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Exact solutions to $\displaystyle{\max_{\|x\|=1} \sum_{i=1}^\infty\|T_i(x)\|^2}$ with applications to Physics, Bioengineering and Statistics

Francisco Javier García-Pacheco, Clemente Cobos-Sánchez, Soledad Moreno-Pulido, Alberto Sánchez-Alzola

TL;DR

A unified MATLAB code is presented for computing generalized supporting vectors of a finite number of matrices and three novel examples are provided to which the technique applies: optimized observable magnitudes by a pure state in a quantum mechanical system, a TMS optimized coil and an optimal location problem using statistics multivariate analysis.

Abstract

The supporting vectors of a matrix A are the solutions of max || x ||_2 =1 {||Ax||_2^2}. The generalized supporting vectors of matrices A_1 , . . . , A_k are the solutions of max || x ||_2 =1 {||A_1x||_2^2 + ||A_2x||_2^2 + ... + ||A_kx||_2^2}. Notice that the previous optimization problem is also a boundary element problem since the maximum is attained on the unit sphere. Many problems in Physics, Statistics and Engineering can be modeled by using generalized supporting vectors. In this manuscript we first raise the generalized supporting vectors to the infinite dimensional case by solving the optimization problem max || x || =1 sum_{i=1}^\infty ||T i (x )||^2 where (T i )_i is a sequence ofbounded linear operators between Hilbert spaces H and K of any dimension. Observe that the previous optimization problem generalizes the first two. Then a unified MATLAB code is presented for computing generalized supporting vectors of a finite number of matrices. Some particular cases are considered and three novel examples are provided to which our technique applies: optimized observable magnitudes by a pure state in a quantum mechanical system, a TMS optimized coil and an optimal location problem using statistics multivariate analysis. These three examples show the wide applicability of our theoretical and computational model.

Exact solutions to $\displaystyle{\max_{\|x\|=1} \sum_{i=1}^\infty\|T_i(x)\|^2}$ with applications to Physics, Bioengineering and Statistics

TL;DR

A unified MATLAB code is presented for computing generalized supporting vectors of a finite number of matrices and three novel examples are provided to which the technique applies: optimized observable magnitudes by a pure state in a quantum mechanical system, a TMS optimized coil and an optimal location problem using statistics multivariate analysis.

Abstract

The supporting vectors of a matrix A are the solutions of max || x ||_2 =1 {||Ax||_2^2}. The generalized supporting vectors of matrices A_1 , . . . , A_k are the solutions of max || x ||_2 =1 {||A_1x||_2^2 + ||A_2x||_2^2 + ... + ||A_kx||_2^2}. Notice that the previous optimization problem is also a boundary element problem since the maximum is attained on the unit sphere. Many problems in Physics, Statistics and Engineering can be modeled by using generalized supporting vectors. In this manuscript we first raise the generalized supporting vectors to the infinite dimensional case by solving the optimization problem max || x || =1 sum_{i=1}^\infty ||T i (x )||^2 where (T i )_i is a sequence ofbounded linear operators between Hilbert spaces H and K of any dimension. Observe that the previous optimization problem generalizes the first two. Then a unified MATLAB code is presented for computing generalized supporting vectors of a finite number of matrices. Some particular cases are considered and three novel examples are provided to which our technique applies: optimized observable magnitudes by a pure state in a quantum mechanical system, a TMS optimized coil and an optimal location problem using statistics multivariate analysis. These three examples show the wide applicability of our theoretical and computational model.
Paper Structure (17 sections, 9 theorems, 98 equations, 4 figures)

This paper contains 17 sections, 9 theorems, 98 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reformulation of Problem \ref{['original']}
  4. Generalized supporting vectors
  5. Supporting vectors of statistically-normalized operators
  6. Basic-normalized matrices of $\mathbb{R}^{m\times 2}$
  7. Basic-normalized matrices of $\mathbb{R}^{m\times n}$
  8. Applications of the proposed model (Theorem \ref{['gsv']})
  9. Pure state that jointly maximizes the modulus of observable magnitudes in a quantum mechanical system, with special attention to the probability density operator
  10. Optimal TMS coils
  11. Optimal location using statistics multivariate analysis
  12. Conclusions
  13. General conclusions
  14. Novelties of this work
  15. Algorithms developed in this work
  16. ...and 2 more sections

Key Result

Theorem 3.1

Let $H$ and $K$ be Hilbert spaces. Let $(T_i)_{i\in\mathbb{N}}$ be a sequence of bounded linear operators from $H$ to $K$, not all zero, such that $\sum_{i=1}^\infty \|T_i\|^2<\infty$. Then

Figures (4)

  • Figure 1: (a) Schematic diagram showing the hemispherical conducting surface with a cylindrical extension along with the region of interest where the stimulation is desired to be maximal, which has been included in a illustrative human head model for sake of clarity. (b) Wirepaths with 16 turns of the TMS coil solution of design problem in Equation \ref{['ffm']}.
  • Figure 2: (a) Colormap of the normalised optimal stream function $\dfrac{\psi}{ \psi_{\max}}$ over the coil surface.
  • Figure 3: Reference frame defined by the three principal components with the variables of Section \ref{['secc_alb']}. In this case, with the variables directly correlated, a supporting vector $x$ indicates the direction of the first component (maximum of the three variables, red arrow). In blue squares, the locations considered in this paper. In blue lines, the standardized variables $m_1^{st}, m_2^{st}, m_3^{st}$ represented in PCA reference frame.
  • Figure 4: (A) Bar diagram with the values of $Mx$ obtained in each province of Spain. The optimal place to locate the private academy shall be where the value of Mx is the highest. (B) Geographic distribution of the $Mx$ values. Green provinces are the best places for our multiobjective problem. Design problem described in Equation \ref{['alb1']}.

Theorems & Definitions (21)

  • Theorem 3.1
  • proof
  • Lemma 4.1
  • proof
  • Theorem 4.2
  • proof
  • Remark 5.1
  • Corollary 5.2
  • proof
  • Remark 5.3
  • ...and 11 more