Data gets lost when reprojecting or mosaicking small areas

It has been observed that small input areas get lost when mosaicking data to a global map.
After serval days of investigating the problem, it turned out that the configured tileSize has an influence on the result. For some configurations, it worked (e.g. 513x513) and for others (e.g. 512x12) not.
The reason is that the data is omitted if the borders of the tile do not intersect with the data. In the attached image this scenario is depicted:

The data in red is surrounded by the tile borders but it does not intersect.

That's how the default JAI implementation of the Warp operation handles this.
We need to switch either to another implementation or provide our own.