Kolmogorov widths and low-rank approximations of parametric elliptic PDEs
Kolmogorov widths and low-rank approximations of parametric elliptic PDEs
复制标题
参数椭圆偏微分方程的柯尔莫哥洛夫宽度和低阶近似
DOI：
10.1090/mcom/3132
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
A. Cohen
中科院分区：
文献类型：
--
作者：
M. Bachmayr;A. Cohen
研究了由扩散系数参数化的椭圆扩散偏微分方程族的柯尔莫哥洛夫 n 宽度和低阶近似。 n 宽度的衰减可以通过使用参数变量中的多项式的最佳 n 项近似实现的误差来控制。然而，我们证明，在某些相关实例中，扩散系数在物理域的分区上是分段常数，n 宽度表现出明显更快的衰减。这反过来又为处理此类参数偏微分方程时简化基或 POD 方法的快速收敛提供了理论依据。我们的结果得到了数值实验的证实，这也揭示了分区几何形状对 n 宽度衰减的影响。
Kolmogorov n-widths and low-rank approximations are studied for families of ellip-tic diffusion PDEs parametrized by the diffusion coefficients. The decay of the n-widths can be controlled by that of the error achieved by best n-term approximations using polynomials in the parametric variable. However, we prove that in certain relevant instances where the diffusion coefficients are piecewise constant over a partition of the physical domain, the n-widths exhibit significantly faster decay. This, in turn, yields a theoretical justification of the fast convergence of reduced basis or POD methods when treating such parametric PDEs. Our results are confirmed by numerical experiments, which also reveal the influence of the partition geometry on the decay of the n-widths.