Differentiable Optimization Techniques for the Recovery of Sharp Features of Solutions to Inverse Problems

[9704899 Lamm]该项目的主要目标是开发局部正则化方法来解决不适定逆问题，例如在逆热传导和图像重建应用中出现的问题。由P.I.发展的局部正则化思想自然导致了一种“预测-校正”类型的解决方法。其思想是首先在解的一小部分域上获得正则化解，然后应用随后的校正以避免过平滑。对于Volterra问题，该方法与标准Tikhonov正则化一样有效，并且计算性能有所提高。此外，该方案自然地提出了一种“可变正则化”方法，其中使用函数正则化参数在域的某些部分应用更多平滑，而在其他部分应用较少平滑。在这个项目中，P.I.计划继续发展Volterra问题的现有理论，并将该理论扩展到图像重建和图像去模糊中出现的非Volterra问题。设计一套理论上合理的功能参数选择自适应过程是本课题的重点和难点。逆问题自然出现在许多科学应用中，包括生物医学成像（磁共振成像和CAT扫描），材料的无损热测试和分析，远程图像的卫星传感，以及地震和储层工程等地球物理应用。这些问题对科学家来说是相当大的挑战，因为它们经常导致非常大的计算问题，这些问题对数据中的错误非常敏感。本课题的目的是开发稳定、高效求解此类逆问题的新方法。这一目标的成功实现将对诸如生物技术、远程成像和材料科学等应用产生重要影响。

9704899 Lamm The main goal of this project is to develop local regularization methods for the solution of ill-posed inverse problems such as those arising in applications of inverse heat conduction and image reconstruction. The ideas of local regularization developed by the P.I. naturally lead to a "predictor-corrector" type of solution method. The idea is that a regularized solution is first obtained on a small part of the domain of the solution, after which subsequent correction is applied in order to avoid oversmoothing. This method has been seen to be as effective as standard Tikhonov regularization for Volterra problems, with improved computational performance. In addition, the scheme naturally suggests a "variable regularization" approach in which a functional regularization parameter is used to apply more smoothing in some parts of the domain and less in others. In this project the P.I. plans to continue to develop the existing theory for Volterra problems, and to extend this theory to non-Volterra problems of the type arising in image reconstruction and image deblurring. The design of a theoretically-sound adaptive process of selecting the functional parameter is an essential and difficult part of this project. Inverse problems naturally arise in numerous scientific applications, including biomedical imaging (Magnetic Resonance Imaging and CAT scans), the nondestructive thermal testing and analysis of materials, satellite sensing of remote images, and geophysical applications such as seismic and reservoir engineering. These problems are of considerable challenge to scientists because they often lead to very large computational problems which are extremely sensitive to errors in data. The aim of this project is to develop new methods for the stable and efficient solution of such inverse problems. The successful attainment of this goal would have an important impact on applications such as biotechnology, remote imaging, and materials science.