论文标题
混合精度应用程序的扰动runge-kutta方法
Perturbed Runge-Kutta methods for mixed precision applications
论文作者
论文摘要
在这项工作中，我们考虑了一种混合精度方法来加速植入多阶段方法。我们表明，可以设计runge-kutta方法，以便可以将某些昂贵的中间计算作为较低精确的计算执行，而不会不利影响整体解决方案的准确性。特别是，设计经过正确设计的runge-kutta方法会逐渐消除初始阶段中犯下的错误。当我们考虑隐式runge-kutta方法时，这是特别感兴趣的。在这种情况下，如果解决方案的精度较低（或等效地具有较低的公差），则阶段值的隐式计算可以更快。我们提供了一个通用的理论添加框架，用于设计混合精度runge-kutta方法，并使用此框架来得出此类方法的订单条件。接下来，我们展示如何使用这种方法使我们能够利用隐式求解器的低精度计算，同时在整体方法中保留高精度。我们通过数值研究介绍了某些混合精液隐式runge-kutta方法的行为，并证明了数值结果如何与理论框架匹配。这个新颖的混合精液隐式runge-kutta框架为许多此类方法的设计打开了大门。
In this work we consider a mixed precision approach to accelerate the implemetation of multi-stage methods. We show that Runge-Kutta methods can be designed so that certain costly intermediate computations can be performed as a lower-precision computation without adversely impacting the accuracy of the overall solution. In particular, a properly designed Runge-Kutta method will damp out the errors committed in the initial stages. This is of particular interest when we consider implicit Runge-Kutta methods. In such cases, the implicit computation of the stage values can be considerably faster if the solution can be of lower precision (or, equivalently, have a lower tolerance). We provide a general theoretical additive framework for designing mixed precision Runge-Kutta methods, and use this framework to derive order conditions for such methods. Next, we show how using this approach allows us to leverage low precision computation of the implicit solver while retaining high precision in the overall method. We present the behavior of some mixed-precision implicit Runge-Kutta methods through numerical studies, and demonstrate how the numerical results match with the theoretical framework. This novel mixed-precision implicit Runge-Kutta framework opens the door to the design of many such methods.
