Accurate gradient computations at interfaces using finite element methods

New finite element methods are proposed for elliptic interface problems in one and two dimensions. The main motivation is not only to get an accurate solution but also an accurate first order derivative at the interface (from each side). The key in 1D is to use the idea from \cite{wheeler1974galerkin}. For 2D interface problems, the idea is to introduce a small tube near the interface and introduce the gradient as part of unknowns, which is similar to a mixed finite element method, except only at the interface. Thus the computational cost is just slightly higher than the standard finite element method. We present rigorous one dimensional analysis, which show second order convergence order for both of the solution and the gradient in 1D. For two dimensional problems, we present numerical results and observe second order convergence for the solution, and super-convergence for the gradient at the interface

Different normalization strategies for microarray gene expression traits affect the heritability estimation

Abstract Several studies have been conducted to assess the influence of genetic variation on genome-wide gene expression profiles measured by the microarray technologies. Due to substantial noise in microarray-based experiments, it has long been recognized that proper normalization is a crucial step to ensure sensitive and reliable downstream analyses. This is especially true when large number of samples were collected and analyzed. In this study, we investigated the impact of different normalization strategies on genome wide linkage analyses, in particular, the estimation of heritability of gene expression traits. We used the Genetics Analysis Workshop 15 Problem 1 data. We found that there are significant differences in the estimated number of genes showing heritability when different normalization strategies were used. RMA (robust multiarray average) and dChip identify 45% and 13% more genes showing heritability than MAS 5.0, respectively. Our study also reveals that a large number of genes show strong "family effect" in their expression levels but no significant heritability. Analysis of their annotation indicates different types of genes were enriched in this group compared to genes showing strong heritability.http://deepblue.lib.umich.edu/bitstream/2027.42/117370/1/12919_2007_Article_2518.pd

BVFB: Training Behavior Verification Mechanism for Secure Blockchain-Based Federated Learning

There are still two problems of the existing methods of defending against poisoning attacks of the blockchain-based federated learning: 1) It is difficult to accurately identify the nodes under attack; 2) The effect of the model is greatly affected when the number of malicious nodes exceeds a half. So, an innovative secure mechanism is proposed for blockchain-based federated learning, which is called the training behavior verification mechanism. The mechanism describes the consistent training behavior rules of nodes by constructing the training behavior model, and distinguishes honest nodes from malicious nodes by comparing the differences in training behavior models on the training behavior verification algorithm. Experiments show that the new mechanism can effectively resist more than half of the label-flipping attacks and backdoor attacks, and has the advantages of higher stability and higher accuracy than methods such as Krum, Trimmed Mean, and Median