Generalized simple regularization for linear and nonlinear inverse problems

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。研究者和她的同事提出了一种新的不适定逆问题的正则化方法，该方法是经典的“简化正则化方法”（或“Lavrentiev方法”）和更新的“局部正则化”方法的思想的扩展。众所周知，这两种方法都保留了反问题的特殊结构，并导致快速的数值解方法，但它们通常仅限于特定的算子方程（例如，算子是非负自伴随的或Volterra型的）。新方法背后的思想是通过一个“广义简单正则化算子”来近似控制算子（线性或非线性）和局部平滑/平均算子的组合，该算子是第三个算子和一个函数乘以恒等算子的总和。要求比简化正则化通常要求的更严格的近似性质（但这是局部正则化所要求的），因此有希望得到的方法在简化正则化通常获得的数值结果的基础上得到改进。此外，由于新方法允许第三个算子近似原始算子（如上所述），再次与简化的正则化方法相反，新方法有可能应用于许多不满足经典方法和局部正则化所需的限制性假设的算子。数学逆问题出现在许多应用中，从卫星图像重建、生物医学成像（CT扫描、x射线）和地球物理勘探问题，到通过在轨道航天器上进行的测量确定大气中的臭氧水平。研究者和她的同事提出的方法适用于许多这样的问题。特别是，这些想法在研究者和她的数学同事正在研究的臭氧测定新模型的解决方案中发挥了特定的作用，以及研究者和她的临床同事正在研究的成人注意缺陷障碍（ADD）和成瘾的系统级模型。

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.