Both are correct. In Petrovski residence, the name functions as an attributive noun or noun adjunct (see https://en.wikipedia.org/wiki/Noun_adjunct). This is analogous to phrases such as tool box and bus stop. That is, it is a residence that happens to be characterized by the fact that its resident is Petrovski; the fact denoted by the phrase is the answer to the question "who lives in that residence?"
In Petrovski's residence, the use of the possessive indicates a relationship between the two nouns (not necessarily one of actual possession). In other words, the question answered by the phrase is "where does Petrovski live?"
It should be noted that if more than one Petrovski lives in a place, it could be called the Petrovskis' residence, in which case it would be answering the question "where do the Petrovskis live?"
(With regard to the question of actual possession, note that the residence or home might not even be real property. It could be a city or a country, as in, for example, He often traveled to Ukraine, but Russia was Petrovski's home. See also my answer to Use of the apostrophe when possession is not the issue.)