A new matrix-free algorithm for the large-scale trust-region subproblem

We present a new method for the large-scale trust-region subproblem. The method is matrix-free in the sense that only matrix-vector products are required. We recast the trust-region subproblem as a parameterized eigenvalue problem and compute an optimal value for the parameter. We then nd the solution of the trust-region subproblem from the eigenvectors associated with two of the smallest eigenvalues of the parameterized eigenvalue problem corresponding to the optimal parameter. The new algorithm uses a different interpolating scheme than existing methods and introduces a uni ed iteration that naturally includes the so-called hard case. We show that the new iteration is well defined and convergent at a superlinear rate. We present computational results to illustrate convergence properties and robustness of the method.11361164

Perfectly matched layers in transmission lines

The field distribution at the ports of the transmission line structure is computed by applying Maxwell's equations to the structure and solving an eigenvalue problem. The high dimensional sparse system matrix is complex in the presence of losses and Perfectly Matched Layer. A method is presented which preserves sparseness and delivers only the small number of interesting modes out with the smallest attenuation. The modes are found solving a sequence of eigenvalue problems of modified matrices with the aid of the invert mode of the Arnoldi iteration using shifts. A new strategy is described which allows the application of the method, first developed for microwave structures, to optoelectronic devices

On acceleration of Krylov-subspace-based Newton and Arnoldi iterations for incompressible CFD: replacing time steppers and generation of initial guess

We propose two techniques aimed at improving the convergence rate of steady state and eigenvalue solvers preconditioned by the inverse Stokes operator and realized via time-stepping. First, we suggest a generalization of the Stokes operator so that the resulting preconditioner operator depends on several parameters and whose action preserves zero divergence and boundary conditions. The parameters can be tuned for each problem to speed up the convergence of a Krylov-subspace-based linear algebra solver. This operator can be inverted by the Uzawa-like algorithm, and does not need a time-stepping. Second, we propose to generate an initial guess of steady flow, leading eigenvalue and eigenvector using orthogonal projection on a divergence-free basis satisfying all boundary conditions. The approach, including the two proposed techniques, is illustrated on the solution of the linear stability problem for laterally heated square and cubic cavities

Application of the Jacobi Davidson method for spectral low-rank preconditioning in computational electromagnetics problems

