Novel Numerical Approximation Techniques for Non-Standard Sampling Regimes

点击翻译按钮获取中文摘要

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.