Novel Numerical Approximation Techniques for Non-Standard Sampling Regimes

非标准采样制度的新颖数值逼近技术

• 批准号: 1216559
• 负责人: Anne Gelb
• 金额: $ 33.69万
• 依托单位: Arizona State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1216559&HistoricalAwards=false
• 关键词: Novel; Numerical; Approximation; Techniques; Non

Title: Novel Numerical Approximation Techniques for Non-Standard Sampling RegimesAnne Gelb and Rosemary RenautThe PIs build on their recently developed techniques that fuse methodsfrom numerical approximation and inverse theory, such as for purposesof reconstruction fidelity, with those that exploit specific applicationinformation, such as sparsity. Their methods address both theoreticaland practical considerations. Algorithms are proposed for function and/orimage recovery, as well as to characterize and extract important informationfrom a data set, such as edges, or other features, without necessarilydetermining the underlying function. Research objectives include (i)developing novel approximation approaches for functional and/or featurerecovery from data that is deficient with respect to one or multipleperspectives, i.e. data is under-sampled or missing, may be noisy, ismeasured via a dual representation, or is otherwise non-standard withrespect to traditional numerical approximation techniques; (ii) developingnumerical approximation operators that can be directly applied to practicaldata sets, or may provide a feedback to practitioners for improving samplingprotocols; and (iii) integrating techniques from numerical linear algebraand statistical regularization that are specifically pertinent for obtainingrobust but efficient solutions of ill-conditioned problems when handlingpractical data. This research will provide rigorous analysis of all newalgorithms in terms of accuracy, efficiency, and robustness, especiallyin the presence of of noise, perturbations, or otherwise incomplete datainformation.Practical data collection techniques are becoming increasingly moresophisticated. User friendly software packages allow disciplinaryscientists to successfully diagnose, predict, model, and determineimportant characteristics from a plethora of measured data. Yet,recent investigations into various reconstruction algorithms for datacollected under modern magnetic resonance imaging (MRI) protocolshave clearly demonstrated shortcomings that arise when pragmaticalgorithmic modifications are used without considering fundamentalmathematical issues regarding accuracy and measurement error. Somecurrently employed algorithms in fact yield both incorrect diagnosesand additional procedural costs. This project extends the PIs priorresearch and addresses the development of novel mathematical techniquesfor handling issues associated with extracting functional and featureinformation from data acquired by non-standard sampling protocols.

标题：非标准抽样制度的新型数值近似技术，凝胶和迷迭香肾脏pis建立在他们最近开发的技术基础上，这些技术从数值近似和逆理论中融合了方法，例如用于重建忠诚的目的，以及那些利用特定的应用程序intermition，例如诸如Sparsity的那些。 他们的方法解决了理论和实际考虑。提出了用于功能和/Orimage恢复的算法，并从数据集（例如边缘或其他功能）表征和提取重要信息，而无需确定基础函数。 研究目标包括（i）从数据中开发用于功能性和/或特征再发现的新颖近似方法，这些数据与一个或多个角度相对于一个或多个角度缺陷的数据，即数据不足或丢失，可能是嘈杂的，可以通过双重表示，或者是通过双重表示，或者不符合传统的数字近似技术的依赖； （ii）可以直接应用于实用数据集的开发自然近似运算符，或者可以向从业人员提供改善采样概述的反馈； （iii）从数值线性代数和统计正规化中整合技术，这些技术是在处理验证数据时特异性相关的，但有效的解决方案的解决方案。 这项研究将在准确性，效率和鲁棒性方面对所有Newalgorithm进行严格的分析，尤其是在噪声，扰动或其他不完整的数据形式的存在方面。实践数据收集技术正在变得越来越情绪化。 用户友好的软件包允许纪律学者成功诊断，预测，建模和确定来自大量测量数据的特征。 然而，最近对在现代磁共振成像（MRI）协议下进行的各种重建算法进行的研究清楚地证明了当使用实用主义的修改而没有考虑精确性和测量误差的基本关系问题时出现的缺点。 实际上采用算法实际上产生了不正确的诊断和额外的程序成本。 该项目扩展了PIS PriorreSearch，并解决了与从非标准采样协议获取的数据中提取功能和功能信息相关的处理问题的新型数学技术的开发。