We use several integrated on-chip converters as simulation examples
to measure running time and speedup. They include a Buck converter,
-a quasi-resonant flyback converter (shown in Fig.~\ref{fig:flyback}),
+a quasi-resonant flyback converter (shown in Figure~\ref{fig:flyback}),
and two boost converters.
Each converter is directly integrated with on-chip power grid networks,
-since the performance of converters should be studied with their loads and
+since the performance of the converters should be studied with their loads and
we can easily observe the waveforms at different nodes in a power
-grid (see Fig.~\ref{fig:pg} for a simplified power grid structure).
+grid (see Figure~\ref{fig:pg} for a simplified power grid structure).
-Fig.~\ref{fig:flyback_wave}
-and Fig.~\ref{fig:buck_wave}
-shows the waveform at output node of the resonant flyback converter
+Figure~\ref{fig:flyback_wave}
+and Figure~\ref{fig:buck_wave}
+show the waveform at output node of the resonant flyback converter
and the Buck converter.
Note that on the envelope curve, the darker
-dots in separated segments indicate the real simulation points were
+dots in separated segments indicate the real simulation points that were
calculated in those cycles, and the segments without dots are the
envelope jumps where no simulation were done.
It can be verified that the proposed Gear-2 envelope-following method
For the comparison of running time spent in solving
Newton update equation, Table~\ref{table:circuit} lists the time
-costed by direct method, explicit GMRES, matrix-free GMRES,
+cost by direct method, explicit GMRES, matrix-free GMRES,
and GPU matrix-free GMRES. All methods carry out the Gear-2 based
envelope-following method, but they handle the sensitivity and
equation solving in different implementation steps.
It is obvious that as long as the sensitivity matrix is explicitly formed,
-such as the cases in direct method and explicitly GMRES,
+such as in the cases of direct method and explicit GMRES,
the cost is much higher than the implicit methods.
-When matrix-free technique is applied to generate matrix-vector
+When the matrix-free technique is applied to generate matrix-vector
products implicitly, the computation cost is greatly reduced.
Thus, for the same example, implicit GMRES would be one order
of magnitude faster than explicit GMRES. Furthermore, our GPU parallel
