Adaptive Wavelet Methods for SPDEs
Adaptive Wavelet Methods for SPDEs
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SPDE 的自适应小波方法
DOI：
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发表时间：
2014
期刊：
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通讯作者：
R. Schilling
中科院分区：
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作者：
P. A. Cioica;S. Dahlke;N. Döhring;S. Kinzel;F. Lindner;T. Raasch;K. Ritter;R. Schilling
我们回顾了在DFG-SPP1324项目“用于SPDEs的自适应小波方法”的背景下所获得的一系列结果。本课题主要研究了二阶抛物型随机偏微分方程解的自适应小波方法的构造和分析问题。在Besov空间(B_{au，au}^{S}(数学{O}))，1/τ=S/d+1/p，α>；0，p≥2的尺度上对解过程u进行了详细的正则性分析。这种尺度上的正则性是已知的，它决定了自适应小波算法和其他非线性逼近方案可以达到的收敛顺序。结果表明，对于特殊偏微分方程组的解，这一正则性一般超过了决定一致逼近格式收敛顺序的(L{p}(数学{O}))-索博列夫正则性。我们还研究了右端随机的椭圆边值问题在(数学{O})上的非线性小波逼近。当将Rothe方法应用于抛物型随机方程时，这些问题自然会出现。介绍了一种适用于右手边的广义随机小波模型，研究了它的Besov正则性以及线性和非线性逼近。计算结果与实验结果吻合较好。
We review a series of results that have been obtained in the context of the DFG-SPP 1324 project “Adaptive wavelet methods for SPDEs”. This project has been concerned with the construction and analysis of adaptive wavelet methods for second order parabolic stochastic partial differential equations on bounded, possibly nonsmooth domains (mathcal{O}subset mathbb{R}^{d}). A detailed regularity analysis for the solution process u in the scale of Besov spaces (B_{ au, au }^{s}(mathcal{O})), 1∕τ = s∕d + 1∕p, α > 0, p ≥ 2, is presented. The regularity in this scale is known to determine the order of convergence that can be achieved by adaptive wavelet algorithms and other nonlinear approximation schemes. As it turns out, in general, for solutions of SPDEs this regularity exceeds the (L_{p}(mathcal{O}))-Sobolev regularity, which determines the order of convergence for uniform approximation schemes. We also study nonlinear wavelet approximation of elliptic boundary value problems on (mathcal{O}) with random right-hand side. Such problems appear naturally when applying Rothe’s method to the parabolic stochastic equation. A general stochastic wavelet model for the right-hand side is introduced and its Besov regularity as well as linear and nonlinear approximation is studied. The results are matched by computational experiments.