zbMATH — the first resource for mathematics

A multilevel approach for computing the limited-memory Hessian and its inverse in variational data assimilation. (English) Zbl 1348.65074
65F30 Other matrix algorithms (MSC2010)
65F08 Preconditioners for iterative methods
15A09 Theory of matrix inversion and generalized inverses
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
Full Text: DOI
[1] T. Amemiya, Non-linear regression models, in Handbook of Econometrics, North-Holland, Amsterdam, 2002. · Zbl 0582.62054
[2] W. Arnoldi, The principle of minimized iterations in the solution of the matrix eigenvalue problem, Quart. Appl. Math., 9 (1951), pp. 17–29. · Zbl 0042.12801
[3] N. Bakhvalov, On the convergence of a relaxation method with natural constraints on the elliptic operator, USSR Comp. Math. Math. Phys., 6 (1966), pp. 101–113.
[4] A. Borzi and V. Schulz, Multigrid methods for pde optimization, SIAM Rev., 51 (2009), pp. 361–395. · Zbl 1167.35354
[5] A. Brandt, Multi-level adaptive solutions to boundary value problems, Math. Comp., 31 (1977), pp. 333–390. · Zbl 0373.65054
[6] Y. Chen and D. Oliver, Ensemble randomized maximum likelihood method as an iterative ensemble smoother, Math. Geosci., 44 (2012), pp. 1–26.
[7] S. Costiner and S. Ta’asan, Adaptive multigrid techniques for large-scale eigenvalue problems: Solutions of the Schrodinger problem in two and three dimensions, Phys. Rev., 44 (1995), pp. 3704–3717.
[8] P. Courtier, J. Thepaut, and A. Hollingsworth, A strategy for operational implementation of 4D-Var, using an incremental approach, Quart. J. Roy. Meteor. Soc., 120 (1994), pp. 1367–1387.
[9] J. Cullum and R. Willoughby, Lanczos Algorithms for Large Symmetric Eigenvalue Computations: Vol. I: Theory, Classics in Appl. Math., SIAM, Philadelphia, 2002. · Zbl 1013.65033
[10] M. Dashti, K. Law, A. Stuart, and J. Voss, Map estimators and posterior consistency in Bayesian nonparametric inverse problems, Inverse Problems, 29 (2013), 095017. · Zbl 1281.62089
[11] L. Debreu, E. Neveu, E. Simon, F.-X. L. Dimet, and A. Vidard, Multigrid solvers and multigrid preconditioners for the solution of variational data assimilation problems, Quart. J. Roy. Meteor. Soc., 142 (2016), pp. 515–528.
[12] F.-X. L. Dimet and O. Talagrand, Variational algorithms for analysis and assimilation of meteorological observations: Theoretical aspects, Tellus A, 38 (1986), pp. 97–110.
[13] M. Ehrendorfer and J. Tribbia, Optimal prediction of forecast error covariances through singular vectors, J. Atmos. Sci., 54 (1997), pp. 286–313.
[14] E. Epstein, The role of initial uncertainties in prediction, J. Appl. Meteor., 8 (1969), pp. 190–198.
[15] D. Furbish, M. Hussaini, F.-X. L. Dimet, P. Ngnepieba, and Y. Wu, On discretization error and its control in variational data assimilation, Tellus A, 60 (2008), pp. 979–991.
[16] I. Gejadze, F.-X. L. Dimet, and V. Shutyaev, On analysis error covariances in variational data assimilation, SIAM J. Sci. Comput., 30 (2008), pp. 1847–1874. · Zbl 1168.65357
[17] I. Gejadze, F.-X. L. Dimet, and V. Shutyaev, On optimal solution error covariances in variational data assimilation problems, J. Comput. Phys., 229 (2010), pp. 2159–2178. · Zbl 1185.65106
[18] I. Gejadze, F.-X. L. Dimet, and V. Shutyaev, Computation of the analysis error covariance in variational data assimilation problems with nonlinear dynamics, J. Comput. Phys., 230 (2011), pp. 7923–7943. · Zbl 1408.65035
[19] S. Gratton, A. Lawless, and N. Nichols, Approximate Gauss–Newton methods for nonlinear least squares problems, SIAM J. Optim., 18 (2007), pp. 106–132. · Zbl 1138.65046
[20] S. Haben, A. Lawless, and N. Nichols, Conditioning and preconditioning of the variational data assimilation problem, Comput. & Fluids, 46 (2011), pp. 252––256. · Zbl 1433.86007
[21] W. Hackbusch, On the computation of approximate eigenvalues and eigenfunctions of elliptic operators by means of a multigrid method, SIAM J. Numer. Anal., 16 (1979), pp. 201–215. · Zbl 0403.65043
[22] L. Hascoet and V. Pascual, Tapenade 2.1 User’s Guide, Technical Report, Inria Sophia Antipolis, Paris, 2004.
