Analysis and Computation for Inverse Problems in Differential Equations

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Many objects of physical interest cannot be studied directly. Examples include the following: imaging the interior of the body, the determination of cracks within solid objects, and material parameters such as the conductivity of inaccessible objects. When these problems are translated into mathematical terms they take the form of partial differential equations, the lingua franca of the mathematical sciences. However, since one may have additional unknowns in the model, these introduce unknown parameters in the equations that have to be resolved by means of further measurements. Specific problems addressed in this project include the recovery of the location and shape of interior objects from surface measurements or the determination of obstacles from acoustic or electromagnetic scattering data. In this project the PI deals with the practical aspects of such "inverse problems" from a mathematical and computational perspective. The main challenge is when a unique determination can be made from a given amount of data, but as these inverse problems are characterized by often severe "ill-conditioning", meaning that even when there is only one solution to the problem, two very different objects may produce data sets that are infinitesimally close. This lack of stability aspect makes designing and analyzing algorithms for the efficient numerical recovery of the unknowns extremely challenging. The PI will concentrate on developing extremely fast algorithms designed to detect significant features utilizing only minimal data. The PI also looks at inverse spectral problems, and a classic example of which is to be given the vibrational frequencies of a body and seek to determine its internal construction. Here the body can be a metal beam or a star such as the sun. A central theme of this proposal is the investigation of inverse problems for so-called anomalous diffusion models. Classical diffusion is based on Brownian motion and has its roots in 19th century physics together with Einstein's 1905 random walk model. Here a very localised disturbance spreads with the characteristic shape of a Gaussian and, further, the process is Markovian; at a given time step the state depends only on that at the previous time step. While this serves well for a wide range of models, it fails for those that exhibit a "history" or "memory" effect. This includes many materials that been developed over the last twenty years as well as economic forecasting such as stock and commodity market modeling. It turns out that degree of ill-conditioning in anomolous diffusion inverse problems can be very different from those of the classical case suggesting that indeed fundamental new physics is involved. From a mathematical and computational standpoint this comes at a price; the resulting analysis is considerably more complex and challenging. The project also has a significant educational component in the training of graduate and undergraduate students.