Average-distance problem with curvature penalization for data parameterization: regularity of minimizers

2020/12/28 by Xinyang Lu, Lu, Xinyang, Dejan Slepcev +2
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Analysis of PDEs (math.AP) #Composite Material Mechanics #FOS: Mathematics #Mathematical Approximation and Integration #Optimization and Control (math.OC) #math.AP #math.OC

paper · pdf · doi:10.48550/arxiv.2012.14532

arxiv created 2020/12/28 · openalex publication_date 2020/12/28 · arxiv updated 2021/01/01 · openalex created_date 2021/01/05 · openalex updated_date 2026/07/28

Abstract

We propose a model for finding one-dimensional structure in a given measure. Our approach is based on minimizing an objective functional which combines the average-distance functional to measure the quality of the approximation and penalizes the curvature, similarly to the elastica functional. Introducing the curvature penalization overcomes some of the shortcomings of the average-distance functional, in particular the lack of regularity of minimizers. We establish existence, uniqueness and regularity of minimizers of the proposed functional. In particular we establish C^1,1 estimates on the minimizers.

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