On separating a fixed point from zero by invariants

Assume a fixed point v∈V^G can be separated from zero by a homogeneous invariant f∈𝕜[V]^G of degree p^rd, where p>0 is the characteristic of the ground field 𝕜 and p,d are coprime. We show that then v can also be separated from zero by an invariant of degree p^r, which we obtain explicitly from f. It follows that the minimal degree of a homogeneous invariant separating v from zero is a p-power.     

Mixed two‐grid finite difference methods for solving one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations

The aim of this paper is to propose mixed two‐grid finite difference methods to obtain the numerical solution of the one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations. The finite difference equations at all interior grid points form a large‐sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The proposed method is based on applying a family of finite difference methods for discretizing the spatial and time derivatives. The obtained system has been solved by two‐grid method, where the two‐grid method is used for solving the large‐sparse linear systems. Also, in the proposed method, the spectral radius with local Fourier analysis is calculated for different values of h and Δt. The numerical examples show the efficiency of this algorithm for solving the one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations. Copyright © 2016 John Wiley & Sons, Ltd.     

A multigrid perspective on the parallel full approximation scheme in space and time

For the numerical solution of time‐dependent partial differential equations, time‐parallel methods have recently been shown to provide a promising way to extend prevailing strong‐scaling limits of numerical codes. One of the most complex methods in this field is the “Parallel Full Approximation Scheme in Space and Time” (PFASST). PFASST already shows promising results for many use cases and benchmarks. However, a solid and reliable mathematical foundation is still missing. We show that, under certain assumptions, the PFASST algorithm can be conveniently and rigorously described as a multigrid‐in‐time method. Following this equivalence, first steps towards a comprehensive analysis of PFASST using blockwise local Fourier analysis are taken. The theoretical results are applied to examples of diffusive and advective type.     

Multilevel-in-width training for deep neural network regression

A common challenge in regression is that for many problems, the degrees of freedom required for a high-quality solution also allows for overfitting. Regularization is a class of strategies that seek to restrict the range of possible solutions so as to discourage overfitting while still enabling good solutions, and different regularization strategies impose different types of restrictions. In this paper, we present a multilevel regularization strategy that constructs and trains a hierarchy of neural networks, each of which has layers that are wider versions of the previous network's layers. We draw intuition and techniques from the field of Algebraic Multigrid (AMG), traditionally used for solving linear and nonlinear systems of equations, and specifically adapt the Full Approximation Scheme (FAS) for nonlinear systems of equations to the problem of deep learning. Training through V-cycles then encourage the neural networks to build a hierarchical understanding of the problem. We refer to this approach as multilevel-in-width to distinguish from prior multilevel works which hierarchically alter the depth of neural networks. The resulting approach is a highly flexible framework that can be applied to a variety of layer types, which we demonstrate with both fully connected and convolutional layers. We experimentally show with PDE regression problems that our multilevel training approach is an effective regularizer, improving the generalize performance of the neural networks studied.     

Multigrid Models of Composite Materials of Irregular Structure with a Small Filling Ratio

This paper considers composites consisting of a set of typical composite multigrid finite elements whose structures are regular and different. Mean local errors are proposed for multigrid modeling of composites.     

