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

研究者和她的博士后研究员以及一名研究生正在为这项资助集中研究解决线性病态逆问题的新算法的开发，分析和实现，其中测量数据和模型都受到误差污染。该分析将为新算法和现有算法的性能提供改进的见解。一个方向是分析线性支持向量机，它已经被证明是统计模式识别的有效工具。重新制定支持向量机，以解释可能发生在待分类模式特征中的错误，其中一个例子可能是错误污染的微阵列数据，利用正则化总最小二乘工具，将更好地解释数据测量中的这些错误。另一个方向是利用全变分正则化结构化总最小二乘算法，为信号反演的变量模型中保持边缘正则化与误差相结合提供了一种全新的机制。对于多个但同样损坏的信号，并行求解技术将增强信号的反演和恢复。虽然文献中已经提出了几种Tikhonov正则化总最小二乘的方法，但尚未对它们在合成数据和真实数据上的竞争力进行有效比较。这些研究人员将决定哪种算法最适合扩展到实际问题。为此项目开发的所有软件将分发给合作者，供其应用，并在万维网上发布。研究人员在研究和开发可用于许多不同应用的计算工具方面有着良好的记录。这项特殊研究的成功结果将对解决生物医学应用、遗传数据分析和地震层析成像的所谓逆问题产生重大影响。在这些领域，PI正在积极与其他从业人员合作，包括亚利桑那州凤凰城的转化基因组学研究中心的从业人员。在许多生物医学情况下会出现逆向问题：例如，医学成像可用于获取有关内部器官功能的非侵入性信息，这些信息也可能受到恶性或良性肿瘤的影响。另一个方向是设计新的和改进的地震数据分析，目的是增加对地球内部动态性质的理解，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.