Analytical Solutions to Minimum-Norm Problems

Almudena Campos-Jiménez, José Antonio Vílchez-Membrilla, Clemente Cobos-Sánchez and Francisco Javier García-Pacheco
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Almudena Campos-Jiménez: Department of Mathematics, College of Engineering, University of Cadiz, 11519 Puerto Real, Spain
José Antonio Vílchez-Membrilla: Department of Electronics, College of Engineering, University of Cadiz, 11510 Puerto Real, Spain
Clemente Cobos-Sánchez: Department of Electronics, College of Engineering, University of Cadiz, 11510 Puerto Real, Spain
Francisco Javier García-Pacheco: Department of Mathematics, College of Engineering, University of Cadiz, 11519 Puerto Real, Spain

Mathematics, 2022, vol. 10, issue 9, 1-18

Abstract: For G ∈ R m × n and g ∈ R m , the minimization min ∥ G ψ − g ∥ 2 , with ψ ∈ R n , is known as the Tykhonov regularization. We transport the Tykhonov regularization to an infinite-dimensional setting, that is min ∥ T ( h ) − k ∥ , where T : H → K is a continuous linear operator between Hilbert spaces H , K and h ∈ H , k ∈ K . In order to avoid an unbounded set of solutions for the Tykhonov regularization, we transform the infinite-dimensional Tykhonov regularization into a multiobjective optimization problem: min ∥ T ( h ) − k ∥ and min ∥ h ∥ . We call it bounded Tykhonov regularization. A Pareto-optimal solution of the bounded Tykhonov regularization is found. Finally, the bounded Tykhonov regularization is modified to introduce the precise Tykhonov regularization: min ∥ T ( h ) − k ∥ with ∥ h ∥ = α . The precise Tykhonov regularization is also optimally solved. All of these mathematical solutions are optimal for the design of Magnetic Resonance Imaging (MRI) coils.

Keywords: Hilbert space; convex optimization; supporting vector; matrix norm; MRI (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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