Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

An adjoint-free algorithm for conditional nonlinear optimal perturbations (CNOPs) via sampling (2208.00956v5)

Published 1 Aug 2022 in math.OC, cs.LG, nlin.CD, and physics.ao-ph

Abstract: In this paper, we propose a sampling algorithm based on state-of-the-art statistical machine learning techniques to obtain conditional nonlinear optimal perturbations (CNOPs), which is different from traditional (deterministic) optimization methods.1 Specifically, the traditional approach is unavailable in practice, which requires numerically computing the gradient (first-order information) such that the computation cost is expensive, since it needs a large number of times to run numerical models. However, the sampling approach directly reduces the gradient to the objective function value (zeroth-order information), which also avoids using the adjoint technique that is unusable for many atmosphere and ocean models and requires large amounts of storage. We show an intuitive analysis for the sampling algorithm from the law of large numbers and further present a Chernoff-type concentration inequality to rigorously characterize the degree to which the sample average probabilistically approximates the exact gradient. The experiments are implemented to obtain the CNOPs for two numerical models, the Burgers equation with small viscosity and the Lorenz-96 model. We demonstrate the CNOPs obtained with their spatial patterns, objective values, computation times, and nonlinear error growth. Compared with the performance of the three approaches, all the characters for quantifying the CNOPs are nearly consistent, while the computation time using the sampling approach with fewer samples is much shorter. In other words, the new sampling algorithm shortens the computation time to the utmost at the cost of losing little accuracy.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (55)
  1. SQP methods and their application to numerical optimal control. In Variational calculus, optimal control and applications, pages 207–222. Springer, 1998.
  2. Nonmonotone spectral projected gradient methods on convex sets. SIAM Journal on Optimization, 10(4):1196–1211, 2000.
  3. Optimization methods for large-scale machine learning. SIAM Review, 60(2):223–311, 2018.
  4. R. Buizza and T. N. Palmer. The singular-vector structure of the atmospheric global circulation. J. Atmos. Sci., 52(9):1434–1456, 1995.
  5. A SVD-based ensemble projection algorithm for calculating the conditional nonlinear optimal perturbation. Science China Earth Sciences, 58(3):385–394, 2015.
  6. Projected shadowing-based data assimilation. SIAM Journal on Applied Dynamical Systems, 17(4):2446–2477, 2018.
  7. W. Duan and J. Hu. The initial errors that induce a significant “spring predictability barrier” for El Niño events and their implications for target observation: Results from an earth system model. Climate Dynamics, 46(11):3599–3615, 2016.
  8. Exploring the initial errors that cause a significant “spring predictability barrier” for El Niño events. Journal of Geophysical Research: Oceans, 114(C4), 2009.
  9. E. T. Eady. Long waves and cyclone waves. Tellus, 1(3):33–52, 1949.
  10. B. F. Farrell. The initial growth of disturbances in a baroclinic flow. J. Atmos. Sci., 39(8):1663–1686, 1982.
  11. Generalized stability theory. Part I: Autonomous operators. Journal of Atmospheric Sciences, 53(14):2025–2040, 1996a.
  12. Generalized stability theory. Part II: Nonautonomous operators. Journal of Atmospheric Sciences, 53(14):2041–2053, 1996b.
  13. Application of singular vector analysis to the Kuroshio large meander. Journal of Geophysical Research: Oceans, 113(C7), 2008.
  14. E. Kalnay. Atmospheric modeling, data assimilation and predictability. Cambridge university press, 2003.
  15. R. R. Kerswell. Nonlinear nonmodal stability theory. Annual Review of Fluid Mechanics, 50:319–345, 2018.
  16. C.-C. Lin. The theory of hydrodynamic stability. Cambridge University Press, Cambridge, UK., 1955.
  17. D. C. Liu and J. Nocedal. On the limited memory BFGS method for large scale optimization. Mathematical programming, 45(1):503–528, 1989.
  18. E. N. Lorenz. A study of the predictability of a 28-variable atmospheric model. Tellus, 17(3):321–333, 1965.
  19. E. N. Lorenz. Predictability: A problem partly solved. In Proc. Seminar on predictability, volume 1. Reading, 1996.
  20. Optimal sites for supplementary weather observations: Simulation with a small model. Journal of the Atmospheric Sciences, 55(3):399–414, 1998.
  21. A nonlinear multiscale interaction model for atmospheric blocking: The eddy-blocking matching mechanism. Quarterly Journal of the Royal Meteorological Society, 140(683):1785–1808, 2014.
  22. A nonlinear theory of atmospheric blocking: A potential vorticity gradient view. Journal of the Atmospheric Sciences, 76(8):2399–2427, 2019.
  23. M. Mu. Nonlinear singular vectors and nonlinear singular values. Science in China Series D: Earth Sciences, 43(4):375–385, 2000.
  24. M. Mu and J. Wang. Nonlinear fastest growing perturbation and the first kind of predictability. Science in China Series D: Earth Sciences, 44(12):1128–1139, 2001.
