Numerical strategies for filtering partially observed stiff stochastic differential equations

Journal of Computational Physics

Volumn 230, Issue 3, 2011, Pages 744-762

  Harlim, John ^a  

^a NORTH CAROLINA STATE UNIVERSITY   (United States)

Author keywords

Data assimilation; Filtering multiscale systems; Heterogeneous Multiscale Methods; Inverse problems; Stiff SDE

Indexed keywords

DIFFERENTIAL EQUATIONS; NUMERICAL METHODS; STOCHASTIC SYSTEMS;

CONDITION; DATA ASSIMILATION; FILTERING MULTISCALE SYSTEM; HETEROGENEOUS MULTISCALE METHOD; MICRO FILTERS; MULTI-SCALE FEATURES; MULTI-SCALE SYSTEM; NUMERICAL STRATEGIES; STIFF SDE; STOCHASTIC DIFFERENTIAL EQUATIONS;

INVERSE PROBLEMS;

EID: 78649325777     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2010.10.016     Document Type: Article

References (47)

1. Majda A.J., Grote M.J. Explicit off-line criteria for stable accurate time filtering of strongly unstable spatially extended systems. Proc. Natl. Acad. Sci. 2007, 104:1124-1129.
2. Harlim J., Majda A.J. Mathematical strategies for filtering complex systems: regularly spaced sparse observations. J. Comput. Phys. 2008, 227(10):5304-5341.
3. Grote M.J., Majda A.J. Stable time filtering of strongly unstable spatially extended systems. Proc. Natl. Acad. Sci. 2006, 103:7548-7553.
4. Anderson B.D., Moore J.B. Optimal Filtering 1979, Prentice-Hall, Englewood Cliffs, NJ.
5. Chui C., Chen G. Kalman Filtering 1999, Springer, New York.
6. Kalman R.E., Bucy R. A new results in linear prediction and filtering theory. Trans. AMSE J. Basic Eng. 1961, 83D:95-108.
7. Gershgorin B., Majda A.J. A nonlinear test model for filtering slow-fast systems. Commun. Math. Sci. 2008, 6(3):611-649.
8. Gershgorin B., Majda A.J. Filtering a nonlinear slow-fast system with strong fast forcing. Commun. Math. Sci. 2010, 8(1):67-92.
9. Gershgorin B., Harlim J., Majda A.J. Test models for improving filtering with model errors through stochastic parameter estimation. J. Comput. Phys. 2010, 229(1):1-31.
10. Gershgorin B., Harlim J., Majda A.J. Improving filtering and prediction of sparsely observed spatially extended turbulent systems with model errors through stochastic parameter estimation. J. Comput. Phys. 2010, 229(1):32-57.
11. Tuckwell H.C., Rodriguez R. Analytical and simulation results for stochastic FitzHugh-Nagumo neurons and neural networks. J. Comput. Neurosci. 1998, 5:91-113.
12. Pavliotis G.A., Stuart A.M. Parameter estimation for multiscale diffusions. J. Stat. Phys. 2007, 127(4):741-781.
13. Kurtz T.G. Semigroups of conditional shifts and approximations of Markov processes. Ann. Prob. 1975, 3:618-642.
14. Papanicolaou G. Some probabilistic problems and methods in singular perturbations. Rocky Mt. J. Math. 1976, 6:653-673.
15. Majda A.J., Timofeyev I., Vanden-Eijnden E. Models for stochastic climate prediction. Proc. Natl. Acad. Sci. 1999, 96:15687-15691.
16. E W., Engquist B. The heterogeneous multi-scale methods. Commun. Math. Sci. 2003, 1:87-133.
17. E W., Engquist B., Li X., Ren W., Vanden-Eijnden E. The heterogeneous multi-scale method: a review. Commun. Comput. Phys 2007, 2:367-450.
18. Anger G. Inverse Problems in Differential Equations 1990, Kluwer Plenum Press, New York.
19. Vanden-Eijnden E. Numerical techniques for multi-scale dynamical systems with stochastic effects. Commun. Math. Sci. 2003, 1(2):385-391.
20. Evensen G. Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J. Geophys. Res. 1994, 99:10143-10162.
21. Anderson J.L. An ensemble adjustment Kalman filter for data assimilation. Mon. Weather Rev. 2001, 129:2884-2903.