The influence of free convection on soil salinization in arid regions

Evaporation of groundwater in a region with a shallow water table and small natural replenishment causes accumulation of salts near the ground surface. Water in the upper soil layer becomes denser than in the depth. This is a potentially unstable situation which may result in convective currents. When free convection takes place, estimates of the salinity profile, salt precipitation rate, etc., obtained within the framework of a 1-D (vertical) model fail.Very simplified model of the process is proposed, in which the unsaturated zone is represented by a horizontal soil layer at a constant water saturation, and temperature changes are neglected. The purpose of the model is to obtain a rough estimate of the role of natural convection in the salinization process.A linear stability analysis of a uniform vertical flow is given, and the stability limit is determined numerically as a function of evaporation rate, salt concentration in groundwater, and porous medium dispersivity. The loss of stability corresponds to quite realistic Rayleigh numbers. The stability limit depends in nonmonotonic way on the evaporation rate.The developed convective regime was simulated numerically for a 2-D vertical domain, using finite volume element discretization and FAS multigrid solver. The dependence of the average salt concentration in the upper layer on the Rayleigh number was obtained.List of Main Symbols horizontal wavenumber - _L , _T dispersivities (longitudinal and transversal) - D ^* diffusion coefficient (in a porous medium) - g acceleration of gravity - H thickness of the vadoze zone - k permeability - p pressure - Pe Péclet number - q mass flux - Ra Rayleigh number Greek _L , _T dimensionless dispersivities - coefficient of concentration expansion - coefficient of viscosity variation - volumetric fraction of the liquid phase - viscosity - density - stream function - mass fraction of salt in water Vectors and tensors D dispersion coefficient - e unit vector - I unit tensor - J nonadvective salt flux - V liquid phase velocity - x radius-vector     

求解复杂湍流的非线性涡黏性系数模型和代数应力模型

非线性涡黏性系数模型和代数应力模型联系了线性涡黏性系数湍流模型和完整的微分 雷诺应力模型.随着它们受到日益关注,其形式也越来越多样化.本篇综述的目的是对这些模 型加以总结并比较它们之间的共同点及不同之处,指出它们与完整微分雷诺应力模型之间的 关系,以及相对于线性涡黏性系数模型而言它们在预报流场上所具有的优势.     

用代数应力模型模化型腔充填过程中的紊流流动

针对型腔充填过程中的紊流流动，用代数应力模型研究紊流流动现象．成功地将代数应力模型引入ＰＨＯＥＮＩＣＳ软件中，完成了恒温流体充填过程三维时均速度和自由表面的计算模拟．与实验结果相比说明，本文基于ＰＨＯＥＮＩＣＳ软件开发的充填模拟具备了对复杂型腔充填过程的数值计算．     

The Opposing Effect of Viscous Dissipation Allows for a Parallel Free Convection Boundary-Layer Flow Along a Cold Vertical Flat Plate

External free convection boundary-layer flows are usually treated by neglecting the effect of viscous dissipation. This assumption always results in a non-parallel flow, besides a strong parallel component also a weak transversal component of the (steady) velocity field occurs. The present paper shows, however, that the weak opposing effect of the buoyancy forces due to heat release by viscous dissipation, can give rise along a cold vertical plate adjacent to a fluid saturated porous medium to a strictly parallel steady free convection flow. This boundary-layer flow is described by an algebraically decaying exact analytical solution of the basic balance equations.     

An explicit algebraic Reynolds stress and heat flux model for incompressible turbulence: Part I Non-isothermal flow

Tensor representation theory is used to derive an explicit algebraic model that consists of an explicit algebraic stress model (EASM) and an explicit algebraic heat flux model (EAHFM) for two-dimensional (2-D) incompressible non-isothermal turbulent flows. The representation methodology used for the heat flux vector is adapted from that used for the polynomial representation of the Reynolds stress anisotropy tensor. Since the methodology is based on the formation of invariants from either vector or tensor basis sets, it is possible to derive explicit polynomial vector expansions for the heat flux vector. The resulting EAHFM is necessarily coupled with the turbulent velocity field through an EASM for the Reynolds stress anisotropy. An EASM has previously been derived by Jongen and Gatski [10]. Therefore, it is used in conjunction with the derived EAHFM to form the explicit algebraic model for incompressible 2-D flows. This explicit algebraic model is analyzed and compared with previous formulations including its ability to approximate the commonly accepted value for the turbulent Prandtl number. The effect of pressure-scrambling vector model calibration on predictive performance is also assessed. Finally, the explicit algebraic model is validated against a 2-D homogeneous shear flow with a variety of thermal gradients. Dedicated to the memory of the late Professor Charles G. Speziale of Boston University