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Zhi Wang
,
Kelvin K. Droegemeier
,
L. White
, and
I. M. Navon

Abstract

The adjoint Newton algorithm (ANA) is based on the first- and second-order adjoint techniques allowing one to obtain the “Newton line search direction” by integrating a “tangent linear model” backward in time (with negative time steps). Moreover, the ANA provides a new technique to find Newton line search direction without using gradient information. The error present in approximating the Hessian (the matrix of second-order derivatives) of the cost function with respect to the control variables in the quasi-Newton-type algorithm is thus completely eliminated, while the storage problem related to storing the Hessian no longer exists since the explicit Hessian is not required in this algorithm. The ANA is applied here, for the first time, in the framework of 4D variational data assimilation to the adiabatic version of the Advanced Regional Prediction System, a three-dimensional, compressible, nonhydrostatic storm-scale model. The purpose is to assess the feasibility and efficiency of the ANA as a large-scale minimization algorithm in the setting of 4D variational data assimilation.

Numerical results using simulated observations indicate that the ANA can efficiently retrieve high quality model initial conditions. It improves upon the efficiency of the usual adjoint method employing the LBFGS algorithm by more than an order of magnitude in terms of both CPU time and number of iterations for test problems presented here. Numerical results also show that the ANA obtains a fast linear convergence rate.

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X. Zou
,
A. Barcilon
,
I. M. Navon
,
J. Whitaker
, and
D. G. Cacuci

Abstract

This paper presents a new methodology for adjoint sensitivity analysis, previously developed in general terms by Cacuci, into a form directly applicable to meteorological problems. This technique is illustrated by examining the sensitivity of a blocking index in a two-layer isentropic model. The index represents a response function for the sensitivity analysis that, unlike previous meteorological applications, is an operator and not a functional, and thus, extends the scope of adjoint sensitivity to general operator-type responses depending on time and/or space.

The sensitivity of the blocking index to perturbations introduced into the model atmosphere, as well as to model parameters, is discussed. The methodology of generalized adjoint sensitivity analysis described in this paper constitutes a prototype for further applications in the atmospheric and/or oceanic sciences.

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S. Zhang
,
X. Zou
,
J. Ahlquist
,
I. M. Navon
, and
J. G. Sela

Abstract

Cost functions formulated in four-dimensional variational data assimilation (4DVAR) are nonsmooth in the presence of discontinuous physical processes (i.e., the presence of “on–off” switches in NWP models). The adjoint model integration produces values of subgradients, instead of gradients, of these cost functions with respect to the model’s control variables at discontinuous points. Minimization of these cost functions using conventional differentiable optimization algorithms may encounter difficulties. In this paper an idealized discontinuous model and an actual shallow convection parameterization are used, both including on–off switches, to illustrate the performances of differentiable and nondifferentiable optimization algorithms. It was found that (i) the differentiable optimization, such as the limited memory quasi-Newton (L-BFGS) algorithm, may still work well for minimizing a nondifferentiable cost function, especially when the changes made in the forecast model at switching points to the model state are not too large; (ii) for a differentiable optimization algorithm to find the true minimum of a nonsmooth cost function, introducing a local smoothing that removes discontinuities may lead to more problems than solutions due to the insertion of artificial stationary points; and (iii) a nondifferentiable optimization algorithm is found to be able to find the true minima in cases where the differentiable minimization failed. For the case of strong smoothing, differentiable minimization performance is much improved, as compared to the weak smoothing cases.

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