We present an efficient and robust approach in the finite element framework for numerical solutions that exhibit multiscale behavior, with applications to singularly perturbed convection-diffusion problems. The first type of equation we study is the convection-dominated convection-diffusion equation, with periodic or random coefficients; the second type of equation is an elliptic equation with singularities due to discontinuous coefficients and non-smooth boundaries. In both cases, standard methods for purely hyperbolic or elliptic problems perform poorly due to sharp boundary and internal layers in the solution.
We propose a framework in which the finite element basis functions are designed to capture the local small-scale behavior correctly. When the structure of the layers can be determined locally, we apply the multiscale finite element method, in which we solve the corresponding homogeneous equation on each element to capture the small scale features of the differential operator. We demonstrate the effectiveness of this method by computing the enhanced diffusivity scaling for a passive scalar in the cellular flow. We also carry out the asymptotic error analysis for its convergence rate and perform numerical experiments for verification. For a random flow with nonlocal layer structure, we use a variational principle to gain additional information in our attempt to design asymptotic basis functions. We also apply the same framework for elliptic equations with discontinuous coefficients or non-smooth boundaries. In that case, we construct local basis function near singularities using infinite element method in order to resolve extreme singularity. Numerical results on problems with various singularities confirm the efficiency and accuracy of this approach.
BACKGROUND: One of the important challenges in microarray analysis is to take full advantage of previously accumulated data, both from one's own laboratory and from public repositories. Through a comparative analysis on a variety of datasets, a more comprehensive view of the underlying mechanism or structure can be obtained. However, as we discover in this work, continual changes in genomic sequence annotations and probe design criteria make it difficult to compare gene expression data even from different generations of the same microarray platform. RESULTS: We first describe the extent of discordance between the results derived from two generations of Affymetrix oligonucleotide arrays, as revealed in cluster analysis and in identification of differentially expressed genes. We then propose a method for increasing comparability. The dataset we use consists of a set of 14 human muscle biopsy samples from patients with inflammatory myopathies that were hybridized on both HG-U95Av2 and HG-U133A human arrays. We find that the use of the probe set matching table for comparative analysis provided by Affymetrix produces better results than matching by UniGene or LocusLink identifiers but still remains inadequate. Rescaling of expression values for each gene across samples and data filtering by expression values enhance comparability but only for few specific analyses. As a generic method for improving comparability, we select a subset of probes with overlapping sequence segments in the two array types and recalculate expression values based only on the selected probes. We show that this filtering of probes significantly improves the comparability while retaining a sufficient number of probe sets for further analysis. CONCLUSIONS: Compatibility between high-density oligonucleotide arrays is significantly affected by probe-level sequence information. With a careful filtering of the probes based on their sequence overlaps, data from different generations of microarrays can be combined more effectively.
DNA microarray technology has been widely used to simultaneously determine the expression levels of thousands of genes. A variety of approaches have been used, both in the implementation of this technology and in the analysis of the large amount of expression data. However, several practical issues still have not been resolved in a satisfactory manner, and among the most critical is the lack of agreement in the results obtained in different array platforms. In this study, we present a comparison of several microarray platforms [Affymetrix oligonucleotide arrays, custom complementary DNA (cDNA) arrays, and custom oligo arrays printed with oligonucleotides from three different sources] as well as analysis of various methods used for microarray target preparation and the reference design. The results indicate that the pairwise correlations of expression levels between platforms are relative low overall but that the log ratios of the highly expressed genes are strongly correlated, especially between Affymetrix and cDNA arrays. The microarray measurements were compared with quantitative real-time-polymerase chain reaction (QRT-PCR) results for 23 genes, and the varying degrees of agreement for each platform were characterized. We have also developed and tested a double amplification method which allows the use of smaller amounts of starting material. The added round of amplification produced reproducible results as compared to the arrays hybridized with single round amplified targets. Finally, the reliability of using a universal RNA reference for two-channel microarrays was tested and the results suggest that comparisons of multiple experimental conditions using the same control can be accurate.
We demonstrate that the process of identifying differentially expressed genes in microarray studies with small sample sizes can be substantially improved by extracting information from a large number of datasets accumulated in public databases. The improvement comes from more reliable estimates of gene-specific variances based on other datasets. For a two-group comparison with two arrays in each group, for example, the result of our method was comparable to that of a t-test analysis with five samples in each group or to that of a regularized t-test analysis with three samples in each group. Our results are further improved by weighting the results of our approach with the regularized t-test results in a hybrid method.