emerged as a promising alternative to traditional planar structures. Unlike existing 3D ICs that stack 2D layers, a full 3D IC features cubic circuit elements unrestricted by layers, offering greater design freedom. Design problems such as floorplanning, transistor sizing, and interconnect sizing are highly complex due to the 3D nature of the circuits and unavoidably require systematic approaches. We introduce geometric programming to solve these design optimization problems systematically and efficiently. © 2025, CC BY-NC-SA.

关键词： Three dimensional integrated circuits

A flow-kick model of dryland vegetation patterns: the impact of rainfall variability on resilience

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arXiv 2025年

作者： Gandhi, Punit Oline, Matthew Silber, Mary Department of Mathematics and Applied Mathematics Virginia Commonwealth University RichmondVA23284 United States Computational and Applied Mathematics University of Chicago ChicagoIL60637 United States Department of Statistics and Committee on Computational and Applied Mathematics University of Chicago ChicagoIL60637 United States

In many drylands around the globe, vegetation self-organizes into regular spatial patterns in response to aridity stress. We consider the regularly-spaced vegetation bands, on gentle hill-slopes, that survive low rainfall conditions by harvesting additional stormwater from upslope low-infiltration bare zones. We are interested in the robustness of this pattern formation survival mechanism to changes in rainfall variability. For this, we use a flow-kick modeling framework that treats storms as instantaneous kicks to the soil water. The positive feedbacks in the storm-level hydrology, that act to concentrate water within the vegetation bands, are captured through the spatial profiles of the soil water kicks. Between storms, the soil water and vegetation, modeled by a two-component reaction-diffusion system, evolve together. We use a combination of linear stability analysis and numerical simulation to compare predictions of idealized periodic rainfall, with no variability, to predictions when there is randomness in the timing and magnitude of water input from storms. We show that including these random elements leads to a decrease in the parameter range over which patterns appear. This suggests that an increase in storm variability, even with the same mean annual rainfall, may negatively impact the resilience of these pattern-forming dryland ecosystems. Copyright © 2025, The Authors. All rights reserved.

关键词： Vegetation

Effect of Weak Elasticity on the Kelvin-Helmholtz Instability

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SSRN

SSRN 2025年

作者： Xie, Binqiang Guo, Boling Zhao, Bin School of Mathematics and Statistics Guangdong University of Technology Guangzhou510006 China Institute of Applied Physics and Computational Mathematics Beijing100088 China

In this paper, we present an analysis of the Kelvin-Helmholtz instability in two-dimensional ideal compressible elastic flows, providing a rigorous confirmation that weak elasticity has a destabilizing effect on the Kelvin-Helmholtz instability. There are two critical velocities, $U_{\text{low}}$ and $U_{\text{upp}}$, where $U_{\text{low}}$ and $U_{\text{upp}}$ represent the lower and upper critical velocities, respectively. We demonstrate that when the magnitude of the rectilinear solutions satisfies $U_{\text{low}}+c\epsilon_{0}\le |\dot{v}^{+}_{1}| \le U_{\text{upp}}-c\epsilon_{0}$, the linear and nonlinear ill-posedness of the piecewise smooth solutions of the Kelvin-Helmholtz problem for two-dimensional ideal compressible elastic fluids is established uniformly, where $c$ is the sound speed and $\epsilon_{0}$ is some small enough positive constant. © 2025, The Authors. All rights reserved.

关键词： Compressible flow