赌局基函数用于突破隐式方案在处理具有粗糙系数的超几何和抛物型ODEs/PDEs中的复杂性瓶颈
Gamblets for opening the complexity-bottleneck of implicit schemes for hyperbolic and parabolic ODEs/PDEs with rough coefficients
AI总结：
本文提出了一种扩展的赌局基函数，用于高效求解隐式方案中的线性系统，提供近线性复杂度和严谨的误差界。
摘要：
隐式方案是求解时间依赖PDEs（如超几何和抛物型PDEs）的流行方法。然而，每次时间步需求解对应的线性系统构成了处理具有粗糙系数的PDEs的复杂性瓶颈。我们介绍了在\cite{OwhadiMultigrid:2015}中提出的赌局基函数的推广，使得这些隐式系统能够在近线性复杂度下求解，并为超几何和抛物型PDEs的数值近似提供严谨的先验误差界。这些扩展的赌局基函数诱导了解空间的多分辨率分解，该分解适应于底层（超几何和抛物型）PDE（以及由空间离散化得到的ODE系统）和数值方案的时间步长。
英文摘要：
Implicit schemes are popular methods for the integration of time dependent PDEs such as hyperbolic and parabolic PDEs. However the necessity to solve corresponding linear systems at each time step constitutes a complexity bottleneck in their application to PDEs with rough coefficients. We present a generalization of gamblets introduced in \cite{OwhadiMultigrid:2015} enabling the resolution of these implicit systems in near-linear complexity and provide rigorous a-priori error bounds on the resulting numerical approximations of hyperbolic and parabolic PDEs. These generalized gamblets induce a multiresolution decomposition of the solution space that is adapted to both the underlying (hyperbolic and parabolic) PDE (and the system of ODEs resulting from space discretization) and to the time-steps of the numerical scheme.
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