The solution strategy of the space-fractional modified Cahn-Hilliard equation as a tool for the gray value image inpainting model is studied. The existing strategies solve the convexity splitting scheme of the vector-valued Cahn-Hilliard model by Fourier spectral method. In this paper, we constructed a fast solver for the discretized linear systems possessing the saddle-point structure within block-Toeplitz-Toeplitz-block (BTTB) structure arising from the 2D space-fractional modified Cahn-Hilliard equation. The new solver enjoys computational advantage since circulant approximation and fast Fourier transforms (FFTs) can be used for solving the involved linear subsystems. Theoretical analysis shows the spectrum of the preconditioned matrix clusters around 1, which implies the fast convergence rate of the proposed preconditioner. Numerical examples are given to confirm the effectiveness of our method.

研究了空间分数修正Cahn-Hilliard方程作为灰度图像修复模型工具的求解策略。现有策略通过傅里叶谱方法求解向量值Cahn-Hilliard模型的凸性分裂方案。在本文中，我们为离散线性系统构建了一个快速求解器，该系统具有块-托普利茨-托普利茨块 (BTTB) 结构内的鞍点结构，该结构源自二维空间分数修正 Cahn-Hilliard 方程。新的求解器具有计算优势，因为循环逼近和快速傅立叶变换 (FFT) 可用于求解所涉及的线性子系统。理论分析显示预处理矩阵簇的频谱在 1 左右，这意味着所提出的预处理器的收敛速度很快。