Inverse Problems Light: Numerical Differentiation

involving a nonlinear diffusion coefficient a: IR iR+. Problem (1.1) also serves as a model for the saturation of porous media by liquid flow, in which case a (u) is related to the capillary pressure of the pores. In certain industrial applications a numerical simulation may require solving (1.1) for u. We call this the direct problem. In these simulations it is crucial that a coefficient a(u) be used that is not only qualitatively correct but also reasonably accurate. Unfortunately, tabulated values for a (u) from the literature often provide only a rough guess of the true coefficient; in this case simulations are not likely to be reliable. Consequently, identification of the diffusion coefficient a (u) from experimental data (typically, u(x, t) for some abscissa x E (0, 1) and 0 < t < T) is often the first hurdle to clear. This is the associated inverse problem. A standard method to solve the inverse problem is the output least squares method, which tries to match the given data with simulated quantities using a gradient or Newton type method for updating the diffusion coefficient. Alternatively, one can consider (1.1) as a linear equation for a (u). To set up this equation requires numerical differentiation of the data [6]. This approach is called the equation error method. It must be emphasized that inverse problems are often very ill-conditioned: for example, small changes in a (.) have little effect on the solution u in (1.1), and consequently one cannot expect high resolution reconstructions of a in the presence of measurement errors in u. Indeed, small errors in u may cause large errors in the computed a if they are not taken into account appropriately. Numerical differentiation of the data encompasses many subtleties and pitfalls that a complex (linear) inverse problem can exhibit; yet it is very easy to understand and analyze. For this reason one could say that numerical differentiation itself is an ideal model for inverse problems in a basic numerical analysis course. To support this statement we revisit a well-known algorithm for numerical differentiation of noisy data and present a new error bound for it. The method and the error bound can be interpreted as an instance of one of the most important results in regularization theory for ill-posed problems. Still, our presentation is on a very basic level and requires no prior knowledge besides standard n-dimensional calculus and the notion of cubic splines. Groetsch's book [4] presents other realistic inverse problems on an elementary technical level. Further examples and a rigorous introduction to regularization theory for the computation of stable solutions to these examples can be found in [1].