Algorithms for Total Least Squares: Development, Evaluation and Novel Applications

这位研究人员和她的博士后同事以及一名研究生将他们的研究重点放在开发、分析和实施解决线性不适定反问题的新算法上，在这种反问题中，测量数据和模型都被错误污染。该分析将为新算法和现有算法的性能提供更好的洞察力。一个方向是对线性支持向量机的分析，它已经被证明是统计模式识别中的一种有效工具。利用正则化总体最小二乘的工具，重新表述支持向量机，以考虑待分类模式的特征中可能出现的误差，例如可能是受误差污染的微阵列数据，将更好地考虑数据测量中的这些误差。另一个方向是使用全变差正则化结构化总体最小二乘算法来提供一种全新的机制，将边缘保持正则化与信号反演变量模型中的误差相结合。对于多个但类似地被破坏的信号，并行解决技术将增强信号反转和恢复。虽然文献中已经提出了几种Tikhonov正则化总体最小二乘的方法，但还没有对它们在合成数据和真实数据中的竞争力进行有效的比较。这些研究人员将确定哪种算法最适合扩展到现实问题。为该项目开发的所有软件将分发给合作者以供其应用，并在万维网上发布。调查人员有研究和开发可用于许多不同应用的计算工具的记录。这一特定研究的成功结果将对生物医学应用、遗传数据分析和地震层析的所谓逆问题的解决产生重大影响。在这些领域，PI正在积极与其他实践者合作，包括亚利桑那州凤凰城的翻译基因组研究中心的那些实践者。在许多生物医学情况下都会出现逆问题：例如，医学成像可以用来获得关于内部器官功能的非侵入性信息，这些信息可能也会受到恶性肿瘤或良性肿瘤的影响。另一个方向是设计新的和改进的地震数据分析，以便更好地了解地球内部的动态性质，特别是在地表观察到的板块构造过程与内部热化学结构之间的关系。

The investigator and her postdoctoral fellow together with one graduate student are focusing their research for this grant on the development, analysis and implementation of novel algorithms for solution of linear ill-posed inverse problems in which both the measured data and the model are error-contaminated. The analysis will provide improved insight for the performance of new and existing algorithms. One direction is analysis of the linear support vector machine which has already been validated as an effective tool in statistical pattern recognition. Reformulation of the support vector machine to account for errors that may occur in the features of the patterns to be classified, an example of which might be error contaminated microarray data, utilizing the tools of regularized total least squares, will better account for these errors in data measurements. Another direction is the use of a total variation regularized structured total least squares algorithm to provide a completely new mechanism for combining edge preserving regularization with an errors in the variables model of signal inversion. For multiple, but similarly corrupted, signals, a concurrent solution technique will enhance signal inversion and restoration. While several approaches for Tikhonov regularized total least squares have been presented in literature, an effective comparison of their competitiveness for both synthetic and real data has not been performed. These investigators will determine which of algorithms are most suitable for extensions to realistic problems. All software developed for this project will be disseminated to collaborators for their applications and published on the world wide web. The investigators have a track record of studying and developing computational tools which can be utilized for many different applications. The successful outcome of this particular research will have major impact on solution of so-called inverse problems for biomedical applications, genetic data analysis and seismic tomography. These are areas in which the PI is actively collaborating with other practioners, including those at the Translational Genomics Research Center in Phoenix, AZ. Inverse problems arise in many biomedical situations: for example medical imaging can be used to obtain non-invasive information about the function of an internal organ, potentially also impacted by presence of malignant or benign tumor. Another direction is for the design of new and improved analysis of seismic data with the intent to lead to increased understanding of the dynamic nature of the Earth's interior, and in particular the relationship between plate tectonic processes observed at the surface and the thermo-chemical structure of the interior.