论文标题
积极的近乎收集点源的超分辨率
Super-resolution of positive near-colliding point sources
论文作者
论文摘要
在本文中，我们分析了一维积极来源的超分辨率的能力。特别是，我们考虑与[Arxiv：1904.09186V2 [Math.na]]中的相同设置，并将其概括为超出阳性来源的情况。更具体地说，我们考虑使用$ p \ leqslant d $ nodes的$ d $正点源紧密间隔并形成群集，而其余的节点则很好地分开。与[Arxiv：1904.09186V2 [Math.na]类似，我们的结果表明，当噪声级别$ε\ lyssim \ Mathrm \ Mathrm {srf}^{ - 2 P+1} $时节点，重建群集节点的最小值错误率是$ \ frac {1}ω\ mathrm {srf}^{2 p-2}ε$，而用于恢复相应的agplites $ \ left \ left \ weft \ {a_j \ right \} $的速率是2 pp rm} ε$。对于非集群节点，节点和振幅的回收率的相应最小值分别为$ \fracεΩ$和$ε$。我们的数值实验表明，在解决正源时，矩阵铅笔法可以达到上述最佳边界。
In this paper, we analyze the capacity of super-resolution of one-dimensional positive sources. In particular, we consider the same setting as in [arXiv:1904.09186v2 [math.NA]] and generalize the results there to the case of super-resolving positive sources. To be more specific, we consider resolving $d$ positive point sources with $p \leqslant d$ nodes closely spaced and forming a cluster, while the rest of the nodes are well separated. Similarly to [arXiv:1904.09186v2 [math.NA]], our results show that when the noise level $ε\lesssim \mathrm{SRF}^{-2 p+1}$, where $\mathrm{SRF}=(ΩΔ)^{-1}$ with $Ω$ being the cutoff frequency and $Δ$ the minimal separation between the nodes, the minimax error rate for reconstructing the cluster nodes is of order $\frac{1}Ω \mathrm{SRF}^{2 p-2} ε$, while for recovering the corresponding amplitudes $\left\{a_j\right\}$ the rate is of order $\mathrm{SRF}^{2 p-1} ε$. For the non-cluster nodes, the corresponding minimax rates for the recovery of nodes and amplitudes are of order $\fracεΩ$ and $ε$, respectively. Our numerical experiments show that the Matrix Pencil method achieves the above optimal bounds when resolving the positive sources.
