GPU implementation of Krylov solvers for block-tridiagonal eigenvalue problems

The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-32149-3_18In an eigenvalue problem defined by one or two matrices with block-tridiagonal structure, if only a few eigenpairs are required it is interesting to consider iterative methods based on Krylov subspaces, even if matrix blocks are dense. In this context, using the GPU for the associated dense linear algebra may provide high performance. We analyze this in an implementation done in the context of SLEPc, the Scalable Library for Eigenvalue Problem Computations. In the case of a generalized eigenproblem or when interior eigenvalues are computed with shift-and-invert, the main computational kernel is the solution of linear systems with a block-tridiagonal matrix. We explore possible implementations of this operation on the GPU, including a block cyclic reduction algorithm.This work was partially supported by the Spanish Ministry of Economy and Competitiveness under grant TIN2013-41049-P. Alejandro Lamas was supported by the Spanish Ministry of Education, Culture and Sport through grant FPU13-06655.Lamas Daviña, A.; Román Moltó, JE. (2016). GPU implementation of Krylov solvers for block-tridiagonal eigenvalue problems. En Parallel Processing and Applied Mathematics. Springer. 182-191. https://doi.org/10.1007%2F978-3-319-32149-3_18S182191Baghapour, B., Esfahanian, V., Torabzadeh, M., Darian, H.M.: A discontinuous Galerkin method with block cyclic reduction solver for simulating compressible flows on GPUs. Int. J. Comput. Math. 92(1), 110–131 (2014)Bientinesi, P., Igual, F.D., Kressner, D., Petschow, M., Quintana-Ortí, E.S.: Condensed forms for the symmetric eigenvalue problem on multi-threaded architectures. Concur. Comput. Pract. Exp. 23, 694–707 (2011)Haidar, A., Ltaief, H., Dongarra, J.: Toward a high performance tile divide and conquer algorithm for the dense symmetric eigenvalue problem. SIAM J. Sci. Comput. 34(6), C249–C274 (2012)Heller, D.: Some aspects of the cyclic reduction algorithm for block tridiagonal linear systems. SIAM J. Numer. Anal. 13(4), 484–496 (1976)Hernandez, V., Roman, J.E., Vidal, V.: SLEPc: a scalable and flexible toolkit for the solution of eigenvalue problems. ACM Trans. Math. Softw. 31(3), 351–362 (2005)Hirshman, S.P., Perumalla, K.S., Lynch, V.E., Sanchez, R.: BCYCLIC: a parallel block tridiagonal matrix cyclic solver. J. Comput. Phys. 229(18), 6392–6404 (2010)Minden, V., Smith, B., Knepley, M.G.: Preliminary implementation of PETSc using GPUs. In: Yuen, D.A., Wang, L., Chi, X., Johnsson, L., Ge, W., Shi, Y. (eds.) GPU Solutions to Multi-scale Problems in Science and Engineering. Lecture Notes in Earth System Sciences, pp. 131–140. Springer, Heidelberg (2013)NVIDIA: CUBLAS Library V7.0. Technical report, DU-06702-001 $$\_$$ v7.0, NVIDIA Corporation (2015)Park, A.J., Perumalla, K.S.: Efficient heterogeneous execution on large multicore and accelerator platforms: case study using a block tridiagonal solver. J. Parallel and Distrib. Comput. 73(12), 1578–1591 (2013)Reguly, I., Giles, M.: Efficient sparse matrix-vector multiplication on cache-based GPUs. In: Innovative Parallel Computing (InPar), pp. 1–12 (2012)Roman, J.E., Vasconcelos, P.B.: Harnessing GPU power from high-level libraries: eigenvalues of integral operators with SLEPc. In: International Conference on Computational Science. Procedia Computer Science, vol. 18, pp. 2591–2594. Elsevier (2013)Seal, S.K., Perumalla, K.S., Hirshman, S.P.: Revisiting parallel cyclic reduction and parallel prefix-based algorithms for block tridiagonal systems of equations. J. Parallel Distrib. Comput. 73(2), 273–280 (2013)Stewart, G.W.: A Krylov-Schur algorithm for large eigenproblems. SIAM J. Matrix Anal. Appl. 23(3), 601–614 (2001)Tomov, S., Nath, R., Dongarra, J.: Accelerating the reduction to upper Hessenberg, tridiagonal, and bidiagonal forms through hybrid GPU-based computing. Parallel Comput. 36(12), 645–654 (2010)Vomel, C., Tomov, S., Dongarra, J.: Divide and conquer on hybrid GPU-accelerated multicore systems. SIAM J. Sci. Comput. 34(2), C70–C82 (2012)Zhang, Y., Cohen, J., Owens, J.D.: Fast tridiagonal solvers on the GPU. In: Proceedings of the 15th ACM SIGPLAN Symposium on Principles and Practice of Parallel Programming. PPopp 2010, pp. 127–136 (2010

