Multiscale analysis and simulation of waves in strongly heterogeneous media
When a wave propagates through a homogeneous medium it retains its initial shape even over long distances. As it encounters an inhomogeneity, however, the wave interacts with the medium and complicated scattered wave patterns emerge. From that information, it is possible to infer characteristics of the inhomogeneity, such as its shape or density, even when buried deeply inside the medium. Many applications in science, engineering and medicine relie on the particular features of waves propagating through an inhomogeneous medium for remote sensing, such as geophysical imaging, ultrasound, non-destructive testing in material science, mine detection, or the design of meta-materials or photonic crystals. If the variations of the medium occur at a scale much smaller than the size of the domain or the wave length, standard numerical methods become prohibitively expensive due to their need to discretize the entire computational domain down to the smallest scales. Thus we seek heterogeneous multiscale methods (HMM) that permit the simulation of waves propagating through strongly varying heterogenous media, at a computational cost independent of the smaller scales. At later time, as the wave propagates through a strongly heterogeous medium, it develops a secondary wave train of dispersive nature, which is not captured by classical homogenization theory. Therefore we shall devise an HMM scheme for the wave equation in strongly heterogeous media, which applies in more general situations without precomputing the homogenized limit problem. Clearly to explore and discover unknown inhomogeneities deeply buried inside a medium, an efficient forward solver is not sufficient. By comparing the response from the simulation with true measurements, it is possible to iteratively improve upon the initial guess of the medium characteristics and determine the hidden scatterer. Such an inverse medium problem is probably best formulated as a PDE-constrained optimization problem. It is generally ill-posed, contains many (false) local solutions and it is usually significantly more difficult to solve than the forward problem. To overcome these difficulties, we shall devise numerical methods that guarantee superlinear global convergence, include inequality constraints to exclude unphysical false solutions, and handle large ill-conditioned indefinite linear systems.