For a nonswitch time step, FM1 is the same as FM0. Shown in Fig. These two nearly identical numerical solutions will be used as the reference solutions for the verification of the tangent linear solutions in the next section. The numerical solution obtained from FM1 will be used as the observed true state for the cost function defined in section 4.
For the m th time step at which a switch is triggered, GCTLM can be derived by integrating the analytical tangent linear model [see 2. When this approximation is used in the derivation of 4. To the same order of accuracy, 4. GCTLM in 4. Like the GTLM1 operator in 4. The CTLM operator in 4.
As shown in Fig. Note from 5. Thus, including random observational errors in x ob n will not affect the nature of the problem examined in this paper. This is verified by our numerical experiments with imperfect observations not shown , so observational errors are not an issue in this paper.
In association with the cost function in 5. When the gradient computed by CAJM0 is checked by 5. The situation is similar to that in Fig. The gradient computed by GAJM1 can be checked by 5.
By using the above-derived three adjoint models, three minimization methods can be designed. The memoryless quasi-Newton algorithm Liu and Nocedal is used to search for the minimum of the cost function. In this way, the cost function minimum is searched only in the two-dimensional space x 0 , y 0 , so the descending steps can be easily illustrated. For clarity, only the first six iterative steps are shown in Fig.
After the sixth step, the solution is trapped in the vicinity of point 6 and there is no improvement in convergence toward the true minimum. The zigzag patterns in the cost function contours manifest discontinuities caused by noncontinuous dependence of the FM0 solution on the initial state. The gradient computed by CAJM0 follows the step surface locally, but the local geometry can be very different from the global coarse-grain geometry of the cost function. This explains why the solution is trapped in the vicinity of point 6 in Fig.
When the modified discretization is used in FM1, the cost function becomes continuous and smooth as shown in Fig. In this case, the gradients can be correctly computed by GAJM1, and the descending procedure converges to the minimum rapidly. Figure 8 shows that the coarse-grain gradients can be correctly computed by GCAJM, so the descending procedure can converge approximately to the minimum for the FM0-GCAJM method, although the cost function has the same discontinuities as in Fig.
In particular, three different values 0. In the above experiments, the minimization procedures are performed in the two-dimensional space of x 0 , y 0 in order to make clear illustrations see Figs. Additional experiments are also performed with different initial guesses of x 0 and for minimization procedures in the three-dimensional space of the initial state x 0 , y 0 , z 0 of the modified Lorenz model.
The results remain qualitatively the same. The vector equation system of Lorenz is modified and used to study how the generalized adjoint theory and analytical formulations in X96b can be applied to time discrete models. As in XGG98, the analytical model system is discretized in two ways by using the traditional and modified discretization schemes, and the resulting discrete models are denoted by FM0 and FM1, respectively.
Corresponding to FM0 and FM1, three types of discrete tangent linear models are derived: the conventional tangent linear model CTLM0 derived from FM0 by ignoring the perturbation of switch time, the generalized tangent linear model GTLM1 derived from FM1, and the coarse-grain tangent linear model GCTLM derived by directly discretizing the analytical tangent linear equations.
From these tangent linear models, three discrete adjoint models are derived. The results obtained with vector examples in this paper support the principle results summarized in the conclusion section of XGG Vector examples are used in this paper to illustrate problems in the conventional adjoint minimization and to examine how the problem can be solved by the generalized adjoint with modified discretization or by the coarse-grain adjoint without modifying the traditional discretization in the forward model. The results are summarized as follows.
The conventional adjoint can compute the local gradient of the zigzag discontinuous cost function constrained by FM0, but the local gradient can be very different from the global coarse-grain gradient. This can cause the conventional adjoint minimization fail to converge. The above problem can be solved if FM1 is used as the forward model in which the switch time is determined by interpolation as a continuous function of the initial state and the generalized adjoint is used to compute the gradient of the cost function constrained by FM1. Without modifying the traditional discretization, the coarse-grain adjoint model can be used to compute the coarse-grain gradient of the zigzag discontinuous cost function constrained by FM0.
The convergence of the coarse-grain adjoint minimization can be ensured and improved if a small time step is used for the time integrations of the forward model and backward adjoint model. Observational errors are not an issue for the problems examined in this paper, because discontinuities caused by the solution in the cost function are independent of observations. Actually, all the basic findings obtained with perfect observations are confirmed by numerical experiments with imperfect observations not presented in this paper.
Although a complete cost function in data assimilation should include a background term, the nature of the problems examined in this and our previous studies is independent of the neglected background term. As shown by Xu a , section 7 , when the parameterized discontinuity is fitted by a continuous or smooth function of the control variable, the variation of the switch point is implicitly considered by the variation of the control variable, and this makes the switch suitable for the conventional adjoint method.
This type of treatment was used previously by Verlinde and Cotton and Zupanski and Mesinger The method is relatively straightforward but requires that the original threshold condition be modified. The generalized adjoint proposed by Xu b does not change the threshold condition but requires that the traditional discretization be modified only in the forward model. The generalized coarse-grain adjoint does not change the original forward model but requires that the integration time step be sufficiently small.
Each method has certain advantages and disadvantages. Comparisons between these different methods deserve further studies. We are thankful to anonymous reviewers for their comments that improved the presentation of the results. The reference initial state x 0 is the same as in Fig. As in Fig.
The gradient computed at each step numbered sequentially is shown by solid arrow. For clarity, only the first eight steps are shown. Table 1.
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Search for articles by this author. Jidong Gao x. Model equations. Discrete forward models—FM0 and FM1. Discrete adjoint and gradient check. Minimization experiments. Acknowledgments We are thankful to anonymous reviewers for their comments that improved the presentation of the results. View larger version 16K Fig.
View larger version 17K Fig.