On large-scale diagonalization techniques for the Anderson model of localization

We propose efficient preconditioning algorithms for an eigenvalue problem arising in quantum physics, namely the computation of a few interior eigenvalues and their associated eigenvectors for large-scale sparse real and symmetric indefinite matrices of the Anderson model of localization. We compare the Lanczos algorithm in the 1987 implementation by Cullum and Willoughby with the shift-and-invert techniques in the implicitly restarted Lanczos method and in the Jacobi–Davidson method. Our preconditioning approaches for the shift-and-invert symmetric indefinite linear system are based on maximum weighted matchings and algebraic multilevel incomplete LDLT factorizations. These techniques can be seen as a complement to the alternative idea of using more complete pivoting techniques for the highly ill-conditioned symmetric indefinite Anderson matrices. We demonstrate the effectiveness and the numerical accuracy of these algorithms. Our numerical examples reveal that recent algebraic multilevel preconditioning solvers can accelerate the computation of a large-scale eigenvalue problem corresponding to the Anderson model of localization by several orders of magnitude

Thorium and uranium isotopes in a manganese nodule from the Peru basin determined by alpha spectrometry and thermal ionization mass spectrometry (TIMS): Are manganese supply and growth related to climate?

Thorium- and uranium isotopes were measured in a diagenetic manganese nodule from the Peru basin applying alpha- and thermal ionization mass spectrometry (TIMS). Alpha-counting of 62 samples was carried out with a depth resolution of 0.4 mm to gain a high-resolution230Thexcess profile. In addition, 17 samples were measured with TIMS to obtain precise isotope concentrations and isotope ratios. We got values of 0.06–0.59 ppb (230Th), 0.43–1.40 ppm (232Th), 0.09–0.49 ppb (234U) and 1.66–8.24 ppm (238U). The uranium activity ratio in the uppermost samples (1–6 mm) and in two further sections in the nodule at 12.5±1.0 mm and 27.3–33.5 mm comes close to the present ocean water value of 1.144±0.004. In two other sections of the nodule, this ratio is significantly higher, probably reflecting incorporation of diagenetic uranium. The upper 25 mm section of the Mn nodule shows a relatively smooth exponential decrease in the230Thexcess concentration (TIMS). The slope of the best fit yields a growth rate of 110 mm/Ma up to 24.5 mm depth. The section from 25 to 30.3 mm depth shows constant230Thexcess concentrations probably due to growth rates even faster than those in the top section of the nodule. From 33 to 50 mm depth, the growth rate is approximately 60 mm/Ma. Two layers in the nodule with distinct laminations (11–15 and 28–33 mm depth) probably formed during the transition from isotopic stage 8 to 7 and in stage 5e, respectively. The Mn/Fe ratio shows higher values during interglacials 5 and 7, and lower ones during glacials 4 and 6. A comparison of our data with data from adjacent sediment cores suggests (a) a variable supply of hydrothermal Mn to sediments and Mn nodules of the Peru basin or (b) suboxic conditions at the water sediment interface during periods with lower Mn/Fe ratios

Whole-organism concentration ratios in wildlife inhabiting Australian uranium mining environments

Wildlife concentration ratios for 226Ra, 210Pb, 210Po and isotopes of Th and U from soil, water, and sediments were evaluated for a range of Australian uranium mining environments. Whole-organism concentration ratios (CRwo-media) were developed for 271 radionuclide-organism pairs within the terrestrial and freshwater wildlife groups. Australian wildlife often has distinct physiological attributes, such as the lower metabolic rates of macropod marsupials as compared with placental mammals. In addition, the Australian CRswo-media originate from tropical and semi-arid climates, rather than from the temperate-dominated climates of Europe and North America from which most (>90%) of internationally available CRwo-media values originate. When compared, the Australian and non-Australian CRs are significantly different for some wildlife categories (e.g. grasses, mammals) but not others (e.g. shrubs). Where differences exist, the Australian values were higher, suggesting that site-, or region-specific CRswo-media should be used in detailed Australian assessments. However, in screening studies, use of the international mean values in the Wildlife Transfer Database (WTD) appears to be appropriate, as long as the values used encompass the Australian 95th percentile values. Gaps in the Australian datasets include a lack of marine parameters, and no CR data are available for freshwater phytoplankton, zooplankton, insects, insect larvae or amphibians; for terrestrial environments, there are no data for amphibians, annelids, ferns, fungi or lichens & bryophytes. The new Australian specific parameters will aide in evaluating remediation plans and ongoing operations at mining and waste sites within Australia. They have also substantially bolstered the body of U- and Th-series CRwo-media data for use internationally

