Greedy algorithms for high-dimensional non-symmetric linear problems

In this article, we present a family of numerical approaches to solve high-dimensional linear non-symmetric problems. The principle of these methods is to approximate a function which depends on a large number of variates by a sum of tensor product functions, each term of which is iteratively computed via a greedy algorithm. There exists a good theoretical framework for these methods in the case of (linear and nonlinear) symmetric elliptic problems. However, the convergence results are not valid any more as soon as the problems considered are not symmetric. We present here a review of the main algorithms proposed in the literature to circumvent this difficulty, together with some new approaches. The theoretical convergence results and the practical implementation of these algorithms are discussed. Their behaviors are illustrated through some numerical examples.Comment: 57 pages, 9 figure

Convergence of a greedy algorithm for high-dimensional convex nonlinear problems

In this article, we present a greedy algorithm based on a tensor product decomposition, whose aim is to compute the global minimum of a strongly convex energy functional. We prove the convergence of our method provided that the gradient of the energy is Lipschitz on bounded sets. The main interest of this method is that it can be used for high-dimensional nonlinear convex problems. We illustrate this method on a prototypical example for uncertainty propagation on the obstacle problem.Comment: 36 pages, 9 figures, accepted in Mathematical Models and Methods for Applied Science

VE-statin/egfl7 Expression in Endothelial Cells Is Regulated by a Distal Enhancer and a Proximal Promoter under the Direct Control of Erg and GATA-2

Angiogenesis is the process by which new blood vessels arise from existing ones by the budding out of endothelial cell capillaries from the luminal side of blood vessels. Blood vessel formation is essential for organ development during embryogenesis and is associated with several physiological and pathological processes, such as wound healing and tumor development. The VE-statin/egfl7 gene is specifically expressed in endothelial cells during embryonic development and in the adult. We studied here the regulatory mechanisms that control this tissue-specific expression. RT-qPCR analyses showed that the specificity of expression of VE-statin/egfl7 in endothelial cells is not shared with its closest neighbor genes notch1 and agpat2 on the mouse chromosome 2. Chromatin-immunoprecipitation analysis of histone modifications at the VE-statin/egfl7 locus showed that the chromatin is specifically opened in endothelial cells, but not in fibroblasts at the transcription start sites. A 13 kb genomic fragment of promoter was cloned and analyzed by gene reporter assays which showed that two conserved regions are important for the specific expression of VE-statin/egfl7 in endothelial cells; a −8409/−7563 enhancer and the −252/+38 region encompassing the exon-1b transcription start site. The latter contains essential GATA and ETS-binding sites, as assessed by linker-scanning analysis and site-directed mutagenesis. An analysis of expression of the ETS and GATA transcription factors showed that Erg, Fli-1 and GATA-2 are the most highly expressed factors in endothelial cells. Erg and GATA-2 directly control the expression of the endogenous VE-statin/egfl7 while Fli-1 probably exerts an indirect control, as assessed by RNA interference and chromatin immunoprecipitation. This first detailed analysis of the mechanisms that govern the expression of the VE-statin/egfl7 gene in endothelial cells pinpoints the specific importance of ETS and GATA factors in the specific regulation of genes in this cell lineage

Publier La Science - Numéro 19

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