Abstract Space fractional diffusion models generally lead to dense discrete matrix operators, which lead to substantial computational challenges when the system size becomes large. For a state of size N , full representation of a fractional diffusion matrix would require O ( N 2 ) memory storage requirement, with a similar estimate for matrix–vector products. In this work, we present H 2 matrix representation and algorithms that are amenable to efficient implementation on GPUs, and that can reduce the cost of storing these operators to O ( N ) asymptotically. Matrix–vector multiplications can be performed in asymptotically linear time as well. Performance of the algorithms is assessed in light of 2D simulations of space fractional diffusion equation with constant diffusivity. Attention is focused on smooth particle approximation of the governing equations, which lead to discrete operators involving explicit radial kernels. The algorithms are first tested using the fundamental solution of the unforced space fractional diffusion equation in an unbounded domain, and then for the steady, forced, fractional diffusion equation in a bounded domain. Both matrix-inverse and pseudo-transient solution approaches are considered in the latter case. Our experiments show that the construction of the fractional diffusion matrix, the matrix–vector multiplication, and the generation of an approximate inverse pre-conditioner all perform very well on a single GPU on 2D problems with N in the range 1 0 5 – 1 0 6 . In addition, the tests also showed that, for the entire range of parameters and fractional orders considered, results obtained using the H 2 approximations were in close agreement with results obtained using dense operators, and exhibited the same spatial order of convergence. Overall, the present experiences showed that the H 2 matrix framework promises to provide practical means to handle large-scale space fractional diffusion models in several space dimensions, at a computational cost that is asymptotically similar to the cost of handling classical diffusion equations.

中文翻译：

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空间分数扩散方程的分层矩阵近似

摘要 空间分数扩散模型通常会导致密集的离散矩阵算子，当系统规模变大时，这会导致大量的计算挑战。对于大小为 N 的状态，分数扩散矩阵的完整表示需要 O ( N 2 ) 内存存储要求，矩阵-向量乘积的估计值类似。在这项工作中，我们提出了 H 2 矩阵表示和算法，它们适合在 GPU 上有效实现，并且可以将这些运算符的存储成本渐近地降低到 O ( N )。矩阵向量乘法也可以在渐近线性时间内执行。根据具有恒定扩散率的空间分数扩散方程的 2D 模拟来评估算法的性能。注意力集中在控制方程的平滑粒子近似上，这导致离散算子涉及显式径向核。这些算法首先使用无界域中非受迫空间分数扩散方程的基本解进行测试，然后使用有界域中的稳定、强制、分数扩散方程进行测试。在后一种情况下，矩阵求逆和伪瞬态求解方法都被考虑在内。我们的实验表明，分数扩散矩阵的构建、矩阵-向量乘法和近似逆预处理器的生成都在单个 GPU 上在 N 范围为 1 0 5 – 1 0 的二维问题上表现良好6 . 此外，测试还表明，对于所考虑的整个参数范围和分数阶数，使用 H 2 近似获得的结果与使用密集算子获得的结果非常一致，并表现出相同的空间收敛顺序。总的来说，目前的经验表明，H 2 矩阵框架有望提供在几个空间维度上处理大规模空间分数扩散模型的实用方法，其计算成本与处理经典扩散方程的成本渐近相似。