论文标题
电源法内核计算平衡度量
Computing Equilibrium Measures with Power Law Kernels
论文作者
论文摘要
We introduce a method to numerically compute equilibrium measures for problems with attractive-repulsive power law kernels of the form $K(x-y) = \frac{|x-y|^α}α-\frac{|x-y|^β}β$ using recursively generated banded and approximately banded operators acting on expansions in ultraspherical polynomial bases.所提出的方法减少了在测量空间上很难在一个或两个变量上固定平衡度量的支持的直接优化问题，因此很难在测量空间上进行优化问题。获得的运算符的结构和快速收敛性能在单个优化步骤中提高计算效率。我们讨论了在Tikhonov正则化下该方法的稳定性和收敛性，并使用实现来展示与分析已知的解决方案以及离散粒子模拟的比较。最后，我们从数值上探讨了关于存在和平衡度量的独特性以及差距的行为的开放性问题，而在势力法的参数范围内，平衡度量的支持分为两个间隔。
We introduce a method to numerically compute equilibrium measures for problems with attractive-repulsive power law kernels of the form $K(x-y) = \frac{|x-y|^α}α-\frac{|x-y|^β}β$ using recursively generated banded and approximately banded operators acting on expansions in ultraspherical polynomial bases. The proposed method reduces what is naively a difficult to approach optimization problem over a measure space to a straightforward optimization problem over one or two variables fixing the support of the equilibrium measure. The structure and rapid convergence properties of the obtained operators results in high computational efficiency in the individual optimization steps. We discuss stability and convergence of the method under a Tikhonov regularization and use an implementation to showcase comparisons with analytically known solutions as well as discrete particle simulations. Finally, we numerically explore open questions with respect to existence and uniqueness of equilibrium measures as well as gap forming behaviour in parameter ranges of interest for power law kernels, where the support of the equilibrium measure splits into two intervals.