22. Bishop C.H., Etherton B., Majumdar S.J. Adaptive sampling with the ensemble transform Kalman filter: Part I: The theoretical aspects. Mon. Weather Rev. 2001, 129:420-436.
23. Tippett M.K., Anderson J.L., Bishop C.H., Hamill T.M., Whitaker J.S. Ensemble square-root filters. Mon. Weather Rev. 2003, 131:1485-1490.
24. Lermusiaux P.F.J. Adaptive modeling, adaptive data assimilation, and adaptive sampling. Physica D 2007, 230:172-196.
25. Sakov P., Oke P.R. Implications of the form of the ensemble transformation in the ensemble square root filters. Mon. Weather Rev. 2008, 136:1042-1053.
26. Yang X., DelSole T. The diffuse ensemble filter. Nonlinear Process. Geophys. 2009, 16:475-486.
27. Majda A.J., Grote M.J. Mathematical test models for superparametrization in anisotropic turbulence. Proc. Natl. Acad. Sci. 2009, 2106(14):5470-5474.
28. Groetsch C.W. Inverse Problems in the Mathematical Sciences 1993, Vieweg.
29. Engl H.W., Groetsch C.W. Optimal parameter choice for ordinary and iterated Tikhonov regularization. Inverse and Ill-Posed Problems 1987, 97-125. Academic Press, New York. H.W. Engl, C.W. Groetsch (Eds.).
30. Tikhonov A.N. Regularization of incorrectly posed problems. Sov. Math. Dokl. 1963, 4:1624-1627.
31. Zupanski M. Maximum likelihood ensemble filter: theoretical aspects. Mon. Weather Rev. 2005, 133:1710-1726.
32. Harlim J., Hunt B.R. A non-Gaussian ensemble filter for assimilating infrequent noisy observations. Tellus A 2007, 59(2):225-237.
33. Bashir O., Wilcox K., Ghattas O., van Bloemen Waanders B., Hill J. Hessian-based model reduction for large-scale systems with initial-condition inputs. Int. J. Numer. Methods Eng. 2007, 73:844-868.
34. Cheng H., Jardak M., Alexe M., Sandhu A. A hybrid approach for estimating error covariances in variational assimilation. Tellus A 2010, 62:288-297.
35. FitzHugh R. Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1961, 1:445-466.
36. Nagumo J.S., Arimoto S., Yoshizawa S. An active pulse transmission line simulating nerve axon. Proc. IRE 1962, 50:2061-2071.
37. Rinzel J. Models in neurobiology. Nonlinear Phenomena in Physics and Biology 1981, 345-367. Plenum Press, New York. R.H. Enns, B.L. Jones, R.M. Miura, S.S. Rangnekar (Eds.).
38. Murray J.D. Mathematical Biology 1989, Springer-Verlag, Berlin Heidelberg.
39. Goldobin D.S., Pikovsky A. Antireliability of noise-driven neurons. Phys. Rev. E 2006, 73:061906.
40. Wang X., Bishop C.H., Julier S.J. Which is better, an ensemble of positive-negative pairs or a centered spherical simplex ensemble?. Mon. Weather Rev. 2004, 132(7):1590-1605.
41. Whitaker J.S., Hamill T.M. Ensemble data assimilation without perturbed observations. Mon. Weather Rev. 2002, 130:1913-1924.
42. Castronovo E., Harlim J., Majda A.J. Mathematical criteria for filtering complex systems: plentiful observations. J. Comput. Phys. 2008, 227(7):3678-3714.
43. Kelley C.T. Iterative Methods for Optimization. Frontiers in Applied Mathematics 1999, vol. 18. SIAM.
44. Hunt B.R., Kalnay E., Kostelich E.J., Ott E., Patil D.J., Sauer T., Szunyogh I., Yorke J.A., Zimin A.V. Four-dimensional ensemble Kalman filtering. Tellus A 2004, 56:273-277.
45. Harlim J., Hunt B.R. Four-dimensional local ensemble transform Kalman filter: numerical experiments with a global circulation model. Tellus A 2007, 59(5):731-748.
46. Katsoulakis M.A., Majda A.J., Sopasakis A. Intermittency, metastability and coarse graining for coupled deterministic-stochastic lattice systems. Nonlinearity 2006, 19(5):1021-1047.
47. Xing Y., Majda A.J., Grabowski W.W. New efficient sparse space-time algorithms for superparameterization on mesoscales. Mon. Weather Rev. 2009, 137:4307-4323.