Accuracy and Stability of Low-rank Linear Solvers
Intervenant(s) : Théo Mary (Manchester)
In many applications requiring the solution of a linear system Ax=b, the matrix A has been shown to have off-diagonal blocks of low numerical rank. Several low-rank matrix formats have been proposed to exploit this property depending on how the block partitioning of the matrix is computed, e.g., the flat block low-rank (BLR) format, its multilevel (MBLR) extension, and hierarchical formats (H, HSS, etc.). In this talk, we will discuss several topics regarding the numerical behavior of these low-rank linear solvers in finite precision arithmetic. First, we will investigate the use of these solvers as low-accuracy preconditioners for iterative methods and present a strategy to improve their convergence when the matrix is ill-conditioned. Then, we will analyze the effect of rounding errors on the numerical stability of the BLR factorization. Finally, we will use a probabilistic model of floating-point computations to improve the traditional worst-case bounds and predict more realistically the average-case numerical behavior of large-scale, low-precision linear solvers.
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