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

The graphical presentation of lead isotope data for environmental source apportionment

Lead isotope ratios are widely used to identify original sources of Pb in the environment. Such source apportionment depends on the ability to distinguish potential sources on the basis of their isotopic composition. However, almost all terrestrial Pb is co-linear in some of the plots i.e. <sup>206</sup>Pb/<sup>208</sup>Pb versus <sup>206</sup>Pb/<sup>207</sup>Pb and <sup>206</sup>Pb/<sup>204</sup>Pb versus <sup>206</sup>Pb/<sup>207</sup>Pb commonly presented in the literature. These diagrams are unable to distinguish more than two sources of environmental Pb. Linear trends in such plots are an inevitable consequence of the co-linearity of terrestrial leads and should not be taken necessarily to indicate simple binary mixing of sources. A more reliable test for multiple source mixing can be obtained from plots involving <sup>206</sup>Pb/<sup>204</sup>Pb, <sup>207</sup>Pb/<sup>204</sup>Pb and <sup>208</sup>Pb/<sup>204</sup>Pb and therefore requires measurements of the minor <sup>204</sup>Pb isotope

Adapted Sparse Approximate Inverse Smoothers in Algebraic Multilevel Methods

In this paper an algebraic multilevel method is discussed that mainly focuses on the use of a sparse approximate inverse smoother. In particular strategies are presented to adapt the sparse approximate inverse smoother to a given problem

An efficient GPU version of the preconditioned GMRES method

[EN] In a large number of scientific applications, the solution of sparse linear systems is the stage that concentrates most of the computational effort. This situation has motivated the study and development of several iterative solvers, among which preconditioned Krylov subspace methods occupy a place of privilege. In a previous effort, we developed a GPU-aware version of the GMRES method included in ILUPACK, a package of solvers distinguished by its inverse-based multilevel ILU preconditioner. In this work, we study the performance of our previous proposal and integrate several enhancements in order to mitigate its principal bottlenecks. The numerical evaluation shows that our novel proposal can reach important run-time reductions.Aliaga, JI.; Dufrechou, E.; Ezzatti, P.; Quintana-Orti, ES. (2019). An efficient GPU version of the preconditioned GMRES method. The Journal of Supercomputing. 75(3):1455-1469. https://doi.org/10.1007/s11227-018-2658-1S14551469753Aliaga JI, Badia RM, Barreda M, Bollhöfer M, Dufrechou E, Ezzatti P, Quintana-Ortí ES (2016) Exploiting task and data parallelism in ILUPACK’s preconditioned CG solver on NUMA architectures and many-core accelerators. Parallel Comput 54:97–107Aliaga JI, Bollhöfer M, Dufrechou E, Ezzatti P, Quintana-Ortí ES (2016) A data-parallel ILUPACK for sparse general and symmetric indefinite linear systems. In: Lecture Notes in Computer Science, 14th Int. Workshop on Algorithms, Models and Tools for Parallel Computing on Heterogeneous Platforms—HeteroPar’16. SpringerAliaga JI, Bollhöfer M, Martín AF, Quintana-Ortí ES (2011) Exploiting thread-level parallelism in the iterative solution of sparse linear systems. Parallel Comput 37(3):183–202Aliaga JI, Bollhöfer M, Martín AF, Quintana-Ortí ES (2012) Parallelization of multilevel ILU preconditioners on distributed-memory multiprocessors. Appl Parallel Sci Comput LNCS 7133:162–172Aliaga JI, Dufrechou E, Ezzatti P, Quintana-Ortí ES (2018) Accelerating a preconditioned GMRES method in massively parallel processors. In: CMMSE 2018: Proceedings of the 18th International Conference on Mathematical Methods in Science and Engineering (2018)Bollhöfer M, Grote MJ, Schenk O (2009) Algebraic multilevel preconditioner for the Helmholtz equation in heterogeneous media. SIAM J Sci Comput 31(5):3781–3805Bollhöfer M, Saad Y (2006) Multilevel preconditioners constructed from inverse-based ILUs. SIAM J Sci Comput 27(5):1627–1650Dufrechou E, Ezzatti P (2018) A new GPU algorithm to compute a level set-based analysis for the parallel solution of sparse triangular systems. In: 2018 IEEE International Parallel and Distributed Processing Symposium, IPDPS 2018, Canada, 2018. IEEE Computer SocietyDufrechou E, Ezzatti P (2018) Solving sparse triangular linear systems in modern GPUs: a synchronization-free algorithm. In: 2018 26th Euromicro International Conference on Parallel, Distributed and Network-Based Processing (PDP), pp 196–203. https://doi.org/10.1109/PDP2018.2018.00034Eijkhout V (1992) LAPACK working note 50: distributed sparse data structures for linear algebra operations. Tech. rep., Knoxville, TN, USAGolub GH, Van Loan CF (2013) Matrix computationsHe K, Tan SXD, Zhao H, Liu XX, Wang H, Shi G (2016) Parallel GMRES solver for fast analysis of large linear dynamic systems on GPU platforms. Integration 52:10–22 http://www.sciencedirect.com/science/article/pii/S016792601500084XLiu W, Li A, Hogg JD, Duff IS, Vinter B (2017) Fast synchronization-free algorithms for parallel sparse triangular solves with multiple right-hand sides. Concurr Comput 29(21)Saad Y (2003) Iterative methods for sparse linear systems, 2nd edn. SIAM, PhiladelphiaSchenk O, Wächter A, Weiser M (2008) Inertia revealing preconditioning for large-scale nonconvex constrained optimization. SIAM J Sci Comput 31(2):939–96

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

Analyzing the wave number dependency of the convergence rate of a multigrid preconditioned Krylov method for the Helmholtz equation with an absorbing layer

This paper analyzes the Krylov convergence rate of a Helmholtz problem preconditioned with Multigrid. The multigrid method is applied to the Helmholtz problem formulated on a complex contour and uses GMRES as a smoother substitute at each level. A one-dimensional model is analyzed both in a continuous and discrete way. It is shown that the Krylov convergence rate of the continuous problem is independent of the wave number. The discrete problem, however, can deviate significantly from this bound due to a pitchfork in the spectrum. It is further shown in numerical experiments that the convergence rate of the Krylov method approaches the continuous bound as the grid distance $h$ gets small

Local Fourier Analysis of the Complex Shifted Laplacian preconditioner for Helmholtz problems

In this paper we solve the Helmholtz equation with multigrid preconditioned Krylov subspace methods. The class of Shifted Laplacian preconditioners are known to significantly speed-up Krylov convergence. However, these preconditioners have a parameter beta, a measure of the complex shift. Due to contradictory requirements for the multigrid and Krylov convergence, the choice of this shift parameter can be a bottleneck in applying the method. In this paper, we propose a wavenumber-dependent minimal complex shift parameter which is predicted by a rigorous k-grid Local Fourier Analysis (LFA) of the multigrid scheme. We claim that, given any (regionally constant) wavenumber, this minimal complex shift parameter provides the reader with a parameter choice that leads to efficient Krylov convergence. Numerical experiments in one and two spatial dimensions validate the theoretical results. It appears that the proposed complex shift is both the minimal requirement for a multigrid V-cycle to converge, as well as being near-optimal in terms of Krylov iteration count.Comment: 20 page