Strong "quantum" chaos in the global ballooning mode spectrum of three-dimensional plasmas

The spectrum of ideal magnetohydrodynamic (MHD) pressure-driven (ballooning) modes in strongly nonaxisymmetric toroidal systems is difficult to analyze numerically owing to the singular nature of ideal MHD caused by lack of an inherent scale length. In this paper, ideal MHD is regularized by using a $k$-space cutoff, making the ray tracing for the WKB ballooning formalism a chaotic Hamiltonian billiard problem. The minimum width of the toroidal Fourier spectrum needed for resolving toroidally localized ballooning modes with a global eigenvalue code is estimated from the Weyl formula. This phase-space-volume estimation method is applied to two stellarator cases.Comment: 4 pages typeset, including 2 figures. Paper accepted for publication in Phys. Rev. Letter

Designs Studies of Low-Aspect Ratio Quasi-Omnigenous Stellarators

Significant progress has been made in the development of new modest-size compact stellarator devices that could test optimization principles for the design of a more attractive reactor. These are 3 and 4 field period low-aspect-ratio quasi-omnigenous (QO) stellarators based on an optimization method that targets improved confinement, stability, ease of coil design, low-aspect-ratio, and low bootstrap current

TBC1D1 Regulates Insulin- and Contraction-Induced Glucose Transport in Mouse Skeletal Muscle

OBJECTIVE: TBC1D1 is a member of the TBC1 Rab-GTPase family of proteins and is highly expressed in skeletal muscle. Insulin and contraction increase TBC1D1 phosphorylation on phospho-Akt substrate motifs (PASs), but the function of TBC1D1 in muscle is not known. Genetic linkage analyses show a TBC1D1 R125W missense variant confers risk for severe obesity in humans. The objective of this study was to determine whether TBC1D1 regulates glucose transport in skeletal muscle. RESEARCH DESIGN AND METHODS: In vivo gene injection and electroporation were used to overexpress wild-type and several mutant TBC1D1 proteins in mouse tibialis anterior muscles, and glucose transport was measured in vivo. RESULTS: Expression of the obesity-associated R125W mutant significantly decreased insulin-stimulated glucose transport in the absence of changes in TBC1D1 PAS phosphorylation. Simultaneous expression of an inactive Rab-GTPase (GAP) domain of TBC1D1 in the R125W mutant reversed this decrease in glucose transport caused by the R125W mutant. Surprisingly, expression of TBC1D1 mutated to Ala on four conserved Akt and/or AMP-activated protein kinase predicted phosphorylation sites (4P) had no effect on insulin-stimulated glucose transport. In contrast, expression of the TBC1D1 4P mutant decreased contraction-stimulated glucose transport, an effect prevented by concomitant disruption of TBC1D1 Rab-GAP activity. There was no effect of the R125W mutation on contraction-stimulated glucose transport. CONCLUSIONS: TBC1D1 regulates both insulin- and contraction-stimulated glucose transport, and this occurs via distinct mechanisms. The R125W mutation of TBC1D1 impairs skeletal muscle glucose transport, which could be a mechanism for the obesity associated with this mutation

Continuity and Change in Howard S. Becker's work: An Interview with Howard S. Becker

Howard S. Becker is one of the foremost sociologists of the second half of the twentieth century. Although he is perhaps best known for research on deviance and his book Outsiders, this constitutes only a very small fraction of his earliest work. This interview looks at some of the continuities and cores of his work over ?fifty years. Becker highlights how his work maintains the same core concerns, although new interests have been added over time. At the core is a concern with 'work' and 'doing things together.' Becker provides many concrete stories from the past and also raises issues about the nature of doing theory and research, how he writes and produces his studies, and the problems attached to the professionalization of sociology. His writing on art and culture can be seen as assuming a major position in his later work, but he does not identify with either postmodernism or cultural studies