As a former landscape designer/contractor I've had some experience with invasive plants. For example, we have a devilish plant called quack grass that is now causing big problems for agriculture. Of course just like the knotweed/false bamboo, some genius brought it to this country because he fancied it would make good forage for cattle. Same with starlings, BTW; some European transplant decided he "missed" their "song."
Anyhow, quack grass looks pretty much like normal turf grasses, but like knotweed, it sprouts from a network of underground rhizomes (like a root, but unlike a normal root, which must remain attached to the main plant in order to live; these things have a life of their own, shoot out in all directions, and sprout new plants). The rhizomes will dive under sidewalks, even across an entire roadway, and then pop up with fresh new leaves. You can eradicate everything, every plant showing, and dig up all the rhizomes you can find, but if you leave even one teensy fragment of the rhizome, a new plant springs up and the spreading starts again.
I agree with the article posted, that a multi-strategy approach is best, and you should prepare for a fight lasting several years.
Adding a couple of things.
I'd be tempted to make that location my burn pile.
Sometimes the best thing is to change the growing conditions. Change the moisture, sun exposure, fertility, drainage, or pH. For example, one thing that actually kills the quack grass is.....baking soda. Apparently it's intolerant to the salt or the pH change. Another example, dollarweed (a southern weed) likes it wet; reduce your watering. Since knotweed can't tolerate shade, once you get at least a partial handle on it, maybe plant some trees.
Finally, a lot of these invasive species hitchhike to your property in potted plants/trees purchased from nurseries. I've started pulling apart the rootball to remove any weeds (and any rhizomes).
Don't let up. Neglect it for a few months and you have wasted all your prior efforts. I imported another nasty rhizome-spreading weed (red sorrel) in some strawberries, which I kept pulling but spread and spread. Finally I smothered the whole area (half my garden) with some compost, thought I was good, but then when I uncovered it, it popped up again bigger than ever and has now taken over most of my laboriously composted garden. I will have to spread a couple feet of compost over the whole thing, plus extending at least two feet from the edges and leave it for at least one year, maybe two.....and dig another garden. Also, since sorrel likes acid soil, I will lime the heck out of it first.
Good luck, and BE CAREFUL WITH STORE-BOUGHT PLANTS.
