Revisiting the minimum-norm problem
Abstract The design of optimal Magnetic Resonance Imaging (MRI) coils is modeled as a minimum-norm problem (MNP), that is, as an optimization problem of the form min x ∈ R ∥ x ∥ $\min_{x\in\mathcal{R}}\|x\|$ , where R $\mathcal{R}$ is a closed and convex subset of a normed space X. This manuscript i...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2022-02-01
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| Series: | Journal of Inequalities and Applications |
| Subjects: | |
| Online Access: | https://doi.org/10.1186/s13660-022-02757-5 |
| _version_ | 1818286753649262592 |
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| author | Soledad Moreno-Pulido Alberto Sánchez-Alzola Francisco Javier García-Pacheco |
| author_facet | Soledad Moreno-Pulido Alberto Sánchez-Alzola Francisco Javier García-Pacheco |
| author_sort | Soledad Moreno-Pulido |
| collection | DOAJ |
| description | Abstract The design of optimal Magnetic Resonance Imaging (MRI) coils is modeled as a minimum-norm problem (MNP), that is, as an optimization problem of the form min x ∈ R ∥ x ∥ $\min_{x\in\mathcal{R}}\|x\|$ , where R $\mathcal{R}$ is a closed and convex subset of a normed space X. This manuscript is aimed at revisiting MNPs from the perspective of Functional Analysis, Operator Theory, and Banach Space Geometry in order to provide an analytic solution to the following MRI problem: min ψ ∈ R ∥ ψ ∥ 2 $\min_{\psi\in\mathcal{R}}\|\psi\|_{2}$ , where R : = { ψ ∈ R n : ∥ A ψ − b ∥ ∞ ∥ b ∥ ∞ ≤ D } $\mathcal{R}:=\{\psi\in \mathbb{R}^{n}:\frac{\|A\psi-b\|_{\infty}}{\|b\|_{\infty}} \leq D\}$ , with A ∈ M m × n ( R ) $A\in\mathcal{M}_{m\times n}(\mathbb{R})$ , D > 0 $D>0$ , and b ∈ R m ∖ { 0 } $b\in\mathbb{R}^{m}\setminus\{0\}$ . |
| first_indexed | 2024-12-13T01:29:36Z |
| format | Article |
| id | doaj.art-8937f39e3ef04a9da39f605f66418537 |
| institution | Directory Open Access Journal |
| issn | 1029-242X |
| language | English |
| last_indexed | 2024-12-13T01:29:36Z |
| publishDate | 2022-02-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-8937f39e3ef04a9da39f605f664185372022-12-22T00:04:02ZengSpringerOpenJournal of Inequalities and Applications1029-242X2022-02-012022111110.1186/s13660-022-02757-5Revisiting the minimum-norm problemSoledad Moreno-Pulido0Alberto Sánchez-Alzola1Francisco Javier García-Pacheco2Department of Mathematics, University of CadizDepartment of Statistics, University of CadizDepartment of Mathematics, University of CadizAbstract The design of optimal Magnetic Resonance Imaging (MRI) coils is modeled as a minimum-norm problem (MNP), that is, as an optimization problem of the form min x ∈ R ∥ x ∥ $\min_{x\in\mathcal{R}}\|x\|$ , where R $\mathcal{R}$ is a closed and convex subset of a normed space X. This manuscript is aimed at revisiting MNPs from the perspective of Functional Analysis, Operator Theory, and Banach Space Geometry in order to provide an analytic solution to the following MRI problem: min ψ ∈ R ∥ ψ ∥ 2 $\min_{\psi\in\mathcal{R}}\|\psi\|_{2}$ , where R : = { ψ ∈ R n : ∥ A ψ − b ∥ ∞ ∥ b ∥ ∞ ≤ D } $\mathcal{R}:=\{\psi\in \mathbb{R}^{n}:\frac{\|A\psi-b\|_{\infty}}{\|b\|_{\infty}} \leq D\}$ , with A ∈ M m × n ( R ) $A\in\mathcal{M}_{m\times n}(\mathbb{R})$ , D > 0 $D>0$ , and b ∈ R m ∖ { 0 } $b\in\mathbb{R}^{m}\setminus\{0\}$ .https://doi.org/10.1186/s13660-022-02757-5Minimum normBanach spaceOptimization |
| spellingShingle | Soledad Moreno-Pulido Alberto Sánchez-Alzola Francisco Javier García-Pacheco Revisiting the minimum-norm problem Journal of Inequalities and Applications Minimum norm Banach space Optimization |
| title | Revisiting the minimum-norm problem |
| title_full | Revisiting the minimum-norm problem |
| title_fullStr | Revisiting the minimum-norm problem |
| title_full_unstemmed | Revisiting the minimum-norm problem |
| title_short | Revisiting the minimum-norm problem |
| title_sort | revisiting the minimum norm problem |
| topic | Minimum norm Banach space Optimization |
| url | https://doi.org/10.1186/s13660-022-02757-5 |
| work_keys_str_mv | AT soledadmorenopulido revisitingtheminimumnormproblem AT albertosanchezalzola revisitingtheminimumnormproblem AT franciscojaviergarciapacheco revisitingtheminimumnormproblem |