Triticum urartu (diploid, AA) is the progenitor of the A subgenome of tetraploid (Triticum turgidum, AABB) and hexaploid (Triticum aestivum, AABBDD) wheat. Genomic studies of T. urartu have been useful for investigating the structure, function and evolution of polyploid wheat genomes. Here we report the generation of a high-quality genome sequence of T. urartu by combining bacterial artificial chromosome (BAC)-by-BAC sequencing, single molecule real-time whole-genome shotgun sequencing , linked reads and optical mapping. We assembled seven chromosome-scale pseudomolecules and identified protein-coding genes, and we suggest a model for the evolution of T. urartu chromosomes. Comparative analyses with genomes of other grasses showed gene loss and amplification in the numbers of transposable elements in the T. urartu genome. Population genomics analysis of 147 T. urartu accessions from across the Fertile Crescent showed clustering of three groups, with differences in altitude and biostress, such as powdery mildew disease. The T. urartu genome assembly provides a valuable resource for studying genetic variation in wheat and related grasses, and promises to facilitate the discovery of genes that could be useful for wheat improvement.

For various applications, it is well-known that a multi-level, in particular twolevel, preconditioned CG (PCG) method is an efficient method for solving large and sparse linear systems with a coefficient matrix that is symmetric positive definite. The corresponding two-level preconditioner combines traditional and projection-type preconditioners to get rid of the effect of both small and large eigenvalues of the coefficient matrix. In the literature, various two-level PCG methods are known, coming from the fields of deflation, domain decomposition and multigrid. Even though these two-level methods differ a lot in their specific components, it can be shown that from an abstract point of view they are closely related to each other. We investigate their equivalences, robustness, spectral and convergence properties, by accounting for their implementation, the effect of roundoff errors and their sensitivity to inexact coarse solves, severe termination criteria and perturbed starting vectors.

We present whole genome profiling (WGP), a novel next-generation sequencing-based physical mapping technology for construction of bacterial artificial chromosome (BAC) contigs of complex genomes, using Arabidopsis thaliana as an example. WGP leverages short read sequences derived from restriction fragments of two-dimensionally pooled BAC clones to generate sequence tags. These sequence tags are assigned to individual BAC clones, followed by assembly of BAC contigs based on shared regions containing identical sequence tags. Following in silico analysis of WGP sequence tags and simulation of a map of Arabidopsis chromosome 4 and maize, a WGP map of Arabidopsis thaliana ecotype Columbia was constructed de novo using a six-genome equivalent BAC library. Validation of the WGP map using the Columbia reference sequence confirmed that 350 BAC contigs (98%) were assembled correctly, spanning 97% of the 102-Mb calculated genome coverage. We demonstrate that WGP maps can also be generated for more complex plant genomes and will serve as excellent scaffolds to anchor genetic linkage maps and integrate whole genome sequence data.

The computation of some entries of a matrix inverse arises in several important applications in practice. This paper presents a probing method for determining the diagonal of the inverse of a sparse matrix in the common situation when its inverse exhibits a decay property, i.e., when many of the entries of the inverse are small. A few simple properties of the inverse suggest a way to determine effective probing vectors based on standard graph theory results. An iterative method is then applied to solve the resulting sequence of linear systems, from which the diagonal of the matrix inverse is extracted. Results of numerical experiments are provided to demonstrate the effectiveness of the probing method.