[EN] We consider the numerical solution of linear systems arising from computational electromagnetics applications. For large scale problems the solution is usually obtained iteratively with a Krylov subspace method. It is well known that for ill conditioned problems the convergence of these methods can be very slow or even it may be impossible to obtain a satisfactory solution. To improve the convergence a preconditioner can be used, but in some cases additional strategies are needed. In this work we study the application of spectral lowrank updates (SLRU) to a previously computed sparse approximate inverse preconditioner.The updates are based on the computation of a small subset of the eigenpairs closest to the origin. Thus, the performance of the SLRU technique depends on the method available to compute the eigenpairs of interest. The SLRU method was first used using the IRA s method implemented in ARPACK. In this work we investigate the use of a Jacobi Davidson method, in particular its JDQR variant. The results of the numerical experiments show that the application of the JDQR method to obtain the spectral low-rank updates can be quite competitive compared with the IRA s method.Mas Marí, J.; Cerdán Soriano, JM.; Malla Martínez, N.; Marín Mateos-Aparicio, J. (2015). Application of the Jacobi Davidson method for spectral low-rank preconditioning in computational electromagnetics problems. Journal of the Spanish Society of Applied Mathematics. 67:39-50. doi:10.1007/s40324-014-0025-6S395067Bergamaschi, L., Pini, G., Sartoretto, F.: Computational experience with sequential, and parallel, preconditioned Jacobi–Davidson for large sparse symmetric matrices. J. Comput. Phys. 188(1), 318–331 (2003)Carpentieri, B.: Sparse preconditioners for dense linear systems from electromagnetics applications. PhD thesis, Institut National Polytechnique de Toulouse, CERFACS (2002)Carpentieri, B., Duff, I.S., Giraud, L.: Sparse pattern selection strategies for robust Frobenius-norm minimization preconditioners in electromagnetism. Numer. Linear Algebr. Appl. 7(7–8), 667–685 (2000)Carpentieri, B., Duff, I.S., Giraud, L.: A class of spectral two-level preconditioners. SIAM J. Sci. Comput. 25(2), 749–765 (2003)Carpentieri, B., Duff, I.S., Giraud, L., Magolu monga Made, M.: Sparse symmetric preconditioners for dense linear systems in electromagnetism. Numer. Linear Algebr. Appl. 11(8–9), 753–771 (2004)Carpentieri, B., Duff, I.S., Giraud, L., Sylvand, G.: Combining fast multipole techniques and an approximate inverse preconditioner for large electromagnetism calculations. SIAM J. Sci. Comput. 27(3), 774–792 (2005)Darve, E.: The fast multipole method I: error analysis and asymptotic complexity. SIAM J. Numer. Anal. 38(1), 98–128 (2000)Fokkema, D.R., Sleijpen, G.L., Van der Vorst, H.A.: Jacobi–Davidson style QR and QZ algorithms for the reduction of matrix pencils. SIAM J. Sci. Comput. 20(1), 94–125 (1998)Greengard, L., Rokhlin, V.: A fast algorithm for particle simulations. J. Comput. Phys. 73(3), 325–348 (1987)Grote, M., Huckle, T.: Parallel preconditioning with sparse approximate inverses. SIAM J. Sci. Comput. 18(3), 838–853 (1997)Harrington, R.: Origin and development of the method of moments for field computation. IEEE Antenna Propag. Mag. (1990)Kunz, K.S., Luebbers, R.J.: The finite difference time domain method for electromagnetics. SIAM J. Sci. Comput. 18(3), 838–853 (1997)Maxwell, J.C.: A dynamical theory of the electromagnetic field. Roy. S. Trans. CLV, (1864). Reprinted in Tricker, R. A. R. The Contributions of Faraday and Maxwell to Electrial Science, Pergamon Press (1966)Marín, J., Malla M.: Some experiments preconditioning via spectral low rank updates for electromagnetism applications. In: Proceedings of the international conference on preconditioning techniques for large sparse matrix problems in scientific and industrial applications (Preconditioning 2007), Toulouse (2007)Meijerink, J.A., van der Vorst, H.A.: An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix. Math. Comput. 31, 148–162 (1977)Sorensen, D.C., Lehoucq, R.B., Yang, C.: ARPACK users’ guide: solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods. SIAM, Philadelphia (1998)Rao, S.M., Wilton, D.R., Glisson, A.W.: Electromagnetic scattering by surfaces of arbitrary shape. IEEE Trans. Antenna Propag. 30, 409–418 (1982)Saad, Y.: Iterative methods for sparse linear systems. PWS Publishing Company, Boston (1996)Silvester, P.P., Ferrari, R.L.: Finite elements for electrical engineers. Cambridge University Press, Cambridge (1990)Sleijpen, S.L., van der Vorst, H.A.: A Jacobi–Davidson iteration method for linear eigenvalue problems. SIAM J. Matrix Anal. Appl. 17, 401–425 (1996)van der Vorst, H.A.: Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of non-symmetric linear systems. SIAM J. Sci. Stat. Comput. 12(6), 631–644 (1992

A Jacobi-Davidson type method with a correction equation tailored for integral operators

