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.     

Feature-driven Cartesian adaptive mesh refinement for vortex-dominated flows

We develop locally normalized feature-detection methods to guide the adaptive mesh refinement (AMR) process for Cartesian grid systems to improve the resolution of vortical features in aerodynamic wakes. The methods include: the Q-criterion [1], the λ₂ method [2], the λ_ci method [3], and the λ₊ method [4]. Specific attention is given to automate the feature identification process by applying a local normalization based upon the shear-strain rate so that they can be applied to a wide range of flow-fields without the need for user intervention. To validate the methods, we assess tagging efficiency and accuracy using a series of static vortex-dominated flow-fields, and use the methods to drive the AMR process for several theoretical and practical simulations. We demonstrate that the adaptive solutions provide comparable accuracy to solutions obtained on uniformly refined meshes at a fraction of the computational cost. Overall, the normalized feature detection methods are shown to be effective in driving the AMR process in an automated and efficient manner.     

Laser resonator with balanced thermal lenses

An adaptive negative thermal lens that compensates for the power-dependent positive thermal lens in a transversally diode-pumped Nd:YAG laser rod is presented. We demonstrate that the proposed technique leads to a reduction of the total thermal lens by more than an order of magnitude.     

Solving Multi-dimensional Evolution Problems with Localized Structures using Second Generation Wavelets

A dynamically adaptive numerical method for solving multi-dimensional evolution problems with localized structures is developed. The method is based on the general class of multi-dimensional second-generation wavelets and is an extension of the second-generation wavelet collocation method of Vasilyev and Bowman to two and higher dimensions and irregular sampling intervals. Wavelet decomposition is used for grid adaptation and interpolation, while O ( N ) hierarchical finite difference scheme, which takes advantage of wavelet multilevel decomposition, is used for derivative calculations. The prowess and computational efficiency of the method are demonstrated for the solution of a number of two-dimensional test problems.     

Adaptive discretization of convex multistage stochastic programs

We propose a new scenario tree reduction algorithm for multistage stochastic programs, which integrates the reduction of a scenario tree into the solution process of the stochastic program. This allows to construct a scenario tree that is highly adapted on the optimization problem. The algorithm starts with a rough approximation of the original tree and locally refines this approximation as long as necessary. Promising numerical results for scenario tree reductions in the settings of portfolio management and power management with uncertain load are presented.     

Stopping criteria for inner iterations in inexact potential reduction methods: a computational study

We focus on the use of adaptive stopping criteria in iterative methods for KKT systems that arise in Potential Reduction methods for quadratic programming. The aim of these criteria is to relate the accuracy in the solution of the KKT system to the quality of the current iterate, to get computational efficiency. We analyze a stopping criterion deriving from the convergence theory of inexact Potential Reduction methods and investigate the possibility of relaxing it in order to reduce as much as possible the overall computational cost. We also devise computational strategies to face a possible slowdown of convergence when an insufficient accuracy is required.     

Searching for a counterfeit coin with two unreliable weighings

We provide a worst-case optimal procedure to identify one unknown heavy coin among N identically looking coins when a balance scale is used and at most two weighing results can be erroneous. The exact minimal number of weighings is determined. This solves completely the weighing problem of the case of two unreliable weighings.     

Adaptive detection technique for optical wireless communication over strong turbulence channels

Optical wireless communication (OWC) systems use the atmosphere as a propagation medium, so the atmospheric turbulence effects lead to fading related with signal intensity. The received signal of OWC over strong turbulence channels is assumed to be a mixture of K-distributed fading and Gaussian distributed thermal noise. Second-order spectral analysis is unable to separately estimate the mixed signal. In order to mitigate the fading induced by turbulence, the decision threshold-updating algorithm based on second and higher order cumulants is proposed and is able to operate in an unknown turbulence environment. The performance of the adaptive processing scheme has been evaluated by means of Monte Carlo simulations. Simulation results show the improvement of the bit error rate (BER) performance.