zbMATH — the first resource for mathematics

Ensemble-type numerical uncertainty information from single model integrations. (English) Zbl 1349.86011
Summary: We suggest an algorithm that quantifies the discretization error of time-dependent physical quantities of interest (goals) for numerical models of geophysical fluid dynamics. The goal discretization error is estimated using a sum of weighted local discretization errors. The key feature of our algorithm is that these local discretization errors are interpreted as realizations of a random process. The random process is determined by the model and the flow state. From a class of local error random processes we select a suitable specific random process by integrating the model over a short time interval at different resolutions. The weights of the influences of the local discretization errors on the goal are modeled as goal sensitivities, which are calculated via automatic differentiation. The integration of the weighted realizations of local error random processes yields a posterior ensemble of goal approximations from a single run of the numerical model. From the posterior ensemble we derive the uncertainty information of the goal discretization error. This algorithm bypasses the requirement of detailed knowledge about the models discretization to generate numerical error estimates. The algorithm is evaluated for the spherical shallow-water equations. For two standard test cases we successfully estimate the error of regional potential energy, track its evolution, and compare it to standard ensemble techniques. The posterior ensemble shares linear-error-growth properties with ensembles of multiple model integrations when comparably perturbed. The posterior ensemble numerical error estimates are of comparable size as those of a stochastic physics ensemble.

86-08 Computational methods for problems pertaining to geophysics
86A05 Hydrology, hydrography, oceanography
86A10 Meteorology and atmospheric physics
76M12 Finite volume methods applied to problems in fluid mechanics
76M20 Finite difference methods applied to problems in fluid mechanics
chammp; CompAD; NAGWare
Full Text: DOI
[1] Becker, R.; Rannacher, R., A feed-back approach to error control in finite element methods: basic analysis and examples, East-West J. Numer. Math., 4, 237-264, (1996) · Zbl 0868.65076
[2] Becker, R.; Rannacher, R., An optimal control approach to a posteriori error estimation in finite element methods, Acta Numer., 10, 1-102, (2002) · Zbl 1105.65349
[3] Chorin, Alexandre J.; Hald, Ole H., Stochastic tools in mathematics and science, Texts in Applied Mathematics, vol. 58, (2013), Springer · Zbl 1384.60001
[4] Ehrendorfer, M.; Tribbia, J. J., Optimal prediction of forecast error covariances through singular vectors, J. Atmos. Sci., 54, 286-313, (1997)
[5] Giles, M. B.; Pierce, N. A.; Sueli, E., Progress in adjoint error correction for integral functionals, Comput. Vis. Sci., 6, 113-121, (2004) · Zbl 1061.65091
[6] Giles, Michael B.; Sueli, Endre, Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality, Acta Numer., 11, 145-236, (2002) · Zbl 1105.65350
[7] Giorgetta, M.; Hundertmark, T.; Korn, P.; Reich, S.; Restelli, M., Conservative space and time regularizations for the ICON model, (Berichte zur Erdsystemforschung, vol. 67, (2009))
[8] Givon, Dror; Kupferman, Raz; Stuart, Andrew, Extracting macroscopic dynamics: model problems and algorithms, Nonlinearity, 17, R55-R127, (2004) · Zbl 1073.82038
[9] Griewank, A., Evaluating derivatives: principles and techniques of algorithmic differentiation, Front. Math. Appl., vol. 19, (2000), SIAM Philadelphia, PA · Zbl 0958.65028
[10] Johnson, C.; Rannacher, R.; Boman, M., Numerics and hydrodynamic stability: toward error control in computational fluid dynamics, SIAM J. Numer. Anal., 32, 4, 1058-1079, (1995) · Zbl 0833.76063
[11] Laeuter, M.; Handorf, D.; Dethloff, K., Unsteady analytical solutions of the spherical shallow water equations, J. Comput. Phys., 210, 535-553, (2005) · Zbl 1078.86001
[12] Marsaglia, G., Normal (Gaussian) random variables for supercomputers, J. Supercomput., 5, 1, 49-55, (1991) · Zbl 1215.65216
[13] Naumann, U.; Riehme, J., Computing adjoints with the nagware Fortran 95 compiler, (Buecker, M.; etal., Automatic Differentiation: Applications, Theory, and Tools, vol. 50, (2006)), 159-170 · Zbl 1270.65090
[14] Oden, J. T.; Prudhomme, Serge, Estimation of modeling error in computational mechanics, J. Comput. Phys., 182, 496-515, (2002) · Zbl 1053.74049
[15] Pierce, N. A.; Giles, M. B., Adjoint recovery of superconvergent functionals from PDE approximations, SIAM Rev., 42, 2, 247-264, (2000) · Zbl 0948.65119
[16] Rauser, F.; Korn, P.; Marotzke, J., Predicting goal error evolution from near-initial-information: a learning algorithm, J. Comput. Phys., 230, 19, 7284-7299, (2011) · Zbl 1408.76085
[17] Rauser, F.; Riehme, J.; Leppke, K.; Korn, P.; Naumann, U., On the use of discrete adjoints in goal error estimation for shallow water equations, Proc. Comput. Sci., 1, 1, (2010)
[18] Ripodas, P.; Gassmann, A.; Foerstner, J.; Majewski, D.; Giorgetta, M.; Korn, P.; Kornblueh, L.; Wan, H.; Zaengl, G.; Bonaventura, L.; Heinze, T., Icosahedral shallow water model (ICOSWM): results of shallow water test cases and sensitivity to model parameters, Geosci. Model Dev. Discuss., 2, 231-251, (2009)
[19] Shutts, G., A kinetic energy backscatter algorithm for use in ensemble prediction systems, Q. J. R. Meteorol. Soc., 131, 612, 3079-3102, (October 2005)
[20] Williamson, D. L.; Drake, J. B.; Hack, J. J.; Jakob, R.; Swarztrauber, P. N., A standard test set for numerical approximations to the shallow water equations in spherical geometry, J. Comput. Phys., 102, 1, 211-224, (1992) · Zbl 0756.76060
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.