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This chapter is devoted to different types of optimal perturbations (OP), deterministic, stochastic, OP in an invariant subspace, and simultaneous stochastic perturbations (SSP). The definitions of OPs are given. It will be shown how the OPs are important for the study on the predictability of behavior of system dynamics, generating ensemble forecasts as well as in the design of a stable filter. A variety of algorithm-based SSP methodology for estimation and decomposition of very high-dimensional (Hd) matrices are presented. Numerical experiments will be presented to illustrate the efficiency and benefice of the perturbation technique.

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On Optimal and Simultaneous Stochastic Perturbations with Application to Estimation of High-Dimensional Matrix and Data Assimilation in High-Dimensional Systems