  25. M. Mu and Q. Wang. Applications of nonlinear optimization approach to atmospheric and oceanic sciences. SCIENTIA SINICA Mathematica, 47(10):1207–1222, 2017.
  26. Conditional nonlinear optimal perturbation and its applications. Nonlinear Processes in Geophysics, 10(6):493–501, 2003.
  27. The sensitivity and stability of the ocean’s thermohaline circulation to finite-amplitude perturbations. Journal of Physical Oceanography, 34(10):2305–2315, 2004.
  28. A kind of initial errors related to “spring predictability barrier” for El Niño events in Zebiak-Cane model. Geophysical Research Letters, 34(3), 2007.
  29. A method for identifying the sensitive areas in targeted observations for tropical cyclone prediction: Conditional nonlinear optimal perturbation. Monthly Weather Review, 137(5):1623–1639, 2009.
  30. Optimal perturbation for two-dimensional vortex systems: route to non-axisymmetric state. Journal of Fluid Mechanics, 855:922–952, 2018.
  31. Problem complexity and method efficiency in optimization. 1983.
  32. A local ensemble kalman filter for atmospheric data assimilation. Tellus A: Dynamic Meteorology and Oceanography, 56(5):415–428, 2004.
  33. Structure of characteristic lyapunov vectors in spatiotemporal chaos. Physical Review E, 78(1):016209, 2008.
  34. Using nonlinear transient growth to construct the minimal seed for shear flow turbulence. Physical review letters, 105(15):154502, 2010.
  35. X. Qin and M. Mu. Influence of conditional nonlinear optimal perturbations sensitivity on typhoon track forecasts. Quarterly Journal of the Royal Meteorological Society, 138(662):185–197, 2012.
  36. L. Rayleigh. On the stability, or instability, of certain fluid motions. Proceedings of the London Mathematical Society, 1(1):57–72, 1879.
  37. R. M. Samelson and E. Tziperman. Instability of the chaotic ENSO: The growth-phase predictability barrier. J. Atmos. Sci., 58(23):3613–3625, 2001.
  38. Optimal initial disturbance of atmospheric blocking: A barotropic view. arXiv preprint arXiv:2210.06011, 2022.
  39. A. E. Sterk and D. L. van Kekem. Predictability of extreme waves in the lorenz-96 model near intermittency and quasi-periodicity. Complexity, 2017, 2017.
  40. G. Sun and M. Mu. A new approach to identify the sensitivity and importance of physical parameters combination within numerical models using the Lund–Potsdam–Jena (LPJ) model as an example. Theoretical and Applied Climatology, 128(3):587–601, 2017.
  41. C. J. Thompson. Initial conditions for optimal growth in a coupled ocean–atmosphere model of ENSO. J. Atmos. Sci., 55(4):537–557, 1998.
  42. A. Trevisan and L. Palatella. On the kalman filter error covariance collapse into the unstable subspace. Nonlinear Processes in Geophysics, 18(2):243–250, 2011.
  43. R. Vershynin. High-dimensional probability: An introduction with applications in data science, volume 47. Cambridge university press, 2018.
  44. B. Wang and X. Tan. Conditional nonlinear optimal perturbations: Adjoint-free calculation method and preliminary test. Monthly Weather Review, 138(4):1043–1049, 2010.
  45. Q. Wang and M. Mu. A new application of conditional nonlinear optimal perturbation approach to boundary condition uncertainty. Journal of Geophysical Research: Oceans, 120(12):7979–7996, 2015.
  46. A useful approach to sensitivity and predictability studies in geophysical fluid dynamics: conditional non-linear optimal perturbation. National Science Review, 7(1):214–223, 2020.
  47. Enoi-iau initialization scheme designed for decadal climate prediction system iap-decpres. Journal of Advances in Modeling Earth Systems, 10(2):342–356, 2018.
  48. Predictability of a coupled model of ENSO using singular vector analysis. Part I: Optimal growth in seasonal background and ENSO cycles. Mon. Wea. Rev., 125(9):2043–2056, 1997a.
  49. Predictability of a coupled model of ENSO using singular vector analysis. Part II: Optimal growth and forecast skill. Mon. Wea. Rev., 125(9):2057–2073, 1997b.
  50. A WRF-based tool for forecast sensitivity to the initial perturbation: The conditional nonlinear optimal perturbations versus the first singular vector method and comparison to MM5. Journal of Atmospheric and Oceanic Technology, 34(1):187–206, 2017.
  51. Parallel cooperative co-evolution based particle swarm optimization algorithm for solving conditional nonlinear optimal perturbation. In International Conference on Neural Information Processing, pages 87–95. Springer, 2015.
  52. Optimal excitation of interannual atlantic meridional overturning circulation variability. Journal of Climate, 24(2):413–427, 2011.
  53. A model El Niño–Southern Oscillation. Monthly Weather Review, 115(10):2262–2278, 1987.
  54. Conditional nonlinear optimal perturbations based on the particle swarm optimization and their applications to the predictability problems. Nonlinear Processes in Geophysics, 24(1):101–112, 2017.
  55. Optimal initial excitations of decadal modification of the atlantic meridional overturning circulation under the prescribed heat and freshwater flux boundary conditions. Journal of Physical Oceanography, 46(7):2029–2047, 2016.
Citations (3)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now
X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets