Parallel Calculation of the Electron Correlation Energy

Calculation of electron correlation energy in molecules involves a very important computational effort, even in the simplest cases. Nowadays, using the new parallel libraries (PETSc and SLEPc) and MPI, we can resolve this calculation faster and with very big molecules. This result is a very important advance in chemical computation.Ramos Peinado, E. (2014). Parallel Calculation of the Electron Correlation Energy. OALib Journal. (1):1-15. doi:10.4236/oalib.1100411S115

Strategies for spectrum slicing based on restarted Lanczos methods

In the context of symmetric-definite generalized eigenvalue problems, it is often required to compute all eigenvalues contained in a prescribed interval. For large-scale problems, the method of choice is the so-called spectrum slicing technique: a shift-and-invert Lanczos method combined with a dynamic shift selection that sweeps the interval in a smart way. This kind of strategies were proposed initially in the context of unrestarted Lanczos methods, back in the 1990's. We propose variations that try to incorporate recent developments in the field of Krylov methods, including thick restarting in the Lanczos solver and a rational Krylov update when moving from one shift to the next. We discuss a parallel implementation in the SLEPc library and provide performance results. © 2012 Springer Science+Business Media, LLC.This work was supported by the Spanish Ministerio de Ciencia e Innovacion under grant TIN2009-07519.Campos González, MC.; Román Moltó, JE. (2012). Strategies for spectrum slicing based on restarted Lanczos methods. Numerical Algorithms. 60(2):279-295. https://doi.org/10.1007/s11075-012-9564-z279295602Amestoy, P.R, Duff, I.S., L’Excellent, J.Y.: Multifrontal parallel distributed symmetric and unsymmetric solvers. Comput. Methods Appl. Mech. Eng. 184(2–4), 501–520 (2000)Balay, S., Brown, J., Buschelman, K., Eijkhout, V., Gropp, W., Kaushik, D., Knepley, M., McInnes, L.C., Smith, B., Zhang, H.: PETSc users manual. Tech. Rep. ANL-95/11 - Revision 3.2, Argonne National Laboratory (2011)Ericsson, T., Ruhe, A.: The spectral transformation Lanczos method for the numerical solution of large sparse generalized symmetric eigenvalue problems. Math. Comput. 35(152), 1251–1268 (1980)Grimes, R.G., Lewis, J.G., Simon, H.D.: A shifted block Lanczos algorithm for solving sparse symmetric generalized eigenproblems. SIAM J. Matrix Anal. Appl. 15(1), 228–272 (1994)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)Hernandez, V., Roman, J.E., Tomas, A.: Parallel Arnoldi eigensolvers with enhanced scalability via global communications rearrangement. Parallel Comput. 33(7–8), 521–540 (2007)Marques, O.A.: BLZPACK: description and user’s guide. Tech. Rep. TR/PA/95/30, CERFACS, Toulouse, France (1995)Meerbergen, K.: Changing poles in the rational Lanczos method for the Hermitian eigenvalue problem. Numer. Linear Algebra Appl. 8(1), 33–52 (2001)Meerbergen, K., Scott, J.: The design of a block rational Lanczos code with partial reorthogonalization and implicit restarting. Tech. Rep. RAL-TR-2000-011, Rutherford Appleton Laboratory (2000)Nour-Omid, B., Parlett, B.N., Ericsson, T., Jensen, P.S.: How to implement the spectral transformation. Math. Comput. 48(178), 663–673 (1987)Olsson, K.H.A., Ruhe, A.: Rational Krylov for eigenvalue computation and model order reduction. BIT Numer. Math. 46, 99–111 (2006)Ruhe, A.: Rational Krylov sequence methods for eigenvalue computation. Linear Algebra Appl. 58, 391–405 (1984)Ruhe, A.: Rational Krylov subspace method. In: Bai, Z., Demmel, J., Dongarra, J., Ruhe, A., van der Vorst, H. (eds.) Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, Society for Industrial and Applied Mathematics, pp. 246–249. Philadelphia (2000)Sorensen, D.C.: Implicit application of polynomial filters in a k-step Arnoldi method. SIAM J. Matrix Anal. Appl. 13, 357–385 (1992)Stewart, G.W.: A Krylov–Schur algorithm for large eigenproblems. SIAM J. Matrix Anal. Appl. 23(3), 601–614 (2001)Vidal, AM., Garcia, V.M., Alonso, P., Bernabeu, M.O.: Parallel computation of the eigenvalues of symmetric Toeplitz matrices through iterative methods. J. Parallel Distrib. Comput. 68(8), 1113–1121 (2008)Wu, K., Simon, H.: Thick-restart Lanczos method for large symmetric eigenvalue problems. SIAM J. Matrix Anal. Appl. 22(2), 602–616 (2000)Zhang, H., Smith, B., Sternberg, M., Zapol, P.: SIPs: Shift-and-invert parallel spectral transformations. ACM Trans. Math. Softw. 33(2), 1–19 (2007

