An improved tri-coloured rooted-tree theory and order conditions for ERKN methods for general multi-frequency oscillatory systems

AbstractIn this work, a version of continuous stage stochastic Runge–Kutta (CSSRK) methods is developed for stochastic differential equations (SDEs). First, a general order theory of these methods is established by the theory of stochastic B-series and multicolored rooted tree. Then the proposed CSSRK methods are applied to three special kinds of SDEs and the corresponding order conditions are derived. In particular, for the single integrand SDEs and SDEs with additive noise, we construct some specific CSSRK methods of high order. Moreover, it is proved that with the help of different numerical quadrature formulas, CSSRK methods can generate corresponding stochastic Runge–Kutta (SRK) methods which have the same order. Thus, some efficient SRK methods are induced. Finally, some numerical experiments are presented to demonstrate those theoretical results.

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Rooted tree analysis of the order conditions of row-type scheme for stochastic differential equations

BIT Numerical Mathematics ◽

10.1007/bf02510172 ◽

1997 ◽

Vol 37 (1) ◽

pp. 43-66 ◽

Cited By ~ 29

Author(s):

Yoshio Komori ◽

Taketomo Mitsui ◽

Hiroshi Sugiura

Keyword(s):

Differential Equations ◽

Stochastic Differential Equations ◽

Rooted Tree ◽

Order Conditions ◽

Tree Analysis

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HTPC: heterogeneous traffic-aware partition coding for random packet spraying in data center networks

Journal of Cloud Computing Advances Systems and Applications ◽

10.1186/s13677-021-00248-4 ◽

2021 ◽

Vol 10 (1) ◽

Author(s):

Jiawei Huang ◽

Shiqi Wang ◽

Shuping Li ◽

Shaojun Zou ◽

Jinbin Hu ◽

...

Keyword(s):

Data Center ◽

Large Scale ◽

Network Performance ◽

Rooted Tree ◽

Heterogeneous Traffic ◽

Data Center Networks ◽

Packet Reordering ◽

Traffic Characteristics ◽

Network Utilization ◽

The Impact

AbstractModern data center networks typically adopt multi-rooted tree topologies such leaf-spine and fat-tree to provide high bisection bandwidth. Load balancing is critical to achieve low latency and high throughput. Although the per-packet schemes such as Random Packet Spraying (RPS) can achieve high network utilization and near-optimal tail latency in symmetric topologies, they are prone to cause significant packet reordering and degrade the network performance. Moreover, some coding-based schemes are proposed to alleviate the problem of packet reordering and loss. Unfortunately, these schemes ignore the traffic characteristics of data center network and cannot achieve good network performance. In this paper, we propose a Heterogeneous Traffic-aware Partition Coding named HTPC to eliminate the impact of packet reordering and improve the performance of short and long flows. HTPC smoothly adjusts the number of redundant packets based on the multi-path congestion information and the traffic characteristics so that the tailing probability of short flows and the timeout probability of long flows can be reduced. Through a series of large-scale NS2 simulations, we demonstrate that HTPC reduces average flow completion time by up to 60% compared with the state-of-the-art mechanisms.

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Computational Analysis of Ca2+ Oscillatory Bio-Signals: Two-Parameter Bifurcation Diagrams

Entropy ◽

10.3390/e23070876 ◽

2021 ◽

Vol 23 (7) ◽

pp. 876

Author(s):

Wieslaw Marszalek ◽

Jan Sadecki ◽

Maciej Walczak

Keyword(s):

Equilibrium Points ◽

Autonomous Systems ◽

Cytosolic Calcium ◽

Calcium Oscillations ◽

Dynamical Model ◽

Bifurcation Diagrams ◽

Step Size ◽

Oscillatory Systems ◽

Two Parameter ◽

Parameter Bifurcation

Two types of bifurcation diagrams of cytosolic calcium nonlinear oscillatory systems are presented in rectangular areas determined by two slowly varying parameters. Verification of the periodic dynamics in the two-parameter areas requires solving the underlying model a few hundred thousand or a few million times, depending on the assumed resolution of the desired diagrams (color bifurcation figures). One type of diagram shows period-n oscillations, that is, periodic oscillations having n maximum values in one period. The second type of diagram shows frequency distributions in the rectangular areas. Each of those types of diagrams gives different information regarding the analyzed autonomous systems and they complement each other. In some parts of the considered rectangular areas, the analyzed systems may exhibit non-periodic steady-state solutions, i.e., constant (equilibrium points), oscillatory chaotic or unstable solutions. The identification process distinguishes the later types from the former one (periodic). Our bifurcation diagrams complement other possible two-parameter diagrams one may create for the same autonomous systems, for example, the diagrams of Lyapunov exponents, Ls diagrams for mixed-mode oscillations or the 0–1 test for chaos and sample entropy diagrams. Computing our two-parameter bifurcation diagrams in practice and determining the areas of periodicity is based on using an appropriate numerical solver of the underlying mathematical model (system of differential equations) with an adaptive (or constant) step-size of integration, using parallel computations. The case presented in this paper is illustrated by the diagrams for an autonomous dynamical model for cytosolic calcium oscillations, an interesting nonlinear model with three dynamical variables, sixteen parameters and various nonlinear terms of polynomial and rational types. The identified frequency of oscillations may increase or decrease a few hundred times within the assumed range of parameters, which is a rather unusual property. Such a dynamical model of cytosolic calcium oscillations, with mitochondria included, is an important model in which control of the basic functions of cells is achieved through the Ca2+ signal regulation.

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