Combined Adaptive Multiscale and Level-Set Parameter Estimation

We propose a solution strategy for parameter estimation, where we combine adaptive multiscale estimation (AME) and level-set estimation (LSE). The approach is applied to the nonlinear inverse problem of recovering a coefficient function in a system of differential equations from spatially sparsely distributed measurement data. The specific equations considered in this paper describe two-phase porous-media flow where a coefficient function defining absolute permeability (fluid conductivity) is estimated based on fluid pressure observations in wells. This inverse problem is known to be ill-posed. The spatial variability of the sought coefficient function is unknown and will typically vary within the porous medium. Due to limited information in the available data, mainly coarse-scale features of the existing variability in the coefficient function will be attainable. In AME, one starts out with a single parameter representation of the sought function, whereafter the domain is successively divided into finer rectangular zones, each representing a constant parameter value of the coefficient function. The strong restrictions on the zone geometry may lead to overparameterization. LSE is a method for moving curves and enables adjustments of the zone structure into more general geometries. In order to perform well, LSE requires a reasonable starting point. In this paper, we have developed a methodology to combine AME and LSE, where at each step either refinements or deformation of the zone structure may be conducted, depending on which method promises the better result. The combined approach seems promising with respect to recovering coarse-scale features of the sought coefficient functions with a low number of parameters.