This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
The investigator and her colleagues propose to develop a new regularization method for ill-posed inverse problems, a method which is an extension of the ideas of the classical "simplified regularization method" (or "Lavrentiev's method") and the newer method of "local regularization". Both of these methods are known to preserve special structures of the inverse problem and lead to fast numerical solution methods, but they are often limited to specialized operator equations (for example, where the operator is nonnegative self-adjoint or of Volterra type). The idea behind the new method is to approximate the composition of the governing (linear or nonlinear) operator and a localized smoothing/averaging operator by a "generalized simple regularization operator" which is the sum of a third operator and a function times the identity operator. Requiring stricter approximation properties than is usually required for simplified regularization (but which is required for local regularization), there is hope that the resulting method improves upon the numerical results usually obtained for simplified regularization. In addition, because the new method allows for approximation of the original operator by a third operator (as mentioned above), again in contrast to the simplified regularization method, there is the potential for the new method to apply to a number of operators which do not satisfy the restrictive assumptions needed for the classical method and for local regularization.
Mathematical inverse problems arise in a wide number of applications, from problems of satellite image reconstruction, biomedical imaging (CT scans, X-rays) and geophysical exploration, to the determination of ozone levels in the atmosphere from measurements taken aboard orbiting spacecraft. The methods proposed by the investigator and her colleagues are applicable to many of these problems. In particular, these ideas play a specific role in the solution of new models for ozone determination under study by the investigator and her mathematical colleagues, as well as proposed systems-level models for attention deficit disorder (ADD) and addiction in adults, models under investigation by the investigator and her clinical colleagues.
