ABSTRACT

In a common inverse problem, we wish to infer about an unknown spatial field x = (x1, . . . , xm)T , given indirect observations y = (y1, . . . , yn)T . The observations, or data, are linked to the unknown field x through some physical system

y = ζ(x) + ,

where ζ(x) denotes the physical system and is an n-vector of observation errors. Examples of such problems include medical imaging (Kaipio and Somersalo, 2004), geologic and hydrologic inversion (Stenerud et al., 2008), and cosmology ( Jimenez et al. 2004). When a forward model, or simulator, of the physical process η(x) is available, one can model the data using the simulator

y = η(x) + e,

where e includes observation error as well as error due to the fact that the simulator η(x) may be systematically different from reality ζ(x) for input condition x. Our goal is to use the observed data y to make inference about the spatial input parameters x-predict x and characterize the uncertainty in the prediction for x. The likelihood L(y|x) is then specified to account for both mismatch and sampling error.

We will assume zero-mean Gaussian errors so that