Bayesian inference of an uncertain generalized diffusion operator

Abstract

This paper defines a novel Bayesian inverse problem to infer an infinite-dimensional uncertain operator appearing in a differential equation, whose action on an observable state variable affects its dynamics. The operator is parametrized using its eigendecomposition, which enables prior information to be incorporated into its formulation. The Bayesian inverse problem is defined in terms of an uncertain, generalized diffusion operator appearing in an evolution equation for a contaminant's transport through a heterogeneous porous medium. Limited observations of the state variable's evolution are used as data for inference, and the dependence on the solution of the inverse problem is studied as a function of the frequency of observations, as well as on whether or not the data is collected as a spatial or time series.

Convergence Rates for Learning Linear Operators from Noisy Data

“…Although quite strong, these assumptions may be realized in practice when, for example, prior knowledge is available that the covariance commutes with the target (hence simultaneously diagonalizable in the same eigenbasis) or that the target obeys certain physical principles (e.g., commutes with translation operators). This allows us to work in coordinates with respect to the eigenbasis and hence reduce inference of the operator to that of its eigenvalue sequence; similar ideas were recently applied to learn a differential operator arising in an advection-diffusion model [48]. Our proof techniques in this linear diagonal setting follow the program set forth in [35], which is concerned with the Bayesian approach to ill-posed inverse problems in a diagonalized problem setting.…”