论文标题
分数扩散算子的基础方法ii
A Reduced Basis Method For Fractional Diffusion Operators II
论文作者
论文摘要
我们提出了一种新颖的数值方案，以近似解决方案映射$ s \ mapsto u（s）：= \ Mathcal {l}^{ - s} f $与涉及分数椭圆运算符的部分微分方程。重新解释$ \ MATHCAL {l}^{ - s} $作为插值操作员允许我们得出$ u（s）$的积分表示，其中包括用于参数化反应反应 - 扩散问题的解决方案。我们建议在有限元方法的基础上降低基础策略，以近似其集成。与先前的作品不同，我们分析地推断出快照的选择。避免进一步离散化，在光谱环境中解释了积分，以直接评估替代物。它的计算归结为矩阵近似$ l $ l $ l $，其逆向低维空间，在该空间中，显式对角线是可行的。基础$ s $独立的简化空间的通用特征允许$（u（s））_ {s \ in（0,1）} $的整体近似。我们证明了指数融合率，并用各种数值示例确认分析。 在这项调查的第二部分中提出了进一步的改进，以避免反转$ L $。取而代之的是，我们将矩阵直接投射到减少空间，并评估其负分数功率。与前任的数值比较突出了其竞争性能。
We present a novel numerical scheme to approximate the solution map $s\mapsto u(s) := \mathcal{L}^{-s}f$ to partial differential equations involving fractional elliptic operators. Reinterpreting $\mathcal{L}^{-s}$ as interpolation operator allows us to derive an integral representation of $u(s)$ which includes solutions to parametrized reaction-diffusion problems. We propose a reduced basis strategy on top of a finite element method to approximate its integrand. Unlike prior works, we deduce the choice of snapshots for the reduced basis procedure analytically. Avoiding further discretization, the integral is interpreted in a spectral setting to evaluate the surrogate directly. Its computation boils down to a matrix approximation $L$ of the operator whose inverse is projected to a low-dimensional space, where explicit diagonalization is feasible. The universal character of the underlying $s$-independent reduced space allows the approximation of $(u(s))_{s\in(0,1)}$ in its entirety. We prove exponential convergence rates and confirm the analysis with a variety of numerical examples. Further improvements are proposed in the second part of this investigation to avoid inversion of $L$. Instead, we directly project the matrix to the reduced space, where its negative fractional power is evaluated. A numerical comparison with the predecessor highlights its competitive performance.
