This lovely fall bloomer is Japanese Anemone (Anemone x hybrida). They do have persistent root systems that are difficult to remove, making them sometimes a thug in a garden, but also a lasting, old-fashioned favorite of many. Here is information from the University of Illinois Extension: http://extension.illinois.edu/hortanswers/plantdetail.cfm?PlantID=703&PlantTypeID=2
If you need to thin or remove them from areas, hand removal is possible with persistent digging over a year or more. Herbicide use is also an option, follow label instructions.