一阶和二阶发展方程的良态直接PinT算法
A well-conditioned direct PinT algorithm for first-and second-order evolutionary equations
摘要：
在本文中，我们提出了一个直接并行时间（PinT）算法的时间依赖的问题，一阶或二阶导数。我们使用二阶边值方法作为时间积分器，导致三对角时间离散矩阵。而不是解决相应的所有在一次系统迭代，我们对角化的时间离散矩阵，从而产生一个直接的并行实现在所有的时间水平。该方法的关键问题是特征向量矩阵$V$的条件数如何随着$n$的增加而增长，其中$n$是时间级别的数量。大的条件数会导致对角化过程中的大的圆度误差，严重污染数值精度。基于特征方程与切比雪夫多项式之间的一种新的联系，我们给出了计算$V$和$V^{-1}$的显式公式，并证明了$\mathrm{Cond}_2(V)=\mathcal{O}(n^{2})$ .这意味着对角化过程是良态的，并且圆形误差仅随着$n$增长而适度增加，因此，与其他直接PinT算法相比，可以使用大得多的$n$来产生令人满意的并行性。并行机上的数值结果支持我们的研究结果，其中超过60倍的加速比实现了256个核心。
英文摘要：
In this paper, we propose a direct parallel-in-time (PinT) algorithm for time-dependent problems with first- or second-order derivative. We use a second-order boundary value method as the time integrator that leads to a tridiagonal time discretization matrix. Instead of solving the corresponding all-at-once system iteratively, we diagonalize the time discretization matrix, which yields a direct parallel implementation across all time levels. A crucial issue on this methodology is how the condition number of the eigenvector matrix $V$ grows as $n$ is increased, where $n$ is the number of time levels. A large condition number leads to large roundoff error in the diagonalization procedure, which could seriously pollute the numerical accuracy. Based on a novel connection between the characteristic equation and the Chebyshev polynomials, we present explicit formulas for computing $V$ and $V^{-1}$, by which we prove that $\mathrm{Cond}_2(V)=\mathcal{O}(n^{2})$. This implies that the diagonalization process is well-conditioned and the roundoff error only increases moderately as $n$ grows and thus, compared to other direct PinT algorithms, a much larger $n$ can be used to yield satisfactory parallelism. Numerical results on parallel machine are given to support our findings, where over 60 times speedup is achieved with 256 cores.
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