混合精度$s$-步Lanczos算法和共轭梯度算法
Mixed Precision $s$-step Lanczos and Conjugate Gradient Algorithms
摘要：
与经典的Lanczos算法相比，$s$步Lanczos变体具有通过渐进地降低每次迭代的同步成本来提高性能的潜力。然而，这是有代价的。尽管在数学上是等效的，但已知$s$-步变体在有限精度中表现得非常不同，相对于经典算法，具有更大的精度损失和收敛速度降低的可能性。先前已经表明，在$s$步骤版本中发生的错误遵循与经典算法中的错误相同的结构，但是增加了取决于在每个外循环中计算的$O(s)-$维Krylov基的条件数的平方的放大因子。由于这些$s$-步碱基的条件数随$s$增长（在某些情况下非常快），这限制了可以选择的参数$s$，从而限制了可以实现的性能。在这项工作中，我们表明，如果选择几个计算在$s$-步骤Lanczos进行了两倍的工作精度，误差项，然后只依赖于线性条件的$s$-步骤基地。这有可能大大改善算法的数值行为，对每次迭代性能的影响很小。我们的数值实验表明，改进的数值行为可能与混合精度的方法，也表明，这种改进的行为扩展到$s$-步CG算法在混合精度。
英文摘要：
Compared to the classical Lanczos algorithm, the $s$-step Lanczos variant has the potential to improve performance by asymptotically decreasing the synchronization cost per iteration. However, this comes at a cost. Despite being mathematically equivalent, the $s$-step variant is known to behave quite differently in finite precision, with potential for greater loss of accuracy and a decrease in the convergence rate relative to the classical algorithm. It has previously been shown that the errors that occur in the $s$-step version follow the same structure as the errors in the classical algorithm, but with the addition of an amplification factor that depends on the square of the condition number of the $O(s)-$dimensional Krylov bases computed in each outer loop. As the condition number of these $s$-step bases grows (in some cases very quickly) with $s$, this limits the parameter $s$ that can be chosen and thus limits the performance that can be achieved. In this work we show that if a select few computations in $s$-step Lanczos are performed in double the working precision, the error terms then depend only linearly on the conditioning of the $s$-step bases. This has the potential for drastically improving the numerical behavior of the algorithm with little impact on per-iteration performance. Our numerical experiments demonstrate the improved numerical behavior possible with the mixed precision approach, and also show that this improved behavior extends to the $s$-step CG algorithm in mixed precision.
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