对数正态达西问题递归一阶矩方程的正则性和稀疏逼近
Regularity and sparse approximation of the recursive first moment equations for the lognormal Darcy problem
摘要：
研究了具有对数正态渗透率场的达西边值问题。我们采用摄动方法，将解在系数的名义值附近展开成泰勒级数，并用其泰勒多项式的期望值逼近偏微分方程随机解的期望值.形式上推导了泰勒多项式的期望值所满足的递推确定性方程（一阶矩方程）。在具有混合正则性的索伯列夫空间-值Hölder空间中，证明了该递推式的适定性和正则性.递归的一阶矩方程，然后通过稀疏逼近技术离散，并推导出的收敛速度。
英文摘要：
We study the Darcy boundary value problem with log-normal permeability field. We adopt a perturbation approach, expanding the solution in Taylor series around the nominal value of the coefficient, and approximating the expected value of the stochastic solution of the PDE by the expected value of its Taylor polynomial. The recursive deterministic equation satisfied by the expected value of the Taylor polynomial (first moment equation) is formally derived. Well-posedness and regularity results for the recursion are proved to hold in Sobolev space-valued Hölder spaces with mixed regularity. The recursive first moment equation is then discretized by means of a sparse approximation technique, and the convergence rates are derived.
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