Examination of numerical results from tangent linear and adjoint of discontinuous nonlinear models

The forward model solution and its functional (e.g., the cost function in 4 DVAR) are discontinuous with respect to the model's control variables if th e model contains discontinuous physical processes that occur during the ass imilation window. In such a case, the tangent linear model (the first-order approximation of a finite perturbation) is unable to represent the sharp j umps of the nonlinear model solution. Also, the first-order approximation p rovided by the adjoint model is unable to represent a finite perturbation o f the cost function when the introduced perturbation in the control variabl es crosses discontinuous points. Using an idealized simple model and the Ar akawa-Schubert cumulus parameterization scheme, the authors examined the be havior of a cost function and its gradient obtained by the adjoint model wi th discontinuous model physics. Numerical results show that a cost function involving discontinuous physical processes is zeroth-order discontinuous, but piecewise differentiable. The maximum possible number of involved disco ntinuity points of a cost function increases exponentially as 2(kn), where k is the total number of thresholds associated with on-off switches, and n is the total number of time steps in the assimilation window. A backward ad joint model integration with the proper forcings added at various time step s, similar to the backward adjoint model integration that provides the grad ient of the cost function at a continuous point, produces a one-sided gradi ent (called a subgradient and denoted as del (s)J) at a discontinuous point . An accuracy check of the gradient shows that the adjoint-calculated gradi ent is computed exactly on either side of a discontinuous surface. While a cost function evaluated using a small interval in the control variable spac e oscillates, the distribution of the gradient calculated at the same resol ution not only shows a rather smooth variation, but also is consistent with the general convexity of the original cost function. The gradients of disc ontinuous cost functions are observed roughly smooth since the adjoint inte gration correctly computes the one-sided gradient at either side of discont inuous surface. This implies that, although (del (s)J)(T)deltax may not app roximate deltaJ = J(x + dx) - J(x) well near the discontinuous surface, the subgradient calculated by the adjoint of discontinuous physics may still p rovide useful information for finding the search directions in a minimizati on procedure. While not eliminating the possible need for the use of a nond ifferentiable optimization algorithm for 4DVAR with discontinuous physics, consistency between the computed gradient by adjoints and the convexity of the cost function may explain why a differentiable limited-memory quasi-New ton algorithm still worked well in many 4DVAR experiments that use a diabat ic assimilation model with discontinuous physics.