High Order IMEX Stochastic Galerkin Schemes for Linear Transport Equation with Random Inputs and Diffusive Scalings

In this paper,we consider the high order method for solving the linear transport equations under diffusive scaling and with random inputs.To tackle the randomness in the problem,the stochastic Galerkin method of the generalized polynomial chaos approach has been employed.Besides,the high order implicit-explicit scheme under the micro-macro decomposition framework and the discontinuous Galerkin method have been employed.We provide several numerical experiments to validate the accuracy and the stochastic asymptotic-preserving property.

Stochastic properties of tumor growth with coupling between non-Gaussian and Gaussian noise terms

Dynamical behavior of a tumor-growth model with coupling between non-Gaussian and Gaussian noise terms is investigated. The departure from the Gaussian noise can not only reduce the probability of tumor cells in the active state, induce the minimum of the average tumor-cell population to move toward a smaller non-Gaussian noise, but also decrease the mean first-passage time. The increase of white-noise intensity can increase the tumor-cell population and shorten the mean first-passage time, while the coupling strength between noise terms has opposite effects, and the noise correlation time has a very small effect.

Stochastic properties of solution for a chemostat model with a distributed delay and random disturbance

In this paper,stochastic properties of solution for a chemostat model with a distributed delay and random disturbance are studied,and we use distribution delay to simulate the delay in nutrient conversion.By the linear chain technique,we transform the stochastic chemostat model with weak kernel into an equivalent degenerate system which contains three equations.First,we state that this model has a unique global positive solution for any initial value,which is helpful to explore its stochastic properties.Furthermore,we prove the stochastic ultimate boundness of the solution of system.Then sufficient conditions for solution of the system tending toward the boundary equilibrium point at exponential rate are established,which means the microorganism will be extinct.Moreover,we also obtain some sufficient conditions for ergodicity of solution of this system by constructing some suitable stochastic Lyapunov functions.Finally,we provide some numerical examples to illustrate theoretical results,and some conclusions and analysis are given.

BACKWARD STOCHASTIC VIABILITY AND RELATED PROPERTIES ON Z FOR BSDES WITH APPLICATIONS

This paper investigates some important properties of Z, the martingale integrant of the backward stochastic differential equations, which is the second process of the solution. These include the backward stochastic viability property, bounded property and the comparison theorem. To explain the theoretical results, the authors apply them to study a financial contingent claim pricing problem. The replication portfolio process can be characterized clearly.

Reduction of data amount in data-driven design of linear quadratic regulators

This paper discusses the data-driven design of linear quadratic regulators,i.e.,to obtain the regulators directly from experimental data without using the models of plants.In particular,we aim to improve an existing design method by reducing the amount of the required experimental data.Reducing the data amount leads to the cost reduction of experiments and computation for the data-driven design.We present a simplified version of the existing method,where parameters yielding the gain of the regulator are estimated from only part of the data required in the existing method.We then show that the data amount required in the presented method is less than half of that in the existing method under certain conditions.In addition,assuming the presence of measurement noise,we analyze the relations between the expectations and variances of the estimated parameters and the noise.As a result,it is shown that using a larger amount of the experimental data might mitigate the effects of the noise on the estimated parameters.These results are verified by numerical examples.

Uncertainty propagation in dynamics of composite plates: A semi-analytical non-sampling-based approach

In this study,the influences of spatially varying stochastic properties on free vibration analysis of composite plates were investigated via development of a new approach named the deterministic-stochastic Galerkin-based semi-analytical method.The material properties including tensile modulus,shear modulus,and density of the plate were assumed to be spatially varying and uncertain.Gaussian fields with first-order Markov kernels were utilized to define the aforementioned material properties.The stochastic fields were decomposed via application of the K arhunen-Loeve theorem.A first-order shear deformation theory was assumed,following which the displacement field was defined using admissible trigonometric modes to derive the potential and kinetic energies.The stochastic equations of motion of the plate were obtained using the variational principle.The deterministic-stochastic Galerkin-based method was utilized to find the probability space of natural frequencies,and the corresponding mode shapes of the plate were determined using a polynomial chaos approach.The proposed method significantly reduced the size of the mathematical models of the structure,which is very useful for enhancing the computational efficiency of stochastic simulations.The methodology was verifed using a stochastic finite element method and the available results in literature.The sensitivity of natural frequencies and corresponding mode shapes due to the uncertainty of material properties was investigated,and the results indicated that the higher-order modes are more sensitive to uncertainty propagation in spatially varying properties.

SOME RESULTS ON THE RESIDUAL LIFE AT RANDOM TIME

In this paper, we consider the residual life at random time, i.e. XY=X-Y|X>Y, where X and Y are non-negative random variables. We establish a number of stochastic comparison properties for XY under various assumptions of X and Y. Under the assumption that Y has decreasing reverse hazard rate (DAHR), we show that if X is in any one of the classes IFR, DFR, DMRL or IMRL then XY is in the same class as X. We also obtain some useful bounds for the distribution and the moment of XY. Because the idle time in classical GI/G/1 queuing system can be regarded as the residual life at random time, the results obtained in this paper have applications in the study of such system.