[Paper Review] Greedy energy points with external fields

A. López‐García|arXiv (Cornell University)|Jan 27, 2009

Mathematical Approximation and Integration8 references3 citations

TL;DR

This paper introduces greedy energy point sequences on locally compact metric spaces with external fields, using a minimization algorithm that balances Riesz kernel interactions and external field potentials. The key contribution is proving that such sequences asymptotically equidistribute according to the equilibrium measure under suitable conditions, extending Choquet's theory to weighted energy problems with external fields in $ ^p$.

ABSTRACT

In this paper we introduce several extremal sequences of points on locally compact metric spaces and study their asymptotic properties. These sequences are defined through a greedy algorithm by minimizing a certain energy functional whose expression involves an external field. Some results are also obtained in the context of Euclidian spaces $\mathbb{R}^{p}$, $p\geq 2$. As a particular example, given a closed set $A\subset\mathbb{R}^{p}$, a lower semicontinuous function $f:\mathbb{R}^p o(-\infty,+\infty]$ and an integer $m\geq 2$, we investigate (under suitable conditions on $A$ and $f$) sequences $\{a_{i}\}_{1}^{\infty}\subset A$ that are constructed inductively by selecting the first $m$ points $a_{1},...,a_{m}$ so that the functional \[ \sum_{1\leq i

Motivation & Objective

• To define and analyze greedy energy sequences on locally compact metric spaces using a minimization algorithm that incorporates an external field.
• To extend Choquet's theory of minimal energy configurations to the case with external fields in the context of Riesz kernels.
• To establish the asymptotic equidistribution of greedy energy points toward the equilibrium measure in $ ^p$ under suitable conditions on the external field and domain.
• To prove convergence of the weighted energy of configurations to the infimum energy, and to characterize the limiting measure as the equilibrium measure.
• To generalize results on Leja points and minimal energy configurations to the setting of external fields and Riesz kernels in Euclidean spaces.

Proposed method

• Define a weighted energy functional $ E_f( u) = E( u) + 2(N-1) extstyleigsum_{i=1}^N f(x_i) $, where $ E( u) $ is the Riesz energy with kernel $ k(x,y) = |x-y|^{-s} $, $ s eq p $, and $ f $ is a lower semicontinuous external field.
• Construct greedy sequences inductively: at each step, select $ m $ new points minimizing the energy functional that includes interaction with previous points and the external field.
• Use potential-theoretic tools, including the Riesz potential $ U_s^ u(x) = extstyleigint |x-y|^{-s} d u(y) $, and the weighted energy $ I_f( u) = W( u) + 2 extstyleigint f d u $.
• Apply Choquet's theory of minimal energy and the Gauss variational problem to establish existence and uniqueness of the equilibrium measure $ u $ minimizing $ I_f( u) $.
• Prove that the greedy sequence satisfies $ U_s^ u(x) + f(x) = ext{constant} $ quasi-everywhere on the support of $ u $, implying equidistribution.
• Use weak-star convergence of empirical measures to show that the limiting distribution of greedy points is the equilibrium measure $ u $.

Experimental results

Research questions

• RQ1 Under what conditions does a greedy energy sequence with an external field equidistribute according to the equilibrium measure?
• RQ2 How does the inclusion of an external field modify the asymptotic behavior of minimal energy configurations in $ ^p $?
• RQ3 Can Choquet's theory of minimal energy be extended to the case of external fields in locally compact metric spaces?
• RQ4 What is the relationship between the potential $ U_s^ u(x) + f(x) $ and the support of the equilibrium measure $ u $ in the presence of an external field?
• RQ5 How does the greedy algorithm ensure that each new point lies in the set where $ U_s^ u(x) + f(x) $ is minimized?

Key findings

• The greedy energy sequence $ igracevert a_n igracevert_{n=1}^rown $ asymptotically equidistributes according to the equilibrium measure $ u $, i.e., the empirical measures $ rac{1}{N} extstyleigsum_{i=1}^N u_{a_i} $ converge weak-star to $ u $.
• The limiting measure $ u $ is characterized by the property that $ U_s^ u(x) + f(x) = M $ for quasi-every $ x $ in the support of $ u $, where $ M $ is the infimum of $ U_s^ u(x) + f(x) $.
• For $ s eq p $, the weighted energy $ I_f( u) $ is finite and minimized uniquely by the equilibrium measure $ u $, which is the weak-star limit of the empirical measures of the greedy sequence.
• The energy of the $ N $-point greedy configuration satisfies $ rac{1}{N} E_f( u_N) o I_f( u) $ as $ N o rown $, with convergence of the energy per particle to the infimum energy.
• In the case $ p=2, s=0 $, the results extend to logarithmic energy with external fields, and the equilibrium measure still satisfies $ U_s^ u(x) + f(x) = ext{constant} $ q.e. on its support.
• The support of the equilibrium measure $ u $ is contained in the set where $ U_s^ u(x) + f(x) $ achieves its minimum, and $ u $ is the unique measure satisfying this condition.