Computing subdominant unstable modes of turbulent plasma with a parallel Jacobi-Davidson eigensolver

In the numerical solution of large-scale eigenvalue problems, Davidson-type methods are an increasingly popular alternative to Krylov eigensolvers. The main motivation is to avoid the expensive factorizations that are often needed by Krylov solvers when the problem is generalized or interior eigenvalues are desired. In Davidson-type methods, the factorization is replaced by iterative linear solvers that can be accelerated by a smart preconditioner. Jacobi-Davidson is one of the most effective variants. However, parallel implementations of this method are not widely available, particularly for non-symmetric problems. We present a parallel implementation that has been included in SLEPc, the Scalable Library for Eigenvalue Problem Computations, and test it in the context of a highly scalable plasma turbulence simulation code. We analyze its parallel efficiency and compare it with a Krylov-Schur eigensolver. © 2011 John Wiley and Sons, Ltd..The authors are indebted to Florian Merz for providing us with the test cases and for his useful suggestions. The authors acknowledge the computer resources provided by the Barcelona Supercomputing Center (BSC). This work was supported by the Spanish Ministerio de Ciencia e Innovacion under project TIN2009-07519.Romero Alcalde, E.; Román Moltó, JE. (2011). Computing subdominant unstable modes of turbulent plasma with a parallel Jacobi-Davidson eigensolver. Concurrency and Computation: Practice and Experience. 23:2179-2191. https://doi.org/10.1002/cpe.1740S2179219123Hochstenbach, M. E., & Notay, Y. (2009). Controlling Inner Iterations in the Jacobi–Davidson Method. SIAM Journal on Matrix Analysis and Applications, 31(2), 460-477. doi:10.1137/080732110Heuveline, V., Philippe, B., & Sadkane, M. (1997). Numerical Algorithms, 16(1), 55-75. doi:10.1023/a:1019126827697Arbenz, P., Bečka, M., Geus, R., Hetmaniuk, U., & Mengotti, T. (2006). On a parallel multilevel preconditioned Maxwell eigensolver. Parallel Computing, 32(2), 157-165. doi:10.1016/j.parco.2005.06.005Genseberger, M. (2010). Improving the parallel performance of a domain decomposition preconditioning technique in the Jacobi–Davidson method for large scale eigenvalue problems. Applied Numerical Mathematics, 60(11), 1083-1099. doi:10.1016/j.apnum.2009.07.004Stathopoulos, A., & McCombs, J. R. (2010). PRIMME. ACM Transactions on Mathematical Software, 37(2), 1-30. doi:10.1145/1731022.1731031Baker, C. G., Hetmaniuk, U. L., Lehoucq, R. B., & Thornquist, H. K. (2009). Anasazi software for the numerical solution of large-scale eigenvalue problems. ACM Transactions on Mathematical Software, 36(3), 1-23. doi:10.1145/1527286.1527287Hernandez, V., Roman, J. E., & Vidal, V. (2005). SLEPc. ACM Transactions on Mathematical Software, 31(3), 351-362. doi:10.1145/1089014.1089019Romero, E., Cruz, M. B., Roman, J. E., & Vasconcelos, P. B. (2011). A Parallel Implementation of the Jacobi-Davidson Eigensolver