Partitioning Schemes and Non-Integer Box Sizes for the Box-Counting Algorithm in Multifractal Analysis

We compare different partitioning schemes for the box-counting algorithm in the multifractal analysis by computing the singularity spectrum and the distribution of the box probabilities. As model system we use the Anderson model of localization in two and three dimensions. We show that a partitioning scheme which includes unrestricted values of the box size and an average over all box origins leads to smaller error bounds than the standard method using only integer ratios of the linear system size and the box size which was found by Rodriguez et al. (Eur. Phys. J. B 67, 77-82 (2009)) to yield the most reliable results.Comment: 10 pages, 13 figure

Robust Dropping Criteria for F-norm Minimization Based Sparse Approximate Inverse Preconditioning

Dropping tolerance criteria play a central role in Sparse Approximate Inverse preconditioning. Such criteria have received, however, little attention and have been treated heuristically in the following manner: If the size of an entry is below some empirically small positive quantity, then it is set to zero. The meaning of "small" is vague and has not been considered rigorously. It has not been clear how dropping tolerances affect the quality and effectiveness of a preconditioner $M$. In this paper, we focus on the adaptive Power Sparse Approximate Inverse algorithm and establish a mathematical theory on robust selection criteria for dropping tolerances. Using the theory, we derive an adaptive dropping criterion that is used to drop entries of small magnitude dynamically during the setup process of $M$. The proposed criterion enables us to make $M$ both as sparse as possible as well as to be of comparable quality to the potentially denser matrix which is obtained without dropping. As a byproduct, the theory applies to static F-norm minimization based preconditioning procedures, and a similar dropping criterion is given that can be used to sparsify a matrix after it has been computed by a static sparse approximate inverse procedure. In contrast to the adaptive procedure, dropping in the static procedure does not reduce the setup time of the matrix but makes the application of the sparser $M$ for Krylov iterations cheaper. Numerical experiments reported confirm the theory and illustrate the robustness and effectiveness of the dropping criteria.Comment: 27 pages, 2 figure

AxPcoords & parallel AxParafit: statistical co-phylogenetic analyses on thousands of taxa

Background Current tools for Co-phylogenetic analyses are not able to cope with the continuous accumulation of phylogenetic data. The sophisticated statistical test for host-parasite co-phylogenetic analyses implemented in Parafit does not allow it to handle large datasets in reasonable times. The Parafit and DistPCoA programs are the by far most compute-intensive components of the Parafit analysis pipeline. We present AxParafit and AxPcoords (Ax stands for Accelerated) which are highly optimized versions of Parafit and DistPCoA respectively. Results Both programs have been entirely re-written in C. Via optimization of the algorithm and the C code as well as integration of highly tuned BLAS and LAPACK methods AxParafit runs 5–61 times faster than Parafit with a lower memory footprint (up to 35% reduction) while the performance benefit increases with growing dataset size. The MPI-based parallel implementation of AxParafit shows good scalability on up to 128 processors, even on medium-sized datasets. The parallel analysis with AxParafit on 128 CPUs for a medium-sized dataset with an 512 by 512 association matrix is more than 1,200/128 times faster per processor than the sequential Parafit run. AxPcoords is 8–26 times faster than DistPCoA and numerically stable on large datasets. We outline the substantial benefits of using parallel AxParafit by example of a large-scale empirical study on smut fungi and their host plants. To the best of our knowledge, this study represents the largest co-phylogenetic analysis to date. Conclusion The highly efficient AxPcoords and AxParafit programs allow for large-scale co-phylogenetic analyses on several thousands of taxa for the first time. In addition, AxParafit and AxPcoords have been integrated into the easy-to-use CopyCat tool

Long term records of erosional change from marine ferromanganese crusts

Ferromanganese crusts from the Atlantic, Indian and Pacific Oceans record the Nd and Pb isotope compositions of the water masses from which they form as hydrogenous precipitates. The10Be/9Be-calibrated time series for crusts are compared to estimates based on Co-contents, from which the equatorial Pacific crusts studied are inferred to have recorded ca. 60 Ma of Pacific deep water history. Time series of ɛNd show that the oceans have maintained a strong provinciality in Nd isotopic composition, determined by terrigenous inputs, over periods of up to 60 Ma. Superimposed on the distinct basin-specific signatures are variations in Nd and Pb isotope time series which have been particularly marked over the last 5 Ma. It is shown that changes in erosional inputs, particularly associated with Himalayan uplift and the northern hemisphere glaciation have influenced Indian and Atlantic Ocean deep water isotopic compositions respectively. There is no evidence so far for an imprint of the final closure of the Panama Isthmus on the Pb and Nd isotopic composition in either Atlantic or Pacific deep water masses