More Like this

1. Stability of a Subcritical Fluid Model for Fair Bandwidth Sharing with General File Size Distributions

   https://doi.org/10.1287/stsy.2019.0058  

   Fu, Yingjia ; Williams, Ruth J. ( September 2020 , Stochastic Systems)

   null (Ed.)

   This work concerns the asymptotic behavior of solutions to a (strictly) subcritical fluid model for a data communication network, where file sizes are generally distributed and the network operates under a fair bandwidth-sharing policy. Here we consider fair bandwidth-sharing policies that are a slight generalization of the [Formula: see text]-fair policies introduced by Mo and Walrand [Mo J, Walrand J (2000) Fair end-to-end window-based congestion control. IEEE/ACM Trans. Networks 8(5):556–567.]. Since the year 2000, it has been a standing problem to prove stability of the data communications network model of Massoulié and Roberts [Massoulié L, Roberts J (2000) Bandwidth sharing and admission control for elastic traffic. Telecommunication Systems 15(1):185–201.], with general file sizes and operating under fair bandwidth sharing policies, when the offered load is less than capacity (subcritical conditions). A crucial step in an approach to this problem is to prove stability of subcritical fluid model solutions. In 2012, Paganini et al. [Paganini F, Tang A, Ferragut A, Andrew LLH (2012) Network stability under alpha fair bandwidth allocation with general file size distribution. IEEE Trans. Automatic Control 57(3):579–591.] introduced a Lyapunov function for this purpose and gave an argument, assuming that fluid model solutions are sufficiently smooth in time and space that they are strong solutions of a partial differential equation and assuming that no fluid level on any route touches zero before all route levels reach zero. The aim of the current paper is to prove stability of the subcritical fluid model without these strong assumptions. Starting with a slight generalization of the Lyapunov function proposed by Paganini et al., assuming that each component of the initial state of a measure-valued fluid model solution, as well as the file size distributions, have no atoms and have finite first moments, we prove absolute continuity in time of the composition of the Lyapunov function with any subcritical fluid model solution and describe the associated density. We use this to prove that the Lyapunov function composed with such a subcritical fluid model solution converges to zero as time goes to infinity. This implies that each component of the measure-valued fluid model solution converges vaguely on [Formula: see text] to the zero measure as time goes to infinity. Under the further assumption that the file size distributions have finite pth moments for some p > 1 and that each component of the initial state of the fluid model solution has finite pth moment, it is proved that the fluid model solution reaches the measure with all components equal to the zero measure in finite time and that the time to reach this zero state has a uniform bound for all fluid model solutions having a uniform bound on the initial total mass and the pth moment of each component of the initial state. In contrast to the analysis of Paganini et al., we do not need their strong smoothness assumptions on fluid model solutions and we rigorously treat the realistic, but singular situation, where the fluid level on some routes becomes zero, whereas other route levels remain positive. 

   more » « less
2. Long‐time asymptotics and the radiation condition with time‐periodic boundary conditions for linear evolution equations on the half‐line and experiment

   https://doi.org/10.1111/sapm.12665  

   Mao, Yifeng ; Mantzavinos, Dionyssios ; Hoefer, Mark A. ( December 2023 , Studies in Applied Mathematics)

   The asymptotic Dirichlet‐to‐Neumann (D‐N) map is constructed for a class of scalar, constant coefficient, linear, third‐order, dispersive equations with asymptotically time/periodic Dirichlet boundary data and zero initial data on the half‐line, modeling a wavemaker acting upon an initially quiescent medium. The large timetasymptotics for the special cases of the linear Korteweg‐de Vries and linear Benjamin–Bona–Mahony (BBM) equations are obtained. The D‐N map is proven to be unique if and only if the radiation condition that selects the unique wave number branch of the dispersion relation for a sinusoidal, time‐dependent boundary condition holds: (i) for frequencies in a finite interval, the wave number is real and corresponds to positive group velocity, and (ii) for frequencies outside the interval, the wave number is complex with positive imaginary part. For fixed spatial locationx, the corresponding asymptotic solution is (i) a traveling wave or (ii) a spatially decaying, time‐periodic wave. The linearized BBM asymptotics are found to quantitatively agree with viscous core‐annular fluid experiments.

