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：非标准采样机制的新颖数值逼近技术 Anne Gelb 和 Rosemary Renaut PI 建立在他们最近开发的技术之上，这些技术将数值逼近和逆理论的方法（例如用于重建保真度的目的）与利用特定应用信息（例如稀疏性）的方法融合在一起。 他们的方法同时考虑了理论和实践方面的考虑。提出了用于功能和/或图像恢复以及从数据集中表征和提取重要信息（例如边缘或其他特征）的算法，而不必确定底层功能。 研究目标包括（i）开发新的近似方法，用于从一个或多个角度有缺陷的数据中恢复功能和/或特征，即数据采样不足或丢失，可能有噪声，通过对偶表示测量，或者以其他方式相对于传统的数值逼近技术来说是非标准的； (ii) 开发可以直接应用于实际数据集的数值近似算子，或者可以向从业者提供反馈以改进采样协议； (iii) 整合数值线性代数和统计正则化技术，这些技术特别适用于在处理实际数据时获得病态问题的稳健而有效的解决方案。 这项研究将对所有新算法的准确性、效率和鲁棒性进行严格分析，特别是在存在噪声、扰动或其他不完整数据信息的情况下。实用的数据收集技术正变得越来越复杂。 用户友好的软件包使学科科学家能够从大量测量数据中成功诊断、预测、建模和确定重要特征。 然而，最近对现代磁共振成像（MRI）协议下收集的数据的各种重建算法的研究清楚地表明，在不考虑有关准确性和测量误差的基本数学问题的情况下使用实用的算法修改时会出现缺陷。 目前使用的一些算法实际上会产生错误的诊断和额外的程序成本。 该项目扩展了 PI 的先前研究，并致力于开发新颖的数学技术，以处理与从非标准采样协议获取的数据中提取功能和特征信息相关的问题。