Higher strong order methods for linear Itô SDEs on matrix Lie groups

In this paper we present a general procedure for designing higher strong order methods for linear Itô stochastic differential equations on matrix Lie groups and illustrate this strategy with two novel schemes that have a strong convergence order of 1.5. Based on the Runge–Kutta–Munthe–Kaas (RKMK) method for ordinary differential equations on Lie groups, we present a stochastic version of this scheme and derive a condition such that the stochastic RKMK has the same strong convergence order as the underlying stochastic Runge–Kutta method. Further, we show how our higher order schemes can be applied in a mechanical engineering as well as in a financial mathematics setting.

Plankton growth dynamic driven by plankton body size in deterministic and stochastic environments

In this paper, we establish a new phytoplankton-zooplankton model by considering the effects of plankton body size and stochastic environmental fluctuations. Mathematical theory work mainly gives the existence of boundary and positive equilibria, and shows their local as well as global stability in the deterministic model. Additionally, we explore the dynamics of V-geometric ergodicity, stochastic ultimate boundedness, stochastic permanence, persistence in the mean, stochastic extinction and the existence of a unique ergodic stationary distribution in the corresponding stochastic version. Numerical simulation work mainly reveals that plankton body size can generate great influences on the interactions between phytoplankton and zooplankton, which in turn proves the effectiveness of mathematical theory analysis. It is worth emphasizing that for the small value of phytoplankton cell size, the increase of zooplankton body size can not change the phytoplankton density or zooplankton density; for the middle value of phytoplankton cell size, the increase of zooplankton body size can decrease zooplankton density or phytoplankton density; for the large value of phytoplankton body size, the increase of zooplankton body size can increase zooplankton density but decrease phytoplankton density. Besides, it should be noted that the increase of zooplankton body size can not affect the effect of random environmental disturbance, while the increase of phytoplankton cell size can weaken its effect. There results may enrich the dynamics of phytoplankton-zooplankton models.

On stochastic Kaczmarz type methods for solving large scale systems of ill-posed equations

In this article we investigate a family of "stochastic gradient type methods", for solving systems of linear ill-posed equations. The method under consideration is a stochastic version of the projective Landweber-Kaczmarz (PLWK) method in [Leitão/Svaiter, Inv. Probl. 2016] (see also [Leitão/Svaiter, NFAO 2018]). In the case of exact data, mean square convergence to zero of the iteration error is proven. In the noise data case, we couple our method with an a priori stopping rule and characterize it as a regularization method for solving systems of linear ill-posed operator equations. Numerical tests are presented for two linear ill-posed problems: (i) a Hilbert matrix type system with over 10^8 equations; (ii) a Big Data linear regression problem with real data. The obtained results indicate superior performance of the proposed method when compared with other well established iterations. Our preliminary investigation indicates that the proposed iteration is a promising alternative for computing stable approximate solutions of large scale systems of linear ill-posed equations.

On the Stochastic Magnus Expansion and Its Application to SPDEs

We derive the stochastic version of the Magnus expansion for linear systems of stochastic differential equations (SDEs). The main novelty with respect to the related literature is that we consider SDEs in the Itô sense, with progressively measurable coefficients, for which an explicit Itô-Stratonovich conversion is not available. We prove convergence of the Magnus expansion up to a stopping time $$\tau $$ τ and provide a novel asymptotic estimate of the cumulative distribution function of $$\tau $$ τ . As an application, we propose a new method for the numerical solution of stochastic partial differential equations (SPDEs) based on spatial discretization and application of the stochastic Magnus expansion. A notable feature of the method is that it is fully parallelizable. We also present numerical tests in order to asses the accuracy of the numerical schemes.