for Unsymmetric Matrices. High Performance Computing for Computational Science – VECPAR 2010, 380-393. doi:10.1007/978-3-642-19328-6_35Romero, E., & Roman, J. E. (2010). A Parallel Implementation of the Jacobi-Davidson Eigensolver and Its Application in a Plasma Turbulence Code. Lecture Notes in Computer Science, 101-112. doi:10.1007/978-3-642-15291-7_11Über ein leichtes Verfahren die in der Theorie der Säcularstörungen vorkommenden Gleichungen numerisch aufzulösen*). (1846). Journal für die reine und angewandte Mathematik (Crelles Journal), 1846(30), 51-94. doi:10.1515/crll.1846.30.51G. Sleijpen, G. L., & Van der Vorst, H. A. (1996). A Jacobi–Davidson Iteration Method for Linear Eigenvalue Problems. SIAM Journal on Matrix Analysis and Applications, 17(2), 401-425. doi:10.1137/s0895479894270427Fokkema, D. R., Sleijpen, G. L. G., & Van der Vorst, H. A. (1998). Jacobi--Davidson Style QR and QZ Algorithms for the Reduction of Matrix Pencils. SIAM Journal on Scientific Computing, 20(1), 94-125. doi:10.1137/s1064827596300073Morgan, R. B. (1991). Computing interior eigenvalues of large matrices. Linear Algebra and its Applications, 154-156, 289-309. doi:10.1016/0024-3795(91)90381-6Paige, C. C., Parlett, B. N., & van der Vorst, H. A. (1995). Approximate solutions and eigenvalue bounds from Krylov subspaces. Numerical Linear Algebra with Applications, 2(2), 115-133. doi:10.1002/nla.1680020205Stathopoulos, A., Saad, Y., & Wu, K. (1998). Dynamic Thick Restarting of the Davidson, and the Implicitly Restarted Arnoldi Methods. SIAM Journal on Scientific Computing, 19(1), 227-245. doi:10.1137/s1064827596304162Sleijpen, G. L. G., Booten, A. G. L., Fokkema, D. R., & van der Vorst, H. A. (1996). Jacobi-davidson type methods for generalized eigenproblems and polynomial eigenproblems. BIT Numerical Mathematics, 36(3), 595-633. doi:10.1007/bf01731936Balay S Buschelman K Eijkhout V Gropp W Kaushik D Knepley M McInnes LC Smith B Zhang H PETSc users manual 2010Hernandez, V., Roman, J. E., & Tomas, A. (2007). Parallel Arnoldi eigensolvers with enhanced scalability via global communications rearrangement. Parallel Computing, 33(7-8), 521-540. doi:10.1016/j.parco.2007.04.004Dannert, T., & Jenko, F. (2005). Gyrokinetic simulation of collisionless trapped-electron mode turbulence. Physics of Plasmas, 12(7), 072309. doi:10.1063/1.1947447Roman, J. E., Kammerer, M., Merz, F., & Jenko, F. (2010). Fast eigenvalue calculations in a massively parallel plasma turbulence code. Parallel Computing, 36(5-6), 339-358. doi:10.1016/j.parco.2009.12.001Merz, F., & Jenko, F. (2010). Nonlinear interplay of TEM and ITG turbulence and its effect on transport. Nuclear Fusion, 50(5), 054005. doi:10.1088/0029-5515/50/5/054005Simoncini, V., & Szyld, D. B. (2002). Flexible Inner-Outer Krylov Subspace Methods. SIAM Journal on Numerical Analysis, 40(6), 2219-2239. doi:10.1137/s0036142902401074Morgan, R. B. (2002). GMRES with Deflated Restarting. SIAM Journal on Scientific Computing, 24(1), 20-37. doi:10.1137/s106482759936465

