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

9704899 LAMM这个项目的主要目标是发展局部正则化方法来解决不适定反问题，例如在逆热传导和图像重建应用中出现的反问题。P.I.发展的局部正则化思想自然导致了一种“预估-校正”式的求解方法。其思想是首先在解域的一小部分上获得正则化解，然后应用随后的校正以避免过度平滑。这种方法已被认为与标准的Tikhonov正则化方法对Volterra问题同样有效，并提高了计算性能。此外，该方案自然地提出了一种“可变正则化”方法，在该方法中，使用函数正则化参数在区域的某些部分应用更多的平滑，而在其他部分应用较少的平滑。在这个项目中，P.I.计划继续发展现有的Volterra问题理论，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。设计一种理论上合理的自适应功能参数选择过程是本项目的重点和难点。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、卫星遥感遥感图像以及地震和油藏工程等地球物理应用。这些问题对科学家来说具有相当大的挑战性，因为它们经常导致非常大的计算问题，而这些问题对数据中的错误极其敏感。这个项目的目的是开发新的方法来稳定和有效地解决这类反问题。这一目标的成功实现将对生物技术、遥感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.