An Iterative Approximation to the Sensitivity in Calculus of Variations

Abstract

The sensitivity of the solution of an optimization problem with respect to general parameter perturbation (e.g., sensitivity in calculus of variations) is addressed. First, a total variation of the optimal solution is obtained as a by-product of an iterative minimization. Then a general relation between the sensitivity and total variation is used to approximate the sensitivity in calculus of variation. The concept of total variation itself is very useful for tracing the sources of the cost function (forecast aspect) changes back to the initial conditions. For specific choices of the cost function, the total variation may be used to find the sources of a forecast error, the effect of a particular parameterization routine on the optimal solution, etc. The proposed method for calculation of the sensitivity in calculus of variations is approximate but computationally more efficient than existing methods. Its additional benefit is that the realistic calculations using the sophisticated forecast model with physics and real data are possible to accomplish. As an example, the method is applied to find the source of a 24-h forecast error in initial conditions. For gradient calculations, an adjoint model with partial physics (horizontal diffusion, large-scale precipitation, and cumulus convection) is employed. The forecast model is the full-physics regional NMC's eta model. The results show a benefit of multiple iterations and applicability of the method in realistic meteorological situations.