Restarted Q-Arnoldi-type methods exploiting symmetry in quadratic eigenvalue problems

The final publication is available at Springer via http://dx.doi.org/ 10.1007/s10543-016-0601-5.We investigate how to adapt the Q-Arnoldi method for the case of symmetric quadratic eigenvalue problems, that is, we are interested in computing a few eigenpairs of with M, C, K symmetric matrices. This problem has no particular structure, in the sense that eigenvalues can be complex or even defective. Still, symmetry of the matrices can be exploited to some extent. For this, we perform a symmetric linearization , where A, B are symmetric matrices but the pair (A, B) is indefinite and hence standard Lanczos methods are not applicable. We implement a symmetric-indefinite Lanczos method and enrich it with a thick-restart technique. This method uses pseudo inner products induced by matrix B for the orthogonalization of vectors (indefinite Gram-Schmidt). The projected problem is also an indefinite matrix pair. The next step is to write a specialized, memory-efficient version that exploits the block structure of A and B, referring only to the original problem matrices M, C, K as in the Q-Arnoldi method. This results in what we have called the Q-Lanczos method. Furthermore, we define a stabilized variant analog of the TOAR method. We show results obtained with parallel implementations in SLEPc.This work was supported by the Spanish Ministry of Economy and Competitiveness under Grant TIN2013-41049-P. Carmen Campos was supported by the Spanish Ministry of Education, Culture and Sport through an FPU Grant with reference AP2012-0608.Campos, C.; Román Moltó, JE. (2016). Restarted Q-Arnoldi-type methods exploiting symmetry in quadratic eigenvalue problems. BIT Numerical Mathematics. 56(4):1213-1236. https://doi.org/10.1007/s10543-016-0601-5S12131236564Bai, Z., Su, Y.: SOAR: a second-order Arnoldi method for the solution of the quadratic eigenvalue problem. SIAM J. Matrix Anal. Appl. 26(3), 640–659 (2005)Bai, Z., Day, D., Ye, Q.: ABLE: an adaptive block Lanczos method for non-Hermitian eigenvalue problems. SIAM J. Matrix Anal. Appl. 20(4), 1060–1082 (1999)Bai, Z., Ericsson, T., Kowalski, T.: Symmetric indefinite Lanczos method. In: Bai, Z., Demmel, J., Dongarra, J., Ruhe, A., van der Vorst, H. (eds.) Templates for the solution of algebraic eigenvalue problems: a practical guide, pp. 249–260. Society for Industrial and Applied Mathematics, Philadelphia (2000)Balay, S., Abhyankar, S., Adams, M., Brown, J., Brune, P., Buschelman, K., Dalcin, L., Eijkhout, V., Gropp, W., Kaushik, D., Knepley, M., McInnes, L.C., Rupp, K., Smith, B., Zampini, S., Zhang, H.: PETSc users manual. Tech. Rep. ANL-95/11 - Revision 3.6, Argonne National Laboratory (2015)Benner, P., Faßbender, H., Stoll, M.: Solving large-scale quadratic eigenvalue problems with Hamiltonian eigenstructure using a structure-preserving Krylov subspace method. Electron. Trans. Numer. Anal. 29, 212–229 (2008)Betcke, T., Higham, N.J., Mehrmann, V., Schröder, C., Tisseur, F.: NLEVP: a collection of nonlinear eigenvalue problems. ACM Trans. Math. Softw. 39(2), 7:1–7:28 (2013)Campos, C., Roman, J.E.: Parallel Krylov solvers for the polynomial eigenvalue problem in SLEPc (2015, submitted)Day, D.: An efficient implementation of the nonsymmetric Lanczos algorithm. SIAM J. Matrix Anal. Appl. 18(3), 566–589 (1997)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)Hernandez, V., Roman, J.E., Tomas, A.: Parallel Arnoldi eigensolvers with enhanced scalability via global communications rearrangement. Parallel Comput. 33(7–8), 521–540 (2007)Jia, Z., Sun, Y.: A refined variant of SHIRA for the skew-Hamiltonian/Hamiltonian (SHH) pencil eigenvalue problem. Taiwan J. Math. 17(1), 259–274 (2013)Kressner, D., Roman, J.E.: Memory-efficient Arnoldi algorithms for linearizations of matrix polynomials in Chebyshev basis. Numer. Linear Algebra Appl. 21(4), 569–588 (2014)Kressner, D., Pandur, M.M., Shao, M.: An indefinite variant of LOBPCG for definite matrix pencils. Numer. Algorithms 66(4), 681–703 (2014)Lancaster, P.: Linearization of regular matrix polynomials. Electron. J. Linear Algebra 17, 21–27 (2008)Lancaster, P., Ye, Q.: Rayleigh-Ritz and Lanczos methods for symmetric matrix pencils. Linear Algebra Appl. 185, 173–201 (1993)Lu, D., Su, Y.: Two-level orthogonal Arnoldi process for the solution of quadratic eigenvalue problems (2012, manuscript)Meerbergen, K.: The Lanczos method with semi-definite inner product. BIT 41(5), 1069–1078 (2001)Meerbergen, K.: The Quadratic Arnoldi method for the solution of the quadratic eigenvalue problem. SIAM J. Matrix Anal. Appl. 30(4), 1463–1482 (2008)Mehrmann, V., Watkins, D.: Structure-preserving methods for computing eigenpairs of large sparse skew-Hamiltonian/Hamiltonian pencils. SIAM J. Sci. Comput. 22(6), 1905–1925 (2001)Parlett, B.N.: The symmetric Eigenvalue problem. Prentice-Hall, Englewood Cliffs (1980) (reissued with revisions by SIAM, Philadelphia)Parlett, B.N., Chen, H.C.: Use of indefinite pencils for computing damped natural modes. Linear Algebra Appl. 140(1), 53–88 (1990)Parlett, B.N., Taylor, D.R., Liu, Z.A.: A look-ahead Lánczos algorithm for unsymmetric matrices. Math. Comput. 44(169), 105–124 (1985)de Samblanx, G., Bultheel, A.: Nested Lanczos: implicitly restarting an unsymmetric Lanczos algorithm. Numer. Algorithms 18(1), 31–50 (1998)Sleijpen, G.L.G., Booten, A.G.L., Fokkema, D.R., van der Vorst, H.A.: Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems. BIT 36(3), 595–633 (1996)Stewart, G.W.: A Krylov-Schur algorithm for large eigenproblems. 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Stable Sparse Orthogonal Factorization of Ill-Conditioned Banded Matrices for Parallel Computing

