Sensitivity of functionals of variational data assimilation problems

The problem of variational data assimilation for a nonlinear evolutionary model is formulated as an optimal control problem to find simultaneously unknown parameters and the initial state of the model. The response function is considered as a functional of the optimal solution found as a result of assimilation. The sensitivity of the functional to observational data is studied. The gradient of the functional with respect to observations is associated with the solution of a nonstandard problem involving a system of direct and adjoint equations. On the basis of the Hessian of the original cost function, the solvability of the nonstandard problem is studied. An algorithm for calculating the gradient of the response function with respect to observational data is formulated and justified.

Sensitivity analysis with respect to observations in variational data assimilation for parameter estimation

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find unknown parameters of the model. The observation data, and hence the optimal solution, may contain uncertainties. A response function is considered as a functional of the optimal solution after assimilation. Based on the second-order adjoint techniques, the sensitivity of the response function to the observation data is studied. The gradient of the response function is related to the solution of a nonstandard problem involving the coupled system of direct and adjoint equations. The nonstandard problem is studied, based on the Hessian of the original cost function. An algorithm to compute the gradient of the response function with respect to observations is presented. A numerical example is given for the variational data assimilation problem related to sea surface temperature for the Baltic Sea thermodynamics model.

SEMIPARAMETRIC ESTIMATION WITH GENERATED COVARIATES

We study a general class of semiparametric estimators when the infinite-dimensional nuisance parameters include a conditional expectation function that has been estimated nonparametrically using generated covariates. Such estimators are used frequently to e.g., estimate nonlinear models with endogenous covariates when identification is achieved using control variable techniques. We study the asymptotic properties of estimators in this class, which is a nonstandard problem due to the presence of generated covariates. We give conditions under which estimators are root-nconsistent and asymptotically normal, derive a general formula for the asymptotic variance, and show how to establish validity of the bootstrap.

On Spatial Evolution of the Solution of a Nonstandard Problem in Linear Thermo-Microstretch Elasticity

An anisotropic and nonhomogeneous compressible linear thermo-microstretch elastic cylinder is subject to zero body loads and heat supply and zero lateral specific boundary conditions. The motion is induced by a time-dependent displacement, microrotation, microstretch, and temperature variation specified pointwise over the base. Further, the motion is constrained such that the displacement, microrotation, microstretch and temperature variation and their derivatives with respect to time at points in the cylinder and at a prescribed time are given in proportion to, but not identical with, their respective initial values. Two different cases for these proportional constants are treated. It is shown that certain integrals of the solution spatially evolve with respect to the axial variable. Conditions are derived that show that the integrals exhibit alternative behavior and in particular for the semi-infinite cylinder that there is either at least exponential growth or at most exponential decay.

DENSITIES AND DYNAMICS

Let M be a closed surface without boundary, possibly a sphere or a torus. A repetitive process determines a large number of points xn distributed more or less uniformly over M. We cannot observe these points, nor do we know what M is. There is a smooth function f:M→R (not given) and we measure the numbers yn≡f(xn). The problem is to say something about the geometry of M from the limited information that we have. We consider also a question concerning the existence of smooth densities of continuous random variables, and then the original problem reduces to a nonstandard problem in density estimation. This is by way of a mathematical result whose proof is given in Secs. 4, 5, and 6, but very little mathematical expertise is required to apply the method in practice. The method is described in Sec. 3 where we also give some examples.

Supplementing the Graphing Curriculum

Most students, both at the secondary school level and the university level, have a great deal of trouble understanding graphs. Even many of the best students seem to have inadequate intuition and understanding when confronted with a nonstandard problem involving interpretation of data represented graphically (see, e.g., Monk [1988]). Students need improved skill in interpretation of graphical data to respond better to information that assails them daily, both outside and inside the classroom. This article presents some suggestions for supplementing the traditional curriculum that serve to help develop the full power of graphs.