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1. Front location determines convergence rate to traveling waves

   https://doi.org/10.4171/AIHPC/120  

   An, Jing; Henderson, Christopher; Ryzhik, Lenya (May 2024, Annales de l'Institut Henri Poincaré C, Analyse non linéaire)

   We propose a novel method for establishing the convergence rates of solutions to reaction–diffusion equations to traveling waves. The analysis is based on the study of the traveling wave shape defect function introduced in An et. al. [Arch. Ration. Mech. Anal. 247 (2023), no. 5, article no. 88]. It turns out that the convergence rate is controlled by the distance between thephantom front locationfor the shape defect function and the true front location of the solution. Curiously, the convergence to a traveling wave has a pulled nature, regardless of whether the traveling wave itself is of pushed, pulled, or pushmi-pullyu type. In addition to providing new results, this approach dramatically simplifies the proof in the Fisher–KPP case and gives a unified, succinct explanation for the known algebraic rates of convergence in the Fisher–KPP case and the exponential rates in the pushed case. 

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2. Structure aware Runge–Kutta time stepping for spacetime tents

   https://doi.org/10.1007/s42985-020-00020-4  

   Gopalakrishnan, Jay; Schöberl, Joachim; Wintersteiger, Christoph (August 2020, SN Partial Differential Equations and Applications)

   null (Ed.)

   Abstract We introduce a new class of Runge–Kutta type methods suitable for time stepping to propagate hyperbolic solutions within tent-shaped spacetime regions. Unlike standard Runge–Kutta methods, the new methods yield expected convergence properties when standard high order spatial (discontinuous Galerkin) discretizations are used. After presenting a derivation of nonstandard order conditions for these methods, we show numerical examples of nonlinear hyperbolic systems to demonstrate the optimal convergence rates. We also report on the discrete stability properties of these methods applied to linear hyperbolic equations. 

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3. Birth–death dynamics for sampling: global convergence, approximations and their asymptotics

   https://doi.org/10.1088/1361-6544/acf988  

   Lu, Yulong; Slepčev, Dejan; Wang, Lihan (September 2023, Nonlinearity)

   Abstract Motivated by the challenge of sampling Gibbs measures with nonconvex potentials, we study a continuum birth–death dynamics. We improve results in previous works (Liuet al2023Appl. Math. Optim.8748; Luet al2019 arXiv:1905.09863) and provide weaker hypotheses under which the probability density of the birth–death governed by Kullback–Leibler divergence or byχ²divergence converge exponentially fast to the Gibbs equilibrium measure, with a universal rate that is independent of the potential barrier. To build a practical numerical sampler based on the pure birth–death dynamics, we consider an interacting particle system, which is inspired by the gradient flow structure and the classical Fokker–Planck equation and relies on kernel-based approximations of the measure. Using the technique of Γ-convergence of gradient flows, we show that on the torus, smooth and bounded positive solutions of the kernelised dynamics converge on finite time intervals, to the pure birth–death dynamics as the kernel bandwidth shrinks to zero. Moreover we provide quantitative estimates on the bias of minimisers of the energy corresponding to the kernelised dynamics. Finally we prove the long-time asymptotic results on the convergence of the asymptotic states of the kernelised dynamics towards the Gibbs measure. 

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4. Linear Uncertainty Quantification of Graphical Model Inference

   Guo, Chenghua; Yu, Han; Liu, Jiaxin; Chen, Chao; Li, Qi; Xie, Sihong; Zhang, Xi (December 2024, Proceedings of the 38th Conference on Neural Information Processing Systems (NeurIPS 2024))

   Uncertainty Quantification (UQ) is vital for decision makers as it offers insights into the potential reliability of data and model, enabling more informed and risk-aware decision-making. Graphical models, capable of representing data with complex dependencies, are widely used across domains. Existing sampling-based UQ methods are unbiased but cannot guarantee convergence and are time-consuming on large-scale graphs. There are fast UQ methods for graphical models with closed-form solutions and convergence guarantee but with uncertainty underestimation. We propose LinUProp, a UQ method that utilizes a novel linear propagation of uncertainty to model uncertainty among related nodes additively instead of multiplicatively, to offer linear scalability, guaranteed convergence, and closed-form solutions without underestimating uncertainty. Theoretically, we decompose the expected prediction error of the graphical model and prove that the uncertainty computed by LinUProp is the generalized variance component of the decomposition. Experimentally, we demonstrate that LinUProp is consistent with the sampling-based method but with linear scalability and fast convergence. Moreover, LinUProp outperforms competitors in uncertainty-based active learning on four real-world graph datasets, achieving higher accuracy with a lower labeling budget. 

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5. Numerical schemes for a fully nonlinear coagulation–fragmentation model coming from wave kinetic theory

   https://doi.org/10.1098/rspa.2025.0197  

   Das, Arijit; Tran, Minh-Binh (June 2025, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   This article introduces a novel numerical approach, based on finite-volume techniques, for studying fully nonlinear coagulation–fragmentation models, where both the coagulation and fragmentation components of the collision operator are nonlinear. The models come from three-wave kinetic equations, a pivotal framework in wave turbulence theory. Despite the importance of wave turbulence theory in physics and mechanics, there have been very few numerical schemes for three-wave kinetic equations, in which no additional assumptions are manually imposed on the evolution of the solutions, and the current manuscript provides one of the first of such schemes. To the best of our knowledge, this also is the first numerical scheme capable of accurately capturing the long-term asymptotic behaviour of solutions to a fully nonlinear coagulation–fragmentation model. The scheme is implemented on some test problems, demonstrating strong alignment with theoretical predictions of energy cascade rates, rigorously obtained in the work (Soffer & Tran. 2020Commun. Math. Phys.376, 2229–2276. (doi:10.1007/BF01419532)). We further introduce a weighted finite-volume variant to ensure energy conservation across varying degrees of kernel homogeneity. Convergence and first-order consistency are established through theoretical analysis and verified by experimental convergence orders in test cases. 

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