The final publication is available at Springer via http://dx.doi.org/10.1007/s11075-012-9656-9We propose two iterative numerical methods for eigenvalue computations of large dimensional problems arising from finite approximations of integral operators, and describe their parallel implementation. A matrix representation of the problem on a space of moderate dimension, defined from an infinite dimensional one, is computed along with its eigenpairs. These are taken as initial approximations and iteratively refined, by means of a correction equation based on the reduced resolvent operator and performed on the moderate size space, to enhance their quality. Each refinement step requires the prolongation of the correction equation solution back to a higher dimensional space, defined from the infinite dimensional one. This approach is particularly adapted for the computation of eigenpair approximations of integral operators, where prolongation and restriction matrices can be easily built making a bridge between coarser and finer discretizations. We propose two methods that apply a Jacobi–Davidson like correction: Multipower Defect-Correction (MPDC), which uses a single-vector scheme, if the eigenvalues to refine are simple, and Rayleigh–Ritz Defect-Correction (RRDC), which is based on a projection onto an expanding subspace. Their main advantage lies in the fact that the correction equation is performed on a smaller space while for general solvers it is done on the higher dimensional one. We discuss implementation and parallelization details, using the PETSc and SLEPc packages. Also, numerical results on an astrophysics application, whose mathematical model involves a weakly singular integral operator, are presented.This work was partially supported by European Regional Development Fund through COMPETE, FCT-Fundacao para a Ciencia e a Tecnologia through CMUP-Centro de Matematica da Universidade do Porto and Spanish Ministerio de Ciencia e Innovacion under projects TIN2009-07519 and AIC10-D-000600.Vasconcelos, PB.; D'almeida, FD.; Román Moltó, JE. (2013). A Jacobi-Davidson type method with a correction equation tailored for integral operators. Numerical Algorithms. 64(1):85-103. doi:10.1007/s11075-012-9656-9S85103641Absil, P.A., Mahony, R., Sepulchre, R., Dooren, P.V.: A Grassmann–Rayleigh quotient iteration for computing invariant subspaces. SIAM Rev. 44(1), 57–73 (2002)Ahues, M., Largillier, A., Limaye, B.V.: Spectral Computations with Bounded Operators. Chapman and Hall, Boca Raton (2001)Ahues, M., d’Almeida, F.D., Largillier, A., Titaud, O., Vasconcelos, P.: An L 1 refined projection approximate solution of the radiation transfer equation in stellar atmospheres. J. Comput. Appl. Math. 140(1–2), 13–26 (2002)Ahues, M., d’Almeida, F.D., Largillier, A., Vasconcelos, P.B.: Defect correction for spectral computations for a singular integral operator. Commun. Pure Appl. Anal. 5(2), 241–250 (2006)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, Philadelphia (2000)Balay, S., Buschelman, K., Eijkhout, V., Gropp, W.D., Kaushik, D., Knepley, M., McInnes, L.C., Smith, B.F., Zhang, H.: PETSc Users Manual. Tech. Rep. ANL-95/11 - Revision 3.1, Argonne National Laboratory (2010)Chatelin, F.: Spectral Approximation of Linear Operators. SIAM, Philadelphia (2011)d’Almeida, F.D., Vasconcelos, P.B.: Convergence of multipower defect correction for spectral computations of integral operators. Appl. Math. Comput. 219(4), 1601–1606 (2012)Falgout, R.D., Yang, U.M.: Hypre: A library of high performance preconditioners. In: Sloot, P.M.A., Tan, C.J.K., Dongarra, J., Hoekstra, A.G. (eds.) Computational Science - ICCS 2002, International Conference, Amsterdam, The Netherlands, April 21–24, 2002. Proceedings, Part III, Lecture Notes in Computer Science, vol. 2331, pp. 632–641. Springer (2002)Henson, V.E., Yang, U.M.: BoomerAMG: A parallel algebraic multigrid solver and preconditioner. Appl. Numer. Math. 41(1), 155–177 (2002)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., Vidal, V.: SLEPc Users Manual. Tech. Rep. DSIC-II/24/02 - Revision 3.1, D. Sistemas Informáticos y Computación, Universidad Politécnica de Valencia (2010)Saad, Y.: Iterative methods for sparse linear systems, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia (2003)Simoncini, V., Eldén, L.: Inexact Rayleigh quotient-type methods for eigenvalue computations. BIT 42(1), 159–182 (2002)Sleijpen, G.L.G., van der Vorst, H.A.: A Jacobi–Davidson iteration method for linear eigenvalue problems. SIAM Rev. 42(2), 267–293 (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

Peroxisome Proliferator-Activated Receptor alpha (PPAR alpha) down-regulation in cystic fibrosis lymphocytes

