计算具有幂律核的平衡测度
Computing Equilibrium Measures with Power Law Kernels
AI总结：
提出基于超球多项式基带状算子的数值方法，将幂律核平衡测度优化问题降维至一两个变量，验证了Tikhonov正则化下的稳定收敛性，并数值探讨了测度存在性与间隙形成行为。
摘要：
我们提出了一种数值计算平衡测度的方法，适用于具有吸引-排斥幂律核 $K(x-y) = \ rac{|x-y|^α}α-\ rac{|x-y|^β}β$ 的问题，该方法利用递归生成的带状和近似带状算子作用于超球多项式基展开。该方法将原本在测度空间上难以处理的优化问题，简化为固定平衡测度支撑集的一到两个变量的直接优化问题。所得算子的结构和快速收敛特性使得单个优化步骤具有高计算效率。我们讨论了该方法在 Tikhonov 正则化下的稳定性和收敛性，并利用实现与已知解析解及离散粒子模拟进行了比较。最后，我们数值探讨了平衡测度存在性和唯一性的开放问题，以及幂律核在感兴趣参数范围内支撑集分裂为两个区间的间隙形成行为。
英文摘要：
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. 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.
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