Bayesian Numerical Homogenization | Multiscale Modeling & Simulation | Vol. 13, No. 3 | Society for Industrial and Applied Mathematics
Bayesian Numerical Homogenization | Multiscale Modeling & Simulation | Vol. 13, No. 3 | Society for Industrial and Applied Mathematics
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贝叶斯数值均质化 |
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
H. Owhadi
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作者：
H. Owhadi
数值均匀化，即，具有任意粗糙系数的偏微分方程的解空间的有限维近似，需要精确的基元的识别。这些基本元素通常是在经过艰苦的科学调查和简单的猜测之后发现的。这个识别问题能否得到解决？是否有一个通用的处方/决策框架来指导基本要素的设计？我们建议，上述问题的答案可能是积极的基础上重新制定的数值均匀化作为贝叶斯推理问题，其中一个给定的偏微分方程与粗糙系数（或多尺度算子）被激发与噪声（随机右手边/源项）和一个试图估计的值的解决方案在一个给定的点的基础上有限数量的观察。我们应用这种重新识别基地的数值均匀化的任意积分微分方程，并表明这些基地具有最佳的恢复性能。特别是，我们展示了如何粗糙的多调和样条可以重新发现高斯滤波问题的最佳解决方案。
Numerical homogenization, i.e., the finite-dimensional approximation of solution spaces of PDEs with arbitrary rough coefficients, requires the identification of accurate basis elements. These basis elements are oftentimes found after a laborious process of scientific investigation and plain guesswork. Can this identification problem be facilitated? Is there a general recipe/decision framework for guiding the design of basis elements? We suggest that the answer to the above questions could be positive based on the reformulation of numerical homogenization as a Bayesian inference problem in which a given PDE with rough coefficients (or multiscale operator) is excited with noise (random right-hand side/source term) and one tries to estimate the value of the solution at a given point based on a finite number of observations. We apply this reformulation to the identification of bases for the numerical homogenization of arbitrary integro-differential equations and show that these bases have optimal recovery properties. In particular we show how rough polyharmonic splines can be rediscovered as the optimal solution of a Gaussian filtering problem.