This question is not about grammaticality, since the adverb is used grammatically in all cases. Rather it is about meaning: concretely is normally not used when one really means specifically, even though dictionaries give as a synonym of concrete the words specific and definite.
If a thing is described as concrete, we understand it to be "existing in material or physical form; real or solid; not abstract" [NOAD]. This dictionary goes on to list "specific; definite" as an alternate meaning, but then illustrates the use with this example:
I haven't got any concrete proof.
That still means "real" or "substantial" proof more than it does "specific" or "definite" proof. It means proof that you can physically observe.
Even when you use concretely as in your second example, its meaning shades toward the nuance of something that is not abstract. What if the "certain subset of such problems" is actually more abstract than the problems in general? Then you are simply making a poor word choice, one that militates against your meaning.
Once again, Mark Twain said it best:
The difference between the right word and almost the right word is the difference between lightning and a lightning bug.
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