Novel Numerical Approximation Techniques for Non-Standard Sampling Regimes

标题：非标准采样制度的新型数值近似技术anne Gelb和Rosemary renautpi建立在他们最近开发的技术基础上，这些技术融合了数值近似和逆理论的方法，例如用于重建保真度的方法，以及那些利用特定应用信息的方法，例如稀疏性。他们的方法兼顾了理论和实践两方面的考虑。算法被提出用于功能和/或图像恢复，以及从数据集（如边缘或其他特征）中表征和提取重要信息，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，即数据采样不足或缺失，可能有噪声，通过对偶表示进行测量，或者与传统的数值近似技术相比是非标准的；（ii）开发可直接应用于实际数据集的数值近似算子，或可为从业人员提供改进采样方案的反馈；（iii）整合数值线性代数和统计正则化的技术，这些技术特别适用于在处理实际数据时获得病态问题的鲁棒但有效的解决方案。本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。用户友好的软件包允许学科科学家成功地诊断，预测，模型，并确定从大量的测量数据的重要特征。然而，最近对在现代磁共振成像（MRI）协议下收集的数据的各种重建算法的研究清楚地表明，当使用实用的算法修改而不考虑有关精度和测量误差的基本数学问题时，会出现缺点。一些目前使用的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pi先前的研究，并解决了新的数学技术的发展，用于处理与从非标准采样协议获得的数据中提取功能和特征信息相关的问题。

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.