Sequential and parallel algorithms based on the LU factorization or the QR factorization have been intensely studied and widely used in the problems of computation with large-scale ill-conditioned banded matrices. Great concerns on existing methods include ill-conditioning, sparsity of factor matrices, computational complexity, and scalability. In this dissertation, we study a sparse orthogonal factorization of a banded matrix motivated by parallel computing. Specifically, we develop a process to factorize a banded matrix as a product of a sparse orthogonal matrix and a sparse matrix which can be transformed to an upper triangular matrix by column permutations. We prove that the proposed process requires low complexity, and it is numerically stable, maintaining similar stability results as the modified Gram-Schmidt process. On this basis, we develop a parallel algorithm for the factorization in a distributed computing environment. Through an analysis of its performance, we show that the communication costs reach the theoretical least upper bounds, while its parallel complexity or speedup approaches the optimal bound. For an ill-conditioned banded system, we construct a sequential solver that breaks it down into small-scale underdetermined systems, which are solved by the proposed factorization with high accuracy. We also implement a parallel solver with strategies to treat the memory issue appearing in extra large-scale linear systems of size over one billion. Numerical experiments confirm the theoretical results derived in this thesis, and demonstrate the superior accuracy and scalability of the proposed solvers for ill-conditioned linear systems, comparing to the most commonly used direct solvers

