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

小行星9704899 这个项目的主要目标是开发局部正则化方法， 不适定的反问题的解决方案，如那些出现在 逆热传导和图像重建的应用。 的 私人侦探提出的地方正规化的想法。自然会导致 “预测-校正”型解法。 这个想法是，一个正则化的 解首先在解的域的一小部分上获得，之后 应用随后的校正以避免过度平滑。 这种方法被认为是有效的标准吉洪诺夫 正则化的沃尔泰拉问题，改进的计算 性能 此外，该计划自然提出了一个“变量 正则化”方法，其中函数正则化参数被 用于在域的某些部分应用更多平滑， 他人 在这个项目中，PI。计划继续发展现有的 理论的沃尔泰拉问题，并将这一理论推广到非沃尔泰拉 在图像重建和图像去模糊中出现的问题。 设计了一个理论上合理的自适应过程， 功能参数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），材料的无损热检测和分析，卫星 遥感图像和地球物理应用，如地震和 油藏工程 这些问题对 科学家因为他们经常导致非常大的计算问题 对数据中的错误极其敏感。 该项目的目的是 开发新的方法，稳定和有效的解决方案，这样的逆 问题 成功实现这一目标将对 对生物技术、远程成像和 材料科学

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.