    

   more » « less
3. The mean-field limit of the Lieb-Liniger model

   https://doi.org/10.3934/dcds.2022006  

   Rosenzweig, Matthew ( January 2022 , Discrete and Continuous Dynamical Systems)

   We consider the well-known Lieb-Liniger (LL) model for \begin{document}$ N $\end{document} bosons interacting pairwise on the line via the \begin{document}$ \delta $\end{document} potential in the mean-field scaling regime. Assuming suitable asymptotic factorization of the initial wave functions and convergence of the microscopic energy per particle, we show that the time-dependent reduced density matrices of the system converge in trace norm to the pure states given by the solution to the one-dimensional cubic nonlinear Schrödinger equation (NLS) with an explict rate of convergence. In contrast to previous work [3] relying on the formalism of second quantization and coherent states and without an explicit rate, our proof is based on the counting method of Pickl [65,66,67] and Knowles and Pickl [44]. To overcome difficulties stemming from the singularity of the \begin{document}$ \delta $\end{document} potential, we introduce a new short-range approximation argument that exploits the Hölder continuity of the \begin{document}$ N $\end{document}-body wave function in a single particle variable. By further exploiting the \begin{document}$ L^2 $\end{document}-subcritical well-posedness theory for the 1D cubic NLS, we can prove mean-field convergence when the limiting solution to the NLS has finite mass, but only for a very special class of \begin{document}$ N $\end{document}-body initial states.

    

   more » « less
4. Microphysically modified magnetosonic modes in collisionless, high-β plasmas

   https://doi.org/10.1017/S0022377823000429  

   Majeski, S. ; Kunz, M.W. ; Squire, J. ( June 2023 , Journal of Plasma Physics)

   With the support of hybrid-kinetic simulations and analytic theory, we describe the nonlinear behaviour of long-wavelength non-propagating (NP) modes and fast magnetosonic waves in high- $\beta$ collisionless plasmas, with particular attention to their excitation of and reaction to kinetic micro-instabilities. The perpendicularly pressure balanced polarization of NP modes produces an excess of perpendicular pressure over parallel pressure in regions where the plasma $\beta$ is increased. For mode amplitudes $|\delta B/B_0| \gtrsim 0.3$ , this excess excites the mirror instability. Particle scattering off these micro-scale mirrors frustrates the nonlinear saturation of transit-time damping, ensuring that large-amplitude NP modes continue their decay to small amplitudes. At asymptotically large wavelengths, we predict that the mirror-induced scattering will be large enough to interrupt transit-time damping entirely, isotropizing the pressure perturbations and morphing the collisionless NP mode into the magnetohydrodynamic (MHD) entropy mode. In fast waves, a fluctuating pressure anisotropy drives both mirror and firehose instabilities when the wave amplitude satisfies $|\delta B/B_0| \gtrsim 2\beta ^{-1}$ . The induced particle scattering leads to delayed shock formation and MHD-like wave dynamics. Taken alongside prior work on self-interrupting Alfvén waves and self-sustaining ion-acoustic waves, our results establish a foundation for new theories of electromagnetic turbulence in low-collisionality, high- $\beta$ plasmas such as the intracluster medium, radiatively inefficient accretion flows and the near-Earth solar wind. 

   more » « less
5. Challenges and opportunities for the simulation of calcium waves on modern multi‐core and many‐core parallel computing platforms

   https://doi.org/10.1002/cnm.3244  

   Barajas, Carlos ; Gobbert, Matthias K. ; Kroiz, Gerson C. ; Peercy, Bradford E. ( September 2019 , International Journal for Numerical Methods in Biomedical Engineering)

   State‐of‐the‐art distributed‐memory computer clusters contain multicore CPUs with 16 and more cores. The second generation of the Intel Xeon Phi many‐core processor has more than 60 cores with 16 GB of high‐performance on‐chip memory. We contrast the performance of the second‐generation Intel Xeon Phi, code‐named Knights Landing (KNL), with 68 computational cores to the latest multicore CPU Intel Skylake with 18 cores. A special‐purpose code solving a system of nonlinear reaction‐diffusion partial differential equations with several thousands of point sources modeled mathematically by Dirac delta distributions serves as realistic test bed. The system is discretized in space by the finite volume method and advanced by fully implicit time‐stepping, with a matrix‐free implementation that allows the complex model to have an extremely small memory footprint. The sample application is a seven variable model of calcium‐induced calcium release (CICR) that models the interplay between electrical excitation, calcium signaling, and mechanical contraction in a heart cell. The results demonstrate that excellent parallel scalability is possible on both hardware platforms, but that modern multicore CPUs outperform the specialized many‐core Intel Xeon Phi KNL architecture for a large class of problems such as systems of parabolic partial differential equations.

    

   more » « less