ABSTRACT
Inverse methods combine observational data with a mathematical model, and there are many reasons for using them. The result of an inversion can be the estimation of parameters in the model, or an "interpolation" of sparse data points. An inverse method can also be used to estimate the errors in the model and data. In addition, if the inversion, or best fit of the model and data, is well conditioned (that is, small changes in our assumptions do not drastically change the results), then we know the observing system with which we collected the data, is efficient.
I will describe the representer method for solving an inverse problem. This method is optimal when the number of unknowns in the model is greater than the number of observational data. This occurs, for example, when the model is a partial differential equation, and the discretization contains many more grids points than there are data values.
The representer method will be illustrated on a problem in physical oceanography. The data are the location of floats in the ocean, over a time period of a few hours. The mathematical model is the shallow water equations (3-D Navier-Stokes simplified to 2-D) in Lagrangian coordinates. This form of the model is used to facilitate the float data, and has not previously been studied. Results will be shown that illustrate that this model can be used to fit simulated float data.