多重网格与粗糙系数的多重分辨率算子分解来自层次信息博弈
Multigrid with rough coefficients and Multiresolution operator decomposition from Hierarchical Information Games
AI总结：
本文提出了一种近线性复杂度的多重网格/多重分辨率方法，用于处理具有粗糙系数的偏微分方程，通过信息博弈理论框架实现精确的先验精度和性能估计。
摘要：
我们介绍了一种近线性复杂度（几何和无网格/代数）的多重网格/多重分辨率方法，用于具有粗糙（L^∞）系数的偏微分方程，具有严格的先验准确性和性能估计。该方法通过决策/博弈理论框架发现，解决三个问题：（1）识别限制和插值算子；（2）基于线性算子图像的范数约束恢复信号；（3）基于解的层次嵌套测量的赌注。所得基本赌注形成一个层次的（确定性）基函数集合H^1_0(Ω)（赌注函数），这些函数（1）在子尺度/子带之间关于由偏微分方程能量范数诱导的标量积正交；（2）在H^1_0(Ω)中实现解空间的稀疏压缩；（3）诱导一个正交的多重分辨率算子分解。多重网格方法的操作图是一个倒置金字塔，其中赌注函数局部计算（由于指数衰减），层次化（从细到粗尺度），并分解为具有均匀有界条件数的独立线性系统。所得算法在空间（通过局部化）和带宽/子尺度（子尺度可以独立计算）上均可并行化。尽管该方法是确定性的，但其在信息博弈框架下具有自然的贝叶斯解释，且多重分辨率逼近相对于由嵌套测量层次诱导的滤波器形成一个鞅。
英文摘要：
We introduce a near-linear complexity (geometric and meshless/algebraic) multigrid/multiresolution method for PDEs with rough ($L^\infty$) coefficients with rigorous a-priori accuracy and performance estimates. The method is discovered through a decision/game theory formulation of the problems of (1) identifying restriction and interpolation operators (2) recovering a signal from incomplete measurements based on norm constraints on its image under a linear operator (3) gambling on the value of the solution of the PDE based on a hierarchy of nested measurements of its solution or source term. The resulting elementary gambles form a hierarchy of (deterministic) basis functions of $H^1_0(Ω)$ (gamblets) that (1) are orthogonal across subscales/subbands with respect to the scalar product induced by the energy norm of the PDE (2) enable sparse compression of the solution space in $H^1_0(Ω)$ (3) induce an orthogonal multiresolution operator decomposition. The operating diagram of the multigrid method is that of an inverted pyramid in which gamblets are computed locally (by virtue of their exponential decay), hierarchically (from fine to coarse scales) and the PDE is decomposed into a hierarchy of independent linear systems with uniformly bounded condition numbers. The resulting algorithm is parallelizable both in space (via localization) and in bandwith/subscale (subscales can be computed independently from each other). Although the method is deterministic it has a natural Bayesian interpretation under the measure of probability emerging (as a mixed strategy) from the information game formulation and multiresolution approximations form a martingale with respect to the filtration induced by the hierarchy of nested measurements.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
