Partial differential equations are used by scientists and engineers to model many different physical phenomenon. For example the wave equation (as its name suggests) can describe sound, light and water waves: d²u/dt ² = c d²u/dx², where c is the wave speed (wavelength/period) which depends on the type of wave being modeled and the medium through which the wave travels. In addition, u(x,t) is the measure of intensity of the wave at a particular location x and time t. Given the wave speed c, initial and boundary conditions, applied mathematicians are typically concerned with estimating u at some future point in time. On the other hand, geophysicists often send seismic or electromagnetic waves through the Earth's subsurface, measure its intensity u and estimate the wave speed c. If they can accurately determine the wave speed, they have learned something about the composition of the Earth's subsurface. Thus while applied mathematicians are typically concerned with solving the partial differential equation, scientists and engineers are in addition concerned with estimating the parameters in the model.

I will discuss some traditional methods for estimating parameters m in the linear model d=Gm, and introduce an approach I am working on for estimating parameters and their uncertainty when the noise in the data is non-necessarily Gaussian.