When you have equation d = G(m)+η, are you interested in the m that gave birth to a particular measured d, when G is nonlinear (η is some small error) and m, d, and η are continuous multidimensional fields?

In this paper, Marzouk and Najm try to answer the above famous nonlinear inverse problem. They introduce powerful statistical and approximation methodologies to computationally solve, in an effective way, representations of G, d, and m. They understand how instances of m and d are exactly correlated in their solution methodologies, and not only do they find the searched m for a model problem (G), but they also understand how their solution methodology is interwoven with various parameters of their process, such as P and K--see below.

Using a Bayesian formulation, the authors write p(m|d) ˜ p(d|m) p_m(m) and then, accepting independent additive errors for η, they write for the likelihood function L := p(d|m) = p_η(d-G(m)). Then, they approximate G with a stochastic forward problem whose solution approximates, via Galerkin projection, the deterministic forward problem model over the support of the prior. They do so by forming Wiener polynomial chaos sums of order P. Finally, by expanding the prior p_m(m) in a Karhunen-Loeve sum of K terms, the authors successfully reduce the degrees of freedom of the parametrization of the unknown field. The output of such a Bayesian formulation is not a single value, but instead a probability distribution p(m|d) that summarizes all available information.

Marzouk and Najm endeavor to simultaneously use many methodologies, in order to speed up--by orders of magnitude, as they have shown--the computation of the posterior p(m|d) and thus make nonlinear inverse problems very tractable.

To demonstrate their approach, they successfully estimate inhomogeneous diffusivity fields in a transient diffusion problem, from noisy measurements (at given spatial and temporal points) of the diffusion profile u(x,t) that obeys a diffusion equation with several localized sources temporarily active and with adiabatic boundaries on the unit interval.