Hopgrid算法：多重网格和小波理论的多级综合
The Hopgrid algorithm: multilevel synthesis of multigrid and wavelet theory
摘要：
多重网格算法是一种多层次的方法来加速离散微分方程的数值解的物理问题，涉及远程相互作用。小波理论的多分辨率分析提供了一个有效的表示功能，表现出本地化的短长度尺度行为的突发。诸如计算分子内部和周围的静电场之类的应用应该从这两种方法中受益。在这项工作中，我们演示了如何一种新的插值小波变换，这本身就是有限元分析和小波理论的合成，可用作数学桥梁连接这两种方法。其结果是一个专门的多重网格算法，可适用于表示在小波基的问题。用这种方法，插值和限制算子和网格的多重网格算法是预先确定的插值多分辨率分析。我们将提出新的方法，并将其效率与标准小波和多重网格方法进行比较。
英文摘要：
The multigrid algorithm is a multilevel approach to accelerate the numerical solution of discretized differential equations in physical problems involving long-range interactions. Multiresolution analysis of wavelet theory provides an efficient representation of functions which exhibit localized bursts of short length-scale behavior. Applications such as computing the electrostatic field in and around a molecule should benefit from both approaches. In this work, we demonstrate how a novel interpolating wavelet transform, which in itself is the synthesis of finite element analysis and wavelet theory, may be used as the mathematical bridge to connect the two approaches. The result is a specialized multigrid algorithm which may be applied to problems expressed in wavelet bases. With this approach, interpolation and restriction operators and grids for the multigrid algorithm are predetermined by an interpolating multiresolution analysis. We will present the new method and contrast its efficiency with standard wavelet and multigrid approaches.
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