DIY Thursday: Getting rid of ivy

I don't mind ivy per se, not in the I hate it and want to rip every shred by its roots and mash it until it dies way. Unless it's taken over my back yard from a neighbor's uncontrolled infestation. I think ivy looks lovely around fences, arbors, pergolas, and the like. In that ornamental greenery kind of way. And until this house, I had that as a vague backyard plan. No more.

For 5 years I've tried to get rid of the ivy that's creeping up my trees, into my garden, and around what little grass...