Robust Stochastic Models for Profit-Maximizing Hub Location Problems

This paper introduces robust stochastic models for profit -maximizing capacitated hub location problems in which two different types of uncertainty, including stochastic demand and uncertain revenue, are simultaneously incorporated into the problem. First, a two-stage stochastic program is presented in which demand and revenue are jointly stochastic. Next, robust stochastic models are developed to better model uncertainty in the revenue while keeping the demand stochastic. Two particular cases are studied based on the dependency between demand and revenue. In the first case, a robust stochastic model with a min-max regret objective is developed assuming a finite set of scenarios that describes uncertainty associated with the revenue under a revenue-elastic demand setting. For the case when demand and revenue are independent, robust stochastic models with a max-min criterion and a min-max regret objective are formulated considering both interval uncertainty and discrete scenarios, respectively. It is proved that the robust stochastic version with max-min criterion can be viewed as a special case of the min-max regret stochastic model. Exact algorithms based on Benders decomposition coupled with a sample average approximation scheme are proposed. Exploiting the repetitive nature of sample average approximation, generic acceleration methodologies are developed to enhance the performance of the algorithms enabling them to solve large-scale intractable instances. Extensive computational experiments are performed to consider the efficiency of the proposed algorithms and also to analyze the effects of uncertainty under different settings. The qualities of the solutions obtained from different modeling approaches are compared under various parameter settings. Computational results justify the need to solve robust stochastic models to embed uncertainty in decision making to design resilient hub networks.

Mathematical model of survival of fractional calculus, critics and their impact: How singular is our world?

Fractional calculus as was predicted by Leibniz to be a paradox, has nowadays evolved to become a centre of interest for many researchers from various backgrounds. As a result, multiple innovative ideas had emerged, which caused significant divisions regarding fractional calculus in the past three years. Therefore, this work is aimed at developing a mathematical model that could be used to depict the survival of fractional calculus. Six classes are herein considered to construct a mathematical model with six ordinary differential equations. All elementary analysis have been performed. Additionally, a new analysis including strength number that accounts for the accelerative information of nonlinear and linear parts of a given epidemiological model is introduced. An analysis of the second derivative of the Lyapunov function as well as an analysis of the second derivative of each class is applied to assess how a wave could be detected. It is strongly believed that this new analysis will particularly open new doors within the field of epidemiological modelling, which will aid researchers to better understand the spread of infectious diseases. The stochastic version of the suggested model was also investigated, and numerical simulations were performed. The obtained reproductive number, strength number, extinction of criticism together with numerical simulation, revealed that the field of fractional calculus will be stable will therefore have no significant effect soon.

Critical behaviour of the stochastic Wilson-Cowan model

Spontaneous brain activity is characterized by bursts and avalanche-like dynamics, with scale-free features typical of critical behaviour. The stochastic version of the celebrated Wilson-Cowan model has been widely studied as a system of spiking neurons reproducing non-trivial features of the neural activity, from avalanche dynamics to oscillatory behaviours. However, to what extent such phenomena are related to the presence of a genuine critical point remains elusive. Here we address this central issue, providing analytical results in the linear approximation and extensive numerical analysis. In particular, we present results supporting the existence of a bona fide critical point, where a second-order-like phase transition occurs, characterized by scale-free avalanche dynamics, scaling with the system size and a diverging relaxation time-scale. Moreover, our study shows that the observed critical behaviour falls within the universality class of the mean-field branching process, where the exponents of the avalanche size and duration distributions are, respectively, 3/2 and 2. We also provide an accurate analysis of the system behaviour as a function of the total number of neurons, focusing on the time correlation functions of the firing rate in a wide range of the parameter space.

Graph-directed random fractal interpolation function

"Barnsley introduced in [1] the notion of fractal interpolation function (FIF). He said that a fractal function is a (FIF) if it possess some interpolation properties. It has the advantage that it can be also combined with the classical methods or real data interpolation. Hutchinson and Ruschendorf [7] gave the stochastic version of fractal interpolation function. In order to obtain fractal interpolation functions with more exibility, Wang and Yu [9] used instead of a constant scaling parameter a variable vertical scaling factor. Also the notion of fractal interpolation can be generalized to the graph-directed case introduced by Deniz and  Ozdemir in [5]. In this paper we study the case of a stochastic fractal interpolation function with graph-directed fractal function."

Stochastic version of Henry type Gronwall’s inequality

We use Young’s and Hölder inequality combined with classical Gronwall’s inequality to derive present a new version of the stochastic of Gronwall’s inequalities with singular kernels.