On the spectral equivalence of hierarchical matrix preconditioners for elliptic problems
On the spectral equivalence of hierarchical matrix preconditioners for elliptic problems
复制标题
椭圆问题层次矩阵预处理器的谱等价性
作者：
M. Bebendorf;M. Bollhöfer;M. Bratsch
。我们将讨论二阶椭圆问题的分层矩阵近似的谱等价性。我们的理论将表明，层次矩阵 Cholesky 分解的修改变体在使用适应的截断阈值时保留测试向量，同时将块截断到较低的等级，将导致谱等效近似。我们的理论还涵盖了通常的分层 Cholesky 分解，它不保留测试向量，但期望明显更具限制性的阈值适应以获得光谱等效近似。数值实验表明，截断参数的自适应对于传统的分层 Cholesky 预处理器获得网格无关收敛是必要的，而保留测试向量的变体在实践中即使使用固定参数也能很好地工作。
. We will discuss the spectral equivalence of hierarchical matrix approximations for second order elliptic problems. Our theory will show that a modified variant of the hierarchical matrix Cholesky decomposition which preserves test vectors while truncating blocks to lower rank will lead to a spectrally equivalent approximation when using an adapted truncation threshold. Our theory also covers the usual hierarchical Cholesky decomposition which does not preserve test vectors but expects a significantly more restrictive threshold adaption to obtain a spectrally equivalent approximation. Numerical experiments indicate that the adaption of the truncation parameter seems to be necessary for the traditional hierarchical Cholesky preconditioner to obtain mesh-independent convergence while the variant which preserves test vectors works in practice quite well even with a fixed parameter.