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别的逆问题方面显示出巨大的前景，通常保留了原始问题的特殊结构，并导致非常快的算法(特别是在Volterra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解的尖锐特征，而不必依赖于不可微或非二次优化方案。到目前为止，Volterra问题的局部正则化的收敛理论仅限于温和的不适定反问题。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。由于局部正则化方法的成本效率，因此一个重要的问题是，是否可以开发出这些方法的新的变种，从而在一般情况下证明其稳定性/收敛。PI建议发展两种新的局部正则化方法，这两种方法在数值试验中表现出很好的前景，并有望建立一个普遍的稳定性/收敛理论。PI还建议开发自适应方案来选择局部正则化方法中的可变正则化参数。变量参数用于在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。在数值试验中，自适应局部正则化技术已被证明在确定可变正则化参数的同时恢复解的局部部分是有效的。由于目前还不存在这种方法的收敛理论，PI建议研究这种自适应方案，并发展一种理论，这将有助于对自适应参数选择方法提出建议。PI还建议将这些想法扩展到非线性Volterra问题和线性非Volterra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(来自卫星或其他来源)和地球物理勘探。Volterra类逆问题出现在确定航天器重新进入地球大气层时的表面温度；此外，Volterra逆问题作为遥感问题的模型出现。虽然有解决这类问题的经典方法，但经典方法往往效率很低，导致求解技术过于昂贵。经典解方法的第二个缺点很容易在成像应用中看到，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建议通过开发基于局部正则化思想的新的解决方法来解决这两个困难。使用这些较新的方法可以显著降低求解一大类实际反问题的成本，并提高解的细节特征的分辨率。

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.