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.     

Finite volume multigrid method of the planar contraction flow of a viscoelastic fluid

This paper reports on a numerical algorithm for the steady flow of viscoelastic fluid. The conservative and constitutive equations are solved using the finite volume method (FVM) with a hybrid scheme for the velocities and first‐order upwind approximation for the viscoelastic stress. A non‐uniform staggered grid system is used. The iterative SIMPLE algorithm is employed to relax the coupled momentum and continuity equations. The non‐linear algebraic equations over the flow domain are solved iteratively by the symmetrical coupled Gauss–Seidel (SCGS) method. In both, the full approximation storage (FAS) multigrid algorithm is used. An Oldroyd‐B fluid model was selected for the calculation. Results are reported for planar 4:1 abrupt contraction at various Weissenberg numbers. The solutions are found to be stable and smooth. The solutions show that at high Weissenberg number the domain must be long enough. The convergence of the method has been verified with grid refinement. All the calculations have been performed on a PC equipped with a Pentium III processor at 550 MHz. Copyright © 2001 John Wiley & Sons, Ltd.     

Enantioselective synthesis of phomallenic acid C, an inhibitor of FAS II pathway

Enantioselective synthesis of phomallenic acid C, an inhibitor of bacterial FAS II pathway, was successful. Allenyldiyne structure was constructed by a direct anti-S_N2′ coupling of propargyl mesylate with diynylindium in the presence of palladium catalyst. Enantiomeric purity was determined by Ohrui-Akasaka method to be 83%ee.     

Analysis of tensides in complex samples with comprehensive two-dimensional gas chromatography coupled with time-of-flight mass spectrometry

Multidimensional gas-chromatographical analysis of various tensides of natural or synthetic origin in cosmetic products is demonstrated. Comprehensive two-dimensional gas chromatography coupled with time-of-flight mass spectrometry allows the qualitative and quantitative determination of alkyl polyglucosides (AG), fatty alcohol ethoxylates (FAEO), fatty alcohol sulfates (FAS), fatty alcohol ether sulfates (FAES) and cocamidopropyl betaines (CAPB) in shower gel and cleaning agents. The samples were aliquoted in two parts. The first part was silylated, diluted and analysed; then, in order to detect anionic tensides (FAES, FAS) too, the second aliquot was hydrolysed before being silylated for analysis. Because of their amphoteric character, the betaines can only be analysed by gas chromatography after thermal decomposition in the injector, which leads to the corresponding amidoamines among other products.     

Multigrid Method for a Two Dimensional Fully Nonlinear Black-Scholes Equation with a Nonlinear Volatility Function

This paper deals with the task of pricing European basket options in the presence of transaction costs. We develop a model that incorporates the illiquidity of the market into the classical two-assets Black-Scholes framework. We perform a numerical simulation using finite difference method. We consider a nonlinear multigrid method in order to reduce computational costs. The objective of this paper is to investigate a deterministic extension for the Barles' and Soner's model and to demonstrate the effectiveness of multigrid approach to solving a fully nonlinear two dimensional Black-Scholes problem.     

β-酮脂酰-ACP合成酶（FabH）抑制剂研究进展

脂肪酸的生物合成对病原细菌的存活至关重要。近年来,在这一生物合成途径中所涉及的关键酶引起了人们的广泛关注。其中β-酮脂酰-ACP合成酶Ⅲ(KAS Ⅲ,FabH)控制着细菌脂肪酸生物合成的起始步骤,普遍存在于病原体中且在人体中无其同源蛋白,成为新型抗菌药物靶标研究的热点。抑制FabH酶活性的小分子抑制剂有望成为对细菌具选择性而对人体无毒的广谱抗菌药。本文对脂肪酸的生物合成,FabH相关结构研究以及目前FabH抑制剂的研究进展进行了综述。     

Optimization of the determination of chemical oxygen demand in wastewaters

Chemical oxygen demand (COD) is one of the most relevant chemical parameters for the management of wastewater treatment facilities including the control of the quality of an effluent. The adequacy of decisions based on COD values relies on the quality of the measurements. Cost effective management of the minor sources of uncertainty can be applied to the analytical procedure without affecting measurement quality. This work presents a detailed assessment of the determination of COD values in wastewaters, according to ISO6060:1989 standard, which can support reduction of both measurement uncertainty and cost of analysis. This assessment includes the definition of the measurement traceability chain and the validation of the measurement procedure supported on sound and objective criteria. Detailed models of the measurement performance, including uncertainty, developed from the Differential Approach, were successfully validated by proficiency tests. The assumption of the measurement function linearity of the uncertainty propagation law was tested through the comparison with the numerical Kragten method. The gathered information supported the definition of strategies for measurement uncertainty or cost reduction. The developed models are available as electronic supplementary material, in an MS-Excel file, to be updated with the user's data.     

A new multigrid algorithm for non-linear equations in conjunction with time-marching procedures

Full approximate storage (FAS) multigrid algorithm is the most commonly used multigrid algorithm for non-linear equations. The algorithm initially developed for steady-state equations was later extended to obtain steady-state solutions employing unsteady equations. In extending the FAS algorithm for the steady-state non-linear equations to unsteady non-linear equations, the FAS algorithm does not to take into account that the governing equations are typically linearized in time before they are solved. Thus, there is a scope to develop a new multigrid algorithm to apply the multigrid technique to the equations linearized in time. In the present work, such an algorithm is developed exploring this possibility and is implemented for two-dimensional incompressible and compressible flows coupled with explicit time marching procedures. The results of the new algorithm compare favourably with those of the FAS multigrid method and single grid.     

Application of Sobolev gradient method to Poisson–Boltzmann system

The idea of a weighted Sobolev gradient, introduced and applied to singular differential equations in [1], is extended to a Poisson–Boltzmann system with discontinuous coefficients. The technique is demonstrated on fully nonlinear and linear forms of the Poisson– Boltzmann equation in one, two, and three dimensions in a finite difference setting. A comparison between the weighted gradient and FAS multigrid is given for large jump size in the coefficient function.     

A finite volume approach for unsteady viscoelastic fluid flows

A finite volume, time‐marching for solving time‐dependent viscoelastic flow in two space dimensions for Oldroyd‐B and Phan Thien–Tanner fluids, is presented. A non‐uniform staggered grid system is used. The conservation and constitutive equations are solved using the finite volume method with an upwind scheme for the viscoelastic stresses and an hybrid scheme for the velocities. To calculate the pressure field, the semi‐implicit method for the pressure linked equation revised method is used. The discretized equations are solved sequentially, using the tridiagonal matrix algorithm solver with under‐relaxation. In both, the full approximation storage multigrid algorithm is used to speed up the convergence rate. Simulations of viscoelastic flows in four‐to‐one abrupt plane contraction are carried out. We will study the behaviour at the entrance corner of the four‐to‐one planar abrupt contraction. Using this solver, we show convergence up to a Weissenberg number We of 20 for the Oldroyd‐B model. No limiting Weissenberg number is observed even though a Phan Thien–Tanner model is used. Several numerical results are presented. Smooth and stable solutions are obtained for high Weissenberg number. Copyright © 2002 John Wiley & Sons, Ltd.