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Analytical Solutions to Minimum-Norm Problems

Author

Listed:
  • Almudena Campos-Jiménez

    (Department of Mathematics, College of Engineering, University of Cadiz, 11519 Puerto Real, Spain
    These authors contributed equally to this work.)

  • José Antonio Vílchez-Membrilla

    (Department of Electronics, College of Engineering, University of Cadiz, 11510 Puerto Real, Spain
    These authors contributed equally to this work.)

  • Clemente Cobos-Sánchez

    (Department of Electronics, College of Engineering, University of Cadiz, 11510 Puerto Real, Spain
    These authors contributed equally to this work.)

  • Francisco Javier García-Pacheco

    (Department of Mathematics, College of Engineering, University of Cadiz, 11519 Puerto Real, Spain
    These authors contributed equally to this work.)

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.

Suggested Citation

  • Almudena Campos-Jiménez & José Antonio Vílchez-Membrilla & Clemente Cobos-Sánchez & Francisco Javier García-Pacheco, 2022. "Analytical Solutions to Minimum-Norm Problems," Mathematics, MDPI, vol. 10(9), pages 1-18, April.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:9:p:1454-:d:802647
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    References listed on IDEAS

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    1. Yuta Susowake & Hasan Masrur & Tetsuya Yabiku & Tomonobu Senjyu & Abdul Motin Howlader & Mamdouh Abdel-Akher & Ashraf M. Hemeida, 2019. "A Multi-Objective Optimization Approach towards a Proposed Smart Apartment with Demand-Response in Japan," Energies, MDPI, vol. 13(1), pages 1-14, December.
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