-Solution time for the heat conduction problem is in itself not very interesting, as it is only a simple model problem. What is interesting for GPU kernels, such as the finite differences kernel, is that increased computational work often comes with a very small price, because the fast computations can be hidden by the relatively slower memory fetches. Therefore, we are able to improve the accuracy of the numerical solution via more accurate finite differences (larger stencil sizes), while improving the computational performance in terms of floating point operations per second (flops). Figure \ref{ch5:fig:stencilperformance} confirms, that larger stencils improve the kernel performance. Notice that even though these performance results are favorable compared to single core systems ($\sim 10$ GFlops double precision on a $2.5$-GHz processor), they are still far from their peak performance, e.g., $\sim 2.4$ TFlops single precision for the GeForce GTX590. The reason is that the kernel is bandwidth bound, i.e., performance is limited by the time it takes to move memory between the global GPU memory and the chip. The Tesla K20 performs better than the GeForce GTX590 because it obtains the highest bandwidth. Being bandwidth bound is a general limitation for matrix-vector-like operations that arise from the discretization of PDE problems. Only matrix-matrix multiplications, which have a high ratio of computations versus memory transactions, are able to reach near-optimal performance results~\cite{ch5:Kirk2010}. These kinds of operators are, however, rarely used to solve PDE problems.
+Solution time for the heat conduction problem is in itself not very interesting, as it is only a simple model problem. What is interesting for GPU kernels, such as the finite differences kernel, is that increased computational work often comes with a very small price, because the fast computations can be hidden by the relatively slower memory fetches. Therefore, we are able to improve the accuracy of the numerical solution via more accurate finite differences (larger stencil sizes), while improving the computational performance in terms of floating point operations per second (flops). Figure \ref{ch5:fig:stencilperformance} confirms, that larger stencils improve the kernel performance. Notice that even though these performance results are favorable compared to single core systems ($\sim 10$ GFlops double-precision on a $2.5$-GHz processor), they are still far from their peak performance, e.g., $\sim 2.4$ TFlops single-precision for the GeForce GTX590. The reason is that the kernel is bandwidth bound, i.e., performance is limited by the time it takes to move memory between the global GPU memory and the chip. The Tesla K20 performs better than the GeForce GTX590 because it obtains the highest bandwidth. Being bandwidth bound is a general limitation for matrix-vector-like operations that arise from the discretization of PDE problems. Only matrix-matrix multiplications, which have a high ratio of computations versus memory transactions, are able to reach near-optimal performance results~\cite{ch5:Kirk2010}. These kinds of operators are, however, rarely used to solve PDE problems.
