A note on least squares methods

Carey, G. F.; Richardson, W. B.

doi:10.1002/cnm.780

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A Least-squares Mixed Methods Example1

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2006

2022

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Cited by 3 publications

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“…Like every numerical technique, this least-squares method has its peculiar strengths and weaknesses [16]. Advantages of least squares include generality and convenience: a symmetric form even for non-self adjoint PDEs, less sensitivity to changes in PDE type (e.g.…”

Section: A Least-squares Mixed Methods Examplementioning

confidence: 99%

Sobolev gradient preconditioning for image‐processing PDEs

Richardson

2006

Commun. Numer. Meth. Engng.

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SUMMARYThe article explores the relationship between Sobolev gradients and H −1 mixed methods for a variety of partial differential equations (PDEs) from image processing. A first-order system least-squares problem is used to introduce the method and compare the Euclidean with the Sobolev gradient. The standard two-term decomposition of an image as f = u + v with u ∈ H 1 and v ∈ L 2 = H 0 yields a second-order linear PDE, while minimizing other L p norms give nonlinear PDEs. Finally, a three-term decomposition f = u + v + w with u ∈ H 1 , v ∈ H −1 , w ∈ H 0 requires the solution of a fourth-order system with the biharmonic operator.

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Section: A Least-squares Mixed Methods Examplementioning

confidence: 99%