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

研究人员和她的博士后研究员以及一名研究生将研究重点放在开发，分析和实施新的算法来解决线性不适定的逆问题，其中测量数据和模型都受到错误污染。分析将提供新的和现有的算法的性能改进的见解。一个方向是分析线性支持向量机，它已经被验证为一个有效的工具，在统计模式识别。 重构的支持向量机，以考虑可能发生在要分类的模式的特征中的错误，其中的一个例子可能是错误污染的微阵列数据，利用正则化的总最小二乘法的工具，将更好地考虑这些错误的数据测量。另一个方向是使用全变分正则化结构化全最小二乘算法来提供用于将边缘保持正则化与信号反演的变量模型中的误差相结合的全新机制。对于多个但类似地被破坏的信号，并行解决方案技术将增强信号反转和恢复。 虽然文献中已经提出了几种吉洪诺夫正则化总最小二乘的方法，但尚未对合成数据和真实的数据进行有效的竞争力比较。这些研究人员将确定哪些算法最适合于扩展到现实问题。为该项目开发的所有软件将分发给合作者供其应用，并在万维网上公布。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成像的所谓逆问题的解决方案产生重大影响。这些领域是PI与其他从业者积极合作的领域，包括位于亚利桑那州凤凰城的翻译基因组学研究中心。 逆问题出现在许多生物医学情况中：例如，医学成像可以用于获得关于内部器官功能的非侵入性信息，其也可能受到恶性或良性肿瘤的存在的影响。另一个方向是设计 对地震数据进行新的和更好的分析，目的是加深对地球内部动力性质的了解，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.