基于Riesz能量最小化的低复杂度流形离散化方法
Low complexity methods for discretizing manifolds via Riesz energy minimization
摘要：
设$A$是嵌入欧氏空间$\RR^p$，$d\le p$中的紧$d$ -可求长集.对于一个给定的连续分布$σ(x)$关于$d$维Hausdorff测度$A$，我们以前的结果提供了一种方法，用于产生$A$上的$N$ -点配置，具有渐近分布$σ(x)$作为$N\to \infty$ ;而且这样的配置是“准均匀”的意义上，覆盖半径的比例，以分离距离是有界的$N$独立。该方法是基于最小化的能量的$N$粒子约束到$A$通过加权幂律势$w(x,y)|x-y|^{-s}$相互作用，其中$s>d$是一个固定的参数和$w(x,y)=\left(σ(x)σ(y)\right)^{-({s}/{2d})}$。在这里，我们表明，可以生成点$A$与上述属性保持在能量和只有那些对点，位于距离最多$r_N=C_N N^{-1/d}$从对方，与$C_N$是一个正序列趋于无穷大任意缓慢。为了做到这一点，我们最小化关于变化的截断权重$v_N(x,y)=Φ\(\left|x-y\right|/r_N\)w(x,y)$的能量，其中$Φ:(0,\infty)\to [0,\infty)$是具有$Φ(t)=0$、$t\geq 1$和$\lim_{t\to 0^+}Φ(t)=1$的有界函数。在适当的假设下，这降低了生成$N$点“低能量”离散化以进行$N C_N^d$计算的复杂性。
英文摘要：
Let $A$ be a compact $d$-rectifiable set embedded in Euclidean space $\RR^p$, $d\le p$. For a given continuous distribution $σ(x)$ with respect to $d$-dimensional Hausdorff measure on $A$, our earlier results provided a method for generating $N$-point configurations on $A$ that have asymptotic distribution $σ(x)$ as $N\to \infty$; moreover such configurations are "quasi-uniform" in the sense that the ratio of the covering radius to the separation distance is bounded independent of $N$. The method is based upon minimizing the energy of $N$ particles constrained to $A$ interacting via a weighted power law potential $w(x,y)|x-y|^{-s}$, where $s>d$ is a fixed parameter and $w(x,y)=\left(σ(x)σ(y)\right)^{-({s}/{2d})}$. Here we show that one can generate points on $A$ with the above mentioned properties keeping in the energy sums only those pairs of points that are located at a distance of at most $r_N=C_N N^{-1/d}$ from each other, with $C_N$ being a positive sequence tending to infinity arbitrarily slowly. To do this we minimize the energy with respect to a varying truncated weight $v_N(x,y)=Φ\(\left|x-y\right|/r_N\)w(x,y)$, where $Φ:(0,\infty)\to [0,\infty)$ is a bounded function with $Φ(t)=0$, $t\geq 1$, and $\lim_{t\to 0^+}Φ(t)=1$. This reduces, under appropriate assumptions, the complexity of generating $N$ point `low energy' discretizations to order $N C_N^d$ computations.
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