Fractional differential problems with numerical anti-reflective boundary conditions: a computational/precision analysis and numerical results

TL;DR

The paper investigates numerical anti-reflective boundary conditions for time-dependent fractional diffusion in 1D open domains, aiming to suppress boundary ringing while maintaining $O(N\log N)$ complexity. It develops two BC families—anti-symmetric and anti-reflective—and derives structured discretizations ${A_{\beta}}^{anti}$ and ${A_{\beta}}^{antiR}$ whose spectra are favorable for iterative solvers; a truncated anti-reflective transform further enables a direct $O(N\log N)$ solver via fast sine transforms. A key theoretical contribution is the decomposition ${A_0}^{anti} = {S_0} + {R_0}^{anti}$ with ${S_0}$ SPD and low-rank boundary corrections, coupled with extensive spectral and GMRES-based experiments showing robust convergence and accuracy across fractional orders $\alpha \in (1,2)$ and coupling parameter $\beta$. The work also demonstrates the practicality of truncation to the anti-reflective algebra, enabling efficient solutions and highlighting the potential for extensions to higher dimensions, non-homogeneous BCs, and non-equispaced grids to further mitigate boundary artifacts in fractional PDEs.

Abstract

Twenty years ago the anti-reflective numerical boundary conditions (BCs) were introduced in a context of signal processing and imaging, for increasing the quality of the reconstruction of a blurred signal/image contaminated by noise and for reducing the overall complexity to that of few fast sine transforms i.e. to $O(N\log N)$ real arithmetic operations, where $N$ is the number of pixels. Here for quality of reconstruction we mean a better accuracy and the elimination of boundary artifacts, called ringing effects. Now we propose numerical anti-reflective BCs in the context of nonlocal problems of fractional differential type: the goals are the same i.e. a smaller approximation error and the reduction of boundary artifacts. In the latter setting, we compare various types of numerical BCs, including the anti-symmetric ones considered in the case of fractional differential problems for modeling reasons. More in detail, given important similarities between anti-symmetric and anti-reflective BCs, we compare them from the perspective of computational efficiency, by considering nontruncated and truncated versions and also other standard numerical BCs such as reflective/Neumann. Several numerical tests, tables, and visualizations are provided and critically discussed. The conclusion is that the truncated numerical anti-reflective BCs perform better, both in terms of accuracy and low computational cost.

Figures (11)

• Figure 1: Eigenvalue distribution of ${A_{0}^\mathrm{anti}}$ with size $16000$ for $\alpha=1.2, 1.5, 1.8$ and absolute error with respect to the generating function $f_{\alpha,T_{0}}(\theta)$.
• Figure 2: Effect of the numerical anti-symmetric and anti-reflective BCs on the matrix structure.
• Figure 3: Truncated anti-symmetric and anti-reflective boundaries effect on the matrix structure.
• Figure 4: Absolute error vs $x$ at $t \in \{2.e-3,1,2\}$ and Maximal absolute error vs $t$ - Implicit Euler, $\alpha=1.2$ - Homogeneus Dirichlet BCs.
• Figure 5: Absolute error vs $x$ at $t \in \{2.e-3,1,2\}$ and Maximal absolute error vs $t$ - Implicit Euler, $\alpha=1.4$ - Homogeneus Dirichlet BCs.