In this paper, we are concerned with a new stochastic system of nonlinear partial differential equations modeling the Lotka–Volterra interactions of predators and preys in the presence of prey-taxis, spatial diffusion, and noises. The spatial and temporal variations of the predator’s velocity are determined by the prey gradient. In the first part, we derive a macroscopic model from stochastic kinetic equations by using the micro–macro decomposition method. In the second part, we sketch the proof of the existence of weak martingale solutions by using a Faedo–Galerkin method. In the last part, we develop a one- and two-dimensional finite volume approximation for the stochastic kinetic and macroscopic models, respectively. Our one-dimensional space numerical scheme is uniformly stable along the transition from kinetic to macroscopic regimes. We close with various numerical tests illustrating the convergence of our numerical method and some features of our stochastic macro-scale system.

This paper develops a supervised learning of overcomplete dictionaries (SLOD) to efficiently perform online analysis and prediction of dynamic stability (DSP) in large power systems. To this end, multiple post-contingency signals with varying faults, load and generator switching are acquired by phasor measurement units (PMUs) based monitoring of all generators to form the training dataset. The SLOD approach then jointly learns a sparse feature dictionary and a stability status classifier using dynamic singular value decomposition (D-KSVD) and orthogonal matching pursuit (OMP). The aim is to learn features of the signals while adding a stability status term to the optimization problem. This enabled stable/unstable classes to be represented. The proposed SLOD manages to improve online DSP shortly after a contingency. It does this without any prior assumptions or signal preprocessing except for dynamic state estimation to get rotor speed from generator terminal PMU data. Test results on the IEEE 2-area 4 machines, 39 and 68 -bus test systems demonstrate that the SLOD gives more than satisfactory performance in all cases (i.e., ∼99.99% accuracy, ∼99.99% reliability, and 100% security) compared to Statistics and Machine Learning Toolbox™ -MATLAB that use time-series-based high-dimensional stability indices as input for DSP.

Fixed and adaptive supervised dictionary learning (SDL) is proposed in this paper for wide-area stability assessment. Single and hybrid fixed structures are developed based on impulse dictionary (ID), discrete Haar transform (DHT), discrete cosine transform (DCT), discrete sine transform (DST), and discrete wavelet transform (DWT) for sparse features extraction and online transient stability prediction. The fixed structures performance is compared with that obtained from transient K-singular value decomposition (TK-SVD) implemented while adding a stability status term to the optimization problem. Stable and unstable dictionary learning are designed based on datasets recorded by simulating thousands of contingencies with varying faults, load, and generator switching on the IEEE 68-bus test system. This separate supervised learning of stable and unstable scenarios allows determining root mean square error (RMSE), useful for online stability status assessment of new scenarios. With respect to the RMSE performance metric in signal reconstruction-based stability prediction, the present analysis demonstrates that [DWT], [DHT|DWT] and [DST|DHT|DCT] are better stability descriptors compared to K-SVD, [DHT], [DCT], [DCT|DWT], [DHT|DCT], [ID|DCT|DST], and [DWT|DHT|DCT] on test datasets. However, the K-SVD approach is faster to execute in both off-line training and real-time playback while yielding satisfactory accuracy in transient stability prediction (i.e., 7.5-cycles decision window after fault-clearing).

This article proposes a hybrid susceptible–infected–removed model which takes into consideration the spatiotemporal dynamics of the individuals. The model is based on a system of discrete stochastic diffusion equations. To build these equations, the two-dimensional diffusion equations coming from a balanced method are coupled with the human displacement probability law pattern through a discretization made by the finite volume method for complex geometries. The validation of this model is applied to COVID-19 spread. Since it is actual and the statistics are available elsewhere. The case of a developed country is used for simulation under some assumptions. Firstly to fit the chosen displacement pattern, then the accuracy of the statistics provided helps to analyze the sensitivity of the parameters of the model. The results explain the influence of the population movement on the evolution of the spread.

We consider the class of linear operator equations with operators admitting self-adjoint positive definite and m-accretive splitting (SAS). This splitting leads to an ADI-like iterative method which is equivalent to a fixed point problem where the operator is a 2 by 2 matrix of operators. An infinite dimensional adaptation of a minimal residual algorithm with Symmetric Gauss-Seidel and polynomial preconditioning is then applied to solve the resulting matrix operator equation. Theoretical analysis shows the convergence of the methods, and upper bounds for the decrease rate of the residual are derived. The convergence of the methods is numerically illustrated with the example of the neutron transport problem in 2-D geometry.

In this paper, we propose a numerical approach to solve a kinetic model of chemotaxis phenomena. This scheme is shown to be uniformly stable with respect to the small parameter, consistent with the uid-di usion limit (Keller-Segel model). Our approach is based on the micro-macro decomposition which leads to an equivalent formulation of the kinetic model that couples a kinetic equation with macroscopic ones. This method is validated by various test cases and compared to other standard methods.