[23] M. Hestenes and E. Stiefel, Methods of conjugate gradients for solving linear systems, J. Res. Nat. Bur. Stand., 49 (1952), pp. 409–436. · Zbl 0048.09901
[24] T. Hwang and I. Parsons, A multigrid method for the generalized symmetric eigenvalue problem: Part I — algorithm and implementation, Internat. J. Numer. Methods Engrg., 35 (1992), pp. 1663–1676. · Zbl 0775.73340
[25] A. Knyazev and K. Neymeyr, Efficient solution of symmetric eigenvalue problems using multigrid preconditioners in the locally optimal block conjugate gradient method, Electron. Trans. Numer. Anal., 15 (2003), pp. 38–55. · Zbl 1031.65126
[26] C. Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Res. Natl. Bur. Stand., 45 (1950), pp. 225–282.
[27] R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, Software Environ. Tools, SIAM, Philadelphia, 1998.
[28] J.-L. Lions, Contrôle Optimal des Systèmes Gouvernés par des Équations aux Dérivées Partielles, Dunod, Paris, 1968.
[29] C. Liu, Q. Xiao, and B. Wang, An ensemble-based four-dimensional variational data assimilation scheme. part i: Technical formulation and preliminary test, Monthly Weather Rev., 136 (2008), pp. 3363–3373.
[30] G. Marchuk, V. Agoshkov, and V. Shutyaev, Adjoint Equations and Perturbation Algorithms in Nonlinear Problems, CRC Press, New York, 1996. · Zbl 0828.47053
[31] I. Mirouze and A. Weaver, Representation of correlation functions in variational assimilation using an implicit diffusion operator, Quart. J. Roy. Meteorol. Soc., 136 (2010), pp. 1421–1443.
[32] M. Moakher, A differential geometric approach to the geometric mean of symmetric positive-definite matrices, SIAM J. Matrix Anal. Appl., 26 (2005), pp. 735–747. · Zbl 1079.47021
[33] L. Nazareth, Recent approaches to solving large residual nonlinear least squares problems, SIAM Rev., 22 (1980), pp. 1–11. · Zbl 0424.65031
[34] B. Neta, F. Giraldo, and I. Navon, Analysis of the Turkel-Zwas scheme for the two-dimensional shallow water equations in spherical coordinates, J. Comput. Phys., 133 (1997), pp. 102–112. · Zbl 0883.76060
[35] S. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing, New York, 1980. · Zbl 0521.76003
[36] F. Rabier and P. Courtier, Four-dimensional assimilation in the presence of baroclinic instability, Quart. J. Roy. Meteorol. Soc., 118 (1992), pp. 649–672.
[37] F. Rabier, H. Järvinen, E. Klinker, J.-F. Mahfouf, and A. Simmons, The ECMWF operational implementation of four-dimensional variational assimilation. I: Experimental results with simplified physics, Quart. J. Roy. Meteorol. Soc., 126 (1992), pp. 1142–1170.
[38] G. Sleijpen and H. V. der Vorst, A Jacobi-Davidson iteration method for linear eigenvalue problems, SIAM J. Matrix Anal. Appl., 17 (1996), pp. 401–425. · Zbl 0860.65023
[39] W. Thacker, The role of the Hessian matrix in fitting models to measurements, J. Geophys. Res., 94 (1989), pp. 6177–6196.
[40] Z. Toth and E. Kalnay, Ensemble forecasting at NMC: The generation of perturbations, Bull. Amer. Meteor. Soc., 74 (1993), pp. 2317–2330.
[41] U. Trottenberg, C. Oosterlee, and A. Schüller, Multigrid, Academic Press, London, 2001.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.