Further results on adaptive full-order and reduced-order observers for Lur’e differential inclusions

This paper is concerned with the adaptive observer design of Lur’e differential inclusions with unknown parameters. Under a relaxed assumption on nonlinear perturbation functions, a sufficient condition for the existence of an adaptive full-order observer is established. Comparing with results in the literature, the present conditions are complemented with a numerically reliable computational approach, which can be checked by means of linear matrix inequalities. Furthermore, it is shown that, under the sufficient condition, the existence of a reduced-order observer is guaranteed. Also, the reduced-order observer is designed. The effectiveness of the proposed design is illustrated via a simulation example.     

Simulation of the effect of learning on the topology of the functional connectivity of neural networks

We introduce a procedure for simulating adaptive learning in neural networks and the effect this learning has on the way in which the functional connections between the nodes of the network are established. The procedure combines two mechanisms: firstly, the gradual dilution of the network through the elimination of synaptic weights in increasing order of magnitude, thus reducing the costs of the network structure. Secondly, to train the network as it is diluted so as not to compromise its performance pursuant to the proposed task. Considering different levels of learning difficulty, we compare the topology of the functional connectivities that result from the application of this procedure with those obtained using fMRI in healthy volunteers. According to our results, the topology of functional connectivities in healthy subjects can be interpreted as the product of a learning process with a specific degree of difficulty.     

An adaptive Newton-method based on a dynamical systems approach

The traditional Newton method for solving nonlinear operator equations in Banach spaces is discussed within the context of the continuous Newton method. This setting makes it possible to interpret the Newton method as a discrete dynamical system and thereby to cast it in the framework of an adaptive step size control procedure. In so doing, our goal is to reduce the chaotic behavior of the original method without losing its quadratic convergence property close to the roots. The performance of the modified scheme is illustrated with various examples from algebraic and differential equations.     

基于网格变动的油水驱动半正定问题迎风混合元方法

在网格随时间变动的有限元空间上研究了不可压缩的两相渗流驱动问题.分别对饱和度方程扩散矩阵正定和半正定的情形,提出了基于网格变动的迎风混合元方法混合元逼进压力方程,饱和度方程的对流项采用迎风格式来处理,扩散项则采用推广的混合元来逼进.在网格任意变动的情形下得到几乎最优的误差估计;对正定问题的格式进行改进,即在两个网格之间投影变化时采取近似解的线性构造,可以得到与固定网格时相同的最优收敛阶.     

A fourth-order approximate projection method for the incompressible Navier–Stokes equations on locally-refined periodic domains

In this follow-up of our previous work [30], the author proposes a high-order semi-implicit method for numerically solving the incompressible Navier–Stokes equations on locally-refined periodic domains. Fourth-order finite-volume stencils are employed for spatially discretizing various operators in the context of structured adaptive mesh refinement (AMR). Time integration adopts a fourth-order, semi-implicit, additive Runge–Kutta method to treat the non-stiff convection term explicitly and the stiff diffusion term implicitly. The divergence-free condition is fulfilled by an approximate projection operator. Altogether, these components yield a simple algorithm for simulating incompressible viscous flows on periodic domains with fourth-order accuracies both in time and in space. Results of numerical tests show that the proposed method is superior to previous second-order methods in terms of accuracy and efficiency. A major contribution of this work is the analysis of a fourth-order approximate projection operator.     

The parameter uniform numerical method for singularly perturbed parabolic reaction–diffusion problems on equidistributed grids

This article presents the study of singularly perturbed parabolic reaction–diffusion problems with boundary layers. To solve these problems, we use a modified backward Euler finite difference scheme on layer adapted nonuniform meshes at each time level. The nonuniform meshes are obtained by equidistribution of a positive monitor function, which involves the second-order spatial derivative of the singular component of the solution. The equidistributing monitor function at each time level allows us to use this technique to non-linear parabolic problems. The truncation error and the stability analysis are obtained. Parameter–uniform error estimates are derived for the numerical solution. To support the theoretical results, numerical experiments are carried out.     

ANALYSIS OF MOVING MESH METHODS BASED ON GEOMETRICAL VARIABLES

1. IntroductionMany methods have been proposed for adapting the mesh to aChieve spatial resolution in thesolution of partial dthereatial equations. In addition to the capability of concedrating sufficientpoints about regions of rapid variation of the solution, a satisfactory mesh equatioll should besample, easy to program, and reasonably insensitive to the choice of its adjustable parameters.The earliest work on adaptive tecboques, based on moving trite element method (MFEM) wdone by Miller …     

On the adaptive control of a class of partially observed Markov decision processes

This paper is concerned with the adaptive control problem, over the infinite horizon, for partially observable Markov decision processes whose transition functions are parameterized by an unknown vector. We treat finite models and impose relatively mild assumptions on the transition function. Provided that a sequence of parameter estimates converging in probability to the true parameter value is available, we show that the certainty equivalence adaptive policy is optimal in the long-run average sense.