Analytical Solutions to Minimum-Norm Problems

Abstract

For G is an element of Rmxn and g is an element of Rm, the minimization min parallel to G psi-g parallel to 2, with psi is an element of Rn, is known as the Tykhonov regularization. We transport the Tykhonov regularization to an infinite-dimensional setting, that is min parallel to T(h)-k parallel to, where T:H -> K is a continuous linear operator between Hilbert spaces H,K and h is an element of H,k is an element of 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 parallel to T(h)-k parallel to andmin parallel to h parallel to. 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 parallel to T(h)-k parallel to with parallel to h parallel to=alpha. The precise Tykhonov regularization is also optimally solved. All of these mathematical solutions are optimal for the design of Magnetic Resonance Imaging (MRI) coils.