| The problem here is estimating an unknown boundary function in linear PDEs given interior or additional boundary data with error and possibly some partial, prior knowledge of the solution itself. A solution method is presented which is based on the method of Tarantola. This method is a modeling framework based on the procedures and constructs of Bayesian statistical inference. An extension of Tarantola's method is presented in which the space of causal parameters, used to characterize the sought boundary function, is not fixed but rather is adaptively refined. This facilitates a finer balance between, on the one hand, stability of the recovered solution with respect to uncertainties in the forward modeling and prior data information, and on the other hand, the function space proximity of the recovered function to the actual function. Numerical examples are given to illustrate the efficacy of the proposed technique. |
