A Multi-Scale DNN Algorithm for Nonlinear Elliptic Equations with Multiple Scales

Algorithms based on deep neural networks (DNNs) have attracted increasing attention from the scientific computing community. DNN based algorithms are easy to implement, natural for nonlinear problems, and have shown great potential to overcome the curse of dimensionality. In this work, we utilize the multi-scale DNN-based algorithm (MscaleDNN) proposed by Liu, Cai and Xu (2020) to solve multi-scale elliptic problems with possible nonlinearity, for example, the p-Laplacian problem. We improve the MscaleDNN algorithm by a smooth and localized activation function. Several numerical examples of multi-scale elliptic problems with separable or non-separable scales in low-dimensional and high-dimensional Euclidean spaces are used to demonstrate the effectiveness and accuracy of the MscaleDNN numerical scheme.     

A Compressible Conserved Discrete Unified Gas-Kinetic Scheme with Unstructured Discrete Velocity Space for Multi-Scale Jet Flow Expanding into Vacuum Environment

The mechanism of jet flow expanding into vacuum environment (or extremely low density environment) is important for the propulsion unit of micro-electro-mechanical systems (MEMS), the thruster of spacecraft, the attitude control system of satellite, etc.. Since its flow field is often composed of local continuum region and local rarefied region, the jet flow into vacuum has noteworthy multi-scale transportation behaviors. Therefore, the numerical study of such flows needs the multi-scale schemes which are valid for both continuum and rarefied flows. In the past few years, a series of unified methods for whole flow regime (from continuum regime to rarefied regime) have been developed from the perspective of the direct modeling, and have been verified by sufficient test cases. In this paper, the compressible conserved discrete unified gas-kinetic scheme is further developed and is utilized for predicting the jet flows into vacuum environment. In order to cover the working conditions of both aerospace and MEMS applications, the jet flows with a wide range of inlet Knudsen (Kn) numbers (from 1E-4 to 100) are considered. The evolution of flow field during the entire startup and shutdown process with Kn number 100 is predicted by the present method, and it matches well with the result of analytical collisionless Boltzmann equation. For Kn numbers from 1E-4 to 10, the flow field properties such as density, momentum, and pressure are investigated, and the results are provided in details, since the published results are not sufficient at the present stage. The extent and intensity of the jet flow influence are especially investigated, because they are strongly related to the plume contamination and momentum impact on objects facing the jet, such as the solar paddles which face the attitude control thruster during the docking process.     

A retinal vessel boundary tracking method based on Bayesian theory and multi-scale line detection

A retinal vessel tracking method based on Bayesian theory and multi-scale line detection is proposed in this paper. The optic disk is located by a PCA method and the initial points of tracking are identified. In each step, candidate points for vessel edges are selected on a semi-ellipse. Three types of vessel structure are considered in the tracking: normal vessel, branching, and crossing. To determine the new pair of edge points, the characteristics of the vessel intensity profiles along both the cross section and the longitudinal direction are considered in the tracking. A Gaussian model is assumed in the cross section and multi-scale line detection is employed in the longitudinal direction. The advantage of the proposed method is that two dimensional vessel information is employed, which makes it work better than methods using one dimensional information only. Our method is tested on the REVIEW database and a comparison study is performed. Experimental results show that the proposed method is precise and robust in tracking vessel edges.     

Combinative multi-scale level set framework for echocardiographic image segmentation

In the automatic segmentation of echocardiographic images, a priori shape knowledge has been used to compensate for poor features in ultrasound images. This shape knowledge is often learned via an off-line training process, which requires tedious human effort and is highly expertise-dependent. More importantly, a learned shape template can only be used to segment a specific class of images with similar boundary shape. In this paper, we present a multi-scale level set framework for segmentation of endocardial boundaries at each frame in a multiframe echocardiographic image sequence. We point out that the intensity distribution of an ultrasound image at a very coarse scale can be approximately modeled by Gaussian. Then we combine region homogeneity and edge features in a level set approach to extract boundaries automatically at this coarse scale. At finer scale levels, these coarse boundaries are used to both initialize boundary detection and serve as an external constraint to guide contour evolution. This constraint functions similar to a traditional shape prior. Experimental results validate this combinative framework.     

Adiabatic layering: A new concept of hierarchical multi-scale optimization

Recurrent neural networks (RNNs) with linearized dynamics have shown great promise in solving continuous valued optimization problems subject to bound constraints. Building on this progress, a novel method of constrained hierarchical multi-scale optimization is developed that applies to a wide range of optimization problems and signal decomposition tasks. Central to the underlying concept is the definition of adiabatic layering. Analytic justification of this model can be regarded as a natural development of the mean-field theory. What emerges is an alternative hierarchical optimization method that promises to improve upon existing hierarchical schemes in combining the accuracy of global optimization with the compact representation of hierarchical optimization. Whereas conventional hierarchical optimization techniques typically tend to average over fine-scale detail when applied to bound-constrained problems, such behavior is avoided by the modified dynamics of the proposed method. Applied to the signal decomposition problem of RBF approximation, the behaviour of the adiabatic layering model is shown to be in close correspondence with the theoretical expectations.     

儿童失神癫(癎)脑电图的多尺度特征

目的:研究儿童失神癫癎脑电图的多尺度定量特征。方法:对15例失神癫癎患儿10次临床发作和20次亚临床癎样放电的脑电图进行子波分析,提取失神癫癎发作过程中脑电信号的多尺度定量典型特征,与发作前10 s及发作后10 s的脑电信号进行比较,并与12例正常同龄儿童脑电图进行比较。结果:研究显示儿童失神癫癎发作过程中脑电信号的多尺度典型特征主要表现为12尺度(对应频率3 Hz)的节律性活动显著增强,发作时20尺度(低频大尺度,对应频率0．12 Hz)结构与频率3 Hz的结构具有非正常的跳跃式尺度关系,3 Hz节律性棘慢复合波与大尺度(频率1 Hz以下)背景低频放电结构共同存在。发作过程中分尺度功率主要集中在20尺度和12尺度,其演变规律为20尺度能量逐渐减低,12尺度能量逐渐增加。10次临床发作的脑电信号均显示上述特征。发作前10 s和后10 s的脑电多尺度信号中仍然存在隐性的3 Hz棘慢复合波成分,与一般认为3 Hz棘慢复合波突起突止不同．而从传统的脑电图上无法分辨出发作前后的这些多尺度细节的定量特征。亚临床癎样放电的多尺度特征与发作期无明显差别,但持续时间短。结论:子波分析作为一种新的信号分析方法,适合于脑电信号的分析,可以获得比传统视觉脑电图更多的定量信息。通过对失神癫癎患儿的脑电信号进行子波分析,得到其发作过程中典型的多尺度定量特征,有助于失神癫癎发作的临床辅助诊断、预后评价以及神经电生理机理的基础研究。