Numerical simulation of a highly underexpanded carbon dioxide jet

The underexpanded jets are present in many processes such as rocket propulsion, mass spectrometry, fuel injection, as well as in the process called rapid expansion of supercritical solutions (RESS). In the RESS process a supercritical solution flows through a capillary nozzle until an expansion chamber where the strong changes in the thermodynamic properties of the solvent are used to encapsulate the solute in very fine particles. The research project was focused on the hydrodynamic modeling of an hypersonic carbon dioxide jet produced in the context of the RESS process. The mathematical modeling of the jet was developed using the set of the compressible Navier-Stokes equations along with the generalized Bender equation of state. This set of PDE was solved using an adaptive discontinuous Galerkin discretization for space and the exponential Rosenbrock-Euler method for the time integration. The numerical solver was implemented in C++ using several libraries such as deal.ii and Sacado-Trilinos

A parallel implementation of Davidson methods for large-scale eigenvalue problems in SLEPc

In the context of large-scale eigenvalue problems, methods of Davidson type such as Jacobi-Davidson can be competitive with respect to other types of algorithms, especially in some particularly difficult situations such as computing interior eigenvalues or when matrix factorization is prohibitive or highly inefficient. However, these types of methods are not generally available in the form of high-quality parallel implementations, especially for the case of non-Hermitian eigenproblems. We present our implementation of various Davidson-type methods in SLEPc, the Scalable Library for Eigenvalue Problem Computations. The solvers incorporate many algorithmic variants for subspace expansion and extraction, and cover a wide range of eigenproblems including standard and generalized, Hermitian and non-Hermitian, with either real or complex arithmetic. We provide performance results on a large battery of test problems.This work was supported by the Spanish Ministerio de Ciencia e Innovacion under project TIN2009-07519. Author's addresses: E. Romero, Institut I3M, Universitat Politecnica de Valencia, Cami de Vera s/n, 46022 Valencia, Spain), and J. E. Roman, Departament de Sistemes Informatics i Computacio, Universitat Politecnica de Valencia, Cami de Vera s/n, 46022 Valencia, Spain; email: [email protected] Alcalde, E.; Román Moltó, JE. (2014). A parallel implementation of Davidson methods for large-scale eigenvalue problems in SLEPc. ACM Transactions on Mathematical Software. 40(2):13:01-13:29. https://doi.org/10.1145/2543696S13:0113:29402P. Arbenz, M. Becka, R. Geus, U. Hetmaniuk, and T. Mengotti. 2006. On a parallel multilevel preconditioned Maxwell eigensolver. Parallel Comput. 32, 2, 157--165.Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, Eds. 2000. 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Romero, and A. Tomas. 2011. SLEPc users manual. Tech. Rep. DSICII/24/02 - Revision 3.2, D. Sistemes Informàtics i Computació, Universitat Politècnica de València. http://www.grycap.upv.es/slepc.T. Dannert and F. Jenko. 2005. Gyrokinetic simulation of collisionless trapped-electronmode turbulence. Phys. Plasmas 12, 7, 072309.E. R. Davidson. 1975. The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices. J. Comput. Phys. 17, 1, 87--94.T. A. Davis and Y. Hu. 2011. The University of Florida Sparse Matrix Collection. ACM Trans. Math. Softw. 38, 1, 1:1--1:25.H. C. Elman, A. Ramage, and D. J. Silvester. 2007. Algorithm 866: IFISS, a Matlab toolbox for modelling incompressible flow. ACM Trans. Math. Softw. 33, 2. Article 14.T. Ericsson and A. Ruhe. 1980. The spectral transformation Lanczos method for the numerical solution of large sparse generalized symmetric eigenvalue problems. Math. Comp. 35, 152, 1251--1268.M. Ferronato, C. Janna, and G. Pini. 2012. Efficient parallel solution to large-size sparse eigenproblems with block FSAI preconditioning. Numer. Linear Algebra Appl. 19, 5, 797--815.D. R. Fokkema, G. L. G. Sleijpen, and H. A. van der Vorst. 1998. Jacobi--Davidson style QR and QZ algorithms for the reduction of matrix pencils. SIAM J. Sci. Comput. 20, 1, 94--125.M. A. Freitag and A. Spence. 2007. Convergence theory for inexact inverse iteration applied to the generalised nonsymmetric eigenproblem. Electron. Trans. Numer. Anal. 28, 40--64.M. Genseberger. 2010. Improving the parallel performance of a domain decomposition preconditioning technique in the Jacobi-Davidson method for large scale eigenvalue problems. App. Numer. Math. 60, 11, 1083--1099.V. Hernandez, J. E. Roman, and A. Tomas. 2007. Parallel Arnoldi eigensolvers with enhanced scalability via global communications rearrangement. Parallel Comput. 33, 7--8, 521--540.V. Hernandez, J. E. Roman, and V. Vidal. 2005. 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Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method. SIAM J. Sci. Comput. 23, 2, 517--541.A. V. Knyazev, M. E. Argentati, I. Lashuk, and E. E. Ovtchinnikov. 2007. Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX) in HYPRE and PETSc. SIAM J. Sci. Comput. 29, 5, 2224--2239.J. Kopal, M. Rozložník, M. Tuma, and A. Smoktunowicz. 2012. Rounding error analysis of orthogonalization with a non-standard inner product. Numer. Math. 52, 4, 1035--1058.D. Kressner. 2006. Block algorithms for reordering standard and generalized Schur forms. ACM Trans. Math. Softw. 32, 4, 521--532.R. B. Lehoucq, D. C. Sorensen, and C. Yang. 1998. ARPACK Users' Guide, Solution of Large-Scale Eigenvalue Problems by Implicitly Restarted Arnoldi Methods. SIAM, Philadelphia, PA.Z. Li, Y. Saad, and M. Sosonkina. 2003. pARMS: a parallel version of the algebraic recursive multilevel solver. Numer. Linear Algebra Appl. 10, 5--6, 485--509.J. R. 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Toward Understanding the Origin of Mass-Independent Fractionation in Sulfur Allotropes and in Ozone