Background: PPARs exhibit anti-inflammatory capacities and are potential modulators of the inflammatory response. We hypothesized that their expression and/or function may be altered in cystic fibrosis (CF), a disorder characterized by an excessive host inflammatory response. Methods: PPARα, β and γ mRNA levels were measured in peripheral blood cells of CF patients and healthy subjects via RT-PCR. PPARα protein expression and subcellular localization was determined via western blot and immunofluorescence, respectively. The activity of PPARα was analyzed by gel shift assay. Results: In lymphocytes, the expression of PPARα mRNA, but not of PPARβ, was reduced (-37%; p < 0.002) in CF patients compared with healthy persons and was therefore further analyzed. A similar reduction of PPARα was observed at protein level (-26%; p < 0.05). The transcription factor was mainly expressed in the cytosol of lymphocytes, with low expression in the nucleus. Moreover, DNA binding activity of the transcription factor was 36% less in lymphocytes of patients (p < 0.01). For PPARα and PPARβ mRNA expression in monocytes and neutrophils, no significant differences were observed between CF patients and healthy persons. In all cells, PPARγ mRNA levels were below the detection limit. Conclusion: Lymphocytes are important regulators of the inflammatory response by releasing cytokines and antibodies. The diminished lymphocytic expression and activity of PPARα may therefore contribute to the inflammatory processes that are observed in CF

Studies on changes of estimated breeding values of U.S. Holstein bulls for final score from the first to second crop of daughters

The purpose of this study was to find ways of reducing changes of sire predicted transmitting ability for type’s final scores (PTATs) from the first to second crop of daughters. The PTATs were estimated from two datasets: D01 (scores recorded up to 2001) and D05 (scores recorded up to 2005). The PTAT changes were calculated as the difference between the evaluations based on D01 and D05. The PTATs were adjusted to a common genetic base of all evaluated cows born in 1995. The single-trait (ST) animal model included the fixed effects of the herd–year–season–classifier, age by year group at classification, stage of lactation at classification, registry status of animals, and additive genetic and permanent environment random effects. Unknown parent groups (UPGs) were defined based on every other birth year starting from 1972. Modifications to the ST model included the usage of a single record per cow, separate UPGs for first and second crop daughters, separate UPGs for sires and dams, and deepened pedigrees for dams with missing phenotypic records. Also, the multiple-trait (MT) model treated records of registered and grade cows as correlated traits. The mean PTAT change, for all of the sires, was close to zero in all of the models analyzed. The estimated mean PTAT change for 145 sires with 40 to 100 first crop and ≥200 second crop daughters was −0.33, −0.20, −0.13, −0.28, and −0.12 with ST, only first records, only last records, updated pedigrees, and allowing separate parent groups (PGs) for sires and dams after updating the pedigrees, respectively. The percentages of sires showing PTAT decline were reduced from 74.5 (with ST) to 57.3 by using only the last records of cows, and to 56.4 by allowing separate UPGs for sires and dams after updating the pedigrees. Though updating of the pedigrees alone was not effective, separate UPGs for sires together with additional pedigree was helpful in reducing the bias

Bee Venom Induces Unfolded Protein Response in A172 Glioblastoma Cell Line

Background: Glioblastoma is a type of brain tumor with poor response to available therapies, and shows high rate of mortality. Despite remarkable advancements in our knowledge about cytogenetic and pathophysiologic features of glioblastoma, current treatment strategies are mainly based on cytotoxic drugs; however, these therapeutic approaches are facing progressive failure because of the resistant nature of glioblastomas. In the recent years, however, promising results have emerged owing to targeted therapies toward molecular pathways within cancerous cells. Unfolded Protein Response (UPR) is a remarkable signaling pathway that triggers both apoptosis and survival pathways within cells, and therefore induces UPR-related apoptotic pathways in cancer cells by ER stress inducers. Objectives: Recently, the role of Bee venom (Bv), which contains powerful bioactive peptides, in inducing UPR-related apoptosis was revealed in cancer cell lines. Nevertheless, currently there are no reports of Bv potential ability in induction of UPR apoptotic routes in glioblastoma. The aim of current study was to evaluate possible role of Bee venome in inducing of UPR pathway within A172 glioblastoma cell line. Materials and Methods: We treated the A172 glioblastoma cell line with different Bv doses, and assessed UPR-related genes expression by real-time Polymerase Chain Reaction (PCR). Results: The IC50 of Bv for the studied cell line was 28 μg/mL. Furthermore, we observed that Bv can induce UPR target genes (Grp94 and Gadd153) over-expression through a dose-dependent mechanism. Conclusions: Our results suggest the potential role of Bv as a therapeutic agent for glioblastomas. Keywords: Glioblastoma; A172 Cell Line; Unfolded Protein Response; Bee Veno