An Adjoint-Free Algorithm for CNOPs via Sampling

Abstract. In this paper, we propose a sampling algorithm based on statistical machine learning to obtain conditional nonlinear optimal perturbation (CNOP), which is different from traditional deterministic optimization methods. The new approach reduces the expensive gradient (first-order) information directly by the objective value (zeroth-order) information and does not use the adjoint technique that requires large amounts of storage and produces instability due to linearization. An intuitive analysis of the sampling algorithm is shown rigorously within the form of a concentration inequality for the approximate gradient. The numerical experiments of a theoretical model, Burgers equation with small viscosity, are implemented to obtain the CNOPs. The performance of standard spatial structures demonstrates that at the cost of losing accuracy, the sample-based method with fewer samples spends time relatively shorter than the adjoint-based method and directly from the definition. Finally, we show that the nonlinear time evolution of the CNOPs obtained by all the algorithms is nearly consistent with the quantity of norm square of perturbations, their difference and relative difference based on the definition method.