Mysterious isotope effects, found in atmospheric ozone, cannot be explained by the standard mass-dependent statistical model. Similar mass-dependent isotope effects were also uncovered in sulfur deposits older than 2 billion years. In an effort to pinpoint possible reasons of these isotope effects, we build a theoretical description of the recombination reactions in sulfur allotropes and in ozone. No potential energy surface exists for the sulfur allotropes, so electronic structure calculations are also required. Ab initio calculation of two dimensionally reduced (2D and 3D) models of the potential energy surface for the tetrasulfur molecule at CCSD(T)-F12 and MRCI levels of theory are considered here. The 2D model is used to calculate the vibrational states energies up to 2000 cm-1. Normal mode analysis indicates that the two considered modes in S4 represent a significant mixture of conventional bending and stretching motions. Analysis of the bound vibrational state properties in ozone reveals that the ratio between the number of states in asymmetric and symmetric ozone molecules deviates noticeably from the statistical factor of 2, but in different directions for the singly- and doubly-substituted molecules. However, in the upper part of the spectrum both singly- and doubly-substituted species behave in the same way, which can be a factor contributing to the isotope effects in ozone. Rotation-vibration coupling and its implications for the isotope effects have been studied in detail for ozone isotopomers for both bound states and scattering resonances, using uncoupled, partially coupled and fully coupled approached. We found that the effects of rovibrational coupling are minor for low values of J, but become more significant for large values of J. However, these effects are rather uniform for both symmetric and asymmetric ozone isotopomers, therefore we conclude that the Coriolis coupling does not seem to favor the formation of asymmetric ozone molecules and cannot be responsible for symmetry-driven mass-independent fractionation of oxygen isotopes. A general program for calculation of energies and lifetimes of bound rotational-vibrational states and scattering resonances for ABA/AAB-type systems is developed (SpectrumSDT). The data calculated by this program can be useful for spectroscopic analysis and prediction of reaction rates