Local Regularization Methods for Ill-Posed Inverse Problems: Fast Algorithms and Adaptive Parameter Selection

局部正则化方法在解决许多不同类型的反问题方面表现出很大的潜力，通常保留了原始问题的特殊结构，并导致非常快速的算法（特别是在沃尔泰拉问题的情况下）。 此外，数值试验表明，局部正则化方法可以很好地解决尖锐的特点，而不必依赖于不可微或非二次优化计划的解决方案。 迄今为止，沃尔泰拉问题的局部正则化的收敛理论仅限于轻度不适定的反问题。 在更严重的不适定问题的情况下，有数值证据表明，某些局部方法可能缺乏稳定性和/或收敛性。 由于局部正则化方法的成本效益，因此一个重要的问题是，是否可以开发这些方法的新的变化，可以证明在一般情况下的稳定性/收敛性。 PI建议开发两种新的局部正则化方法，这些方法在数值测试中表现出希望，并且有希望建立一个通用的稳定性/收敛性理论。 PI还建议开发用于选择局部正则化方法中的可变正则化参数的自适应方案。 可变参数用于在域的某些部分中应用更多平滑而在其他部分中应用更少平滑。 在数值试验中，自适应局部正则化技术已被证明是有效的，在确定的可变正则化参数的同时，解决方案的局部部分被恢复。 由于没有收敛理论存在于目前的这种方法，PI建议研究这种自适应计划，并制定一个理论，这将是有用的自适应参数选择方法的建议。 PI还建议将这些想法扩展到非线性沃尔泰拉问题和线性非沃尔泰拉问题。逆问题广泛存在于许多应用中，包括生物医学成像（CT扫描和X射线），图像重建（从卫星或其他来源）和地球物理勘探问题。 沃尔泰拉类逆问题出现在确定表面温度的航天器，因为它重新进入地球的大气层;此外，沃尔泰拉逆问题出现的遥感问题的模型。 虽然存在用于解决此类问题的经典方法，但经典方法通常非常低效，并且导致过于昂贵的解决方案技术。 经典解算方法的第二个缺点在成像应用中很容易看到，其中重建的图像可能具有模糊的边缘和不够详细的特征。 PI建议通过基于局部正则化思想开发新的求解方法来解决这两个难题。 使用这些新的方法可以导致成本显着降低的解决方案的广泛的一类实际的反问题，提高分辨率的详细功能的解决方案。

Local regularization methods show great promise for the solution of a number of different classes of inverse problems, often retaining the special structure of the original problem as well as leading to very fast algorithms (especially in the case of Volterra problems). In addition, numerical tests show that local regularization methods can work well to resolve sharp features of solutions without having to rely on nondifferentiable or nonquadratic optimization schemes. To date, the convergence theory for the local regularization of Volterra problems has been limited to only mildly ill-posed inverse problems. In the case of more severely ill-posed problems, there is numerical evidence that certain local methods may suffer from lack of stability and/or convergence. Because of the cost efficiencies of local regularization methods, an important question is therefore whether new variations of these methods can be developed for which stability/convergence can be proven in the general case. The PI proposes to develop two new variations of local regularization methods which show promise in numerical tests and for which there is hope of establishing a general stability/convergence theory. The PI also proposes to develop adaptive schemes for the selection of variable regularization parameters in local regularization methods. Variable parameters are of use in applying more smoothing in some parts of the domain and less in others. In numerical tests adaptive local regularization techniques have been shown to be effective in determining the variable regularization parameter at the same time that local parts of the solution are recovered. Because no convergence theory exists at the present time for such an approach, the PI proposes to study such adaptive schemes and develop a theory which will be useful in making recommendations for adaptive parameter selection methods. The PI also proposes to extend these ideas to nonlinear Volterra problems and to linear non-Volterra problems.Inverse problems occur widely in many applications, including problems of biomedical imaging (CT scans and X-rays), image reconstruction (from satellites or other sources), and geophysical exploration. The Volterra class of inverse problems arises in the determination of the surface temperature of a space vehicle as it re-enters the earth's atmosphere; additionally, Volterra inverse problems appear as models for remote sensing problems. While classical methods exist for for solving such problems, classical methods are often very inefficient and lead to overly expensive solution techniques. A second disadvantage of classical solution methods is readily seen in imaging applications where reconstructed images may have blurred edges and inadequately detailed features. The PI proposes to address both of these difficulties with the development of new solution methods based on the ideas of local regularization. The use of these newer methods can lead to a significant decrease in cost for the solution of a wide class of practical inverse problems, with improved resolution of detailed features of solutions.