The method / path:
1. Shade squares touching "1"s on a side. Shade squares touching two different numbers on a side.
2. For 2's that can only be completed on one of two adjacent sides, shade the square that would prevent that. Fill in what you can around them. Fill in areas on the right edge which no number can reach.
3. The two topmost 20's must both go upward (there is not enough space in any other direction). In particular, the left one must go left, around the 5; the right one must go right. Fill in what you can, including making sure no shaded squares are isolated. Shade in unreachable parts in the upper left.
4. The third-highest 20 must pass downward between the 3 and 2, as there is no other path for it. Fill in what you can in that area. Also, avoid shaded square isolation in the lower right.
5. 2x2 shaded squares must be avoided. In particular, the upper right 20 must go into the upper area and the area off to the right to avoid shaded 2x2s, and also the shaded squares in the upper right must connect below them to avoid being isolated, as there is no other path. Fill in what you can with the upper left 20.
6. Do more avoiding of 2x2 shaded areas, and avoiding isolated shaded areas. This includes one I missed earlier, above the top left 20. Of the two leftmost 20s, the lower one must go down and left at least somewhat, or it will not give the upper one enough space to fill.
7. The leftmost 20 cannot connect to the far left area below the 1 without isolating a shaded area, so this far left part must be connected with the lowest 20. Then, we can reason that the leftmost 20 must travel down and then right, over to the lower right area, as nothing else in that vicinity (the 4, or 5, or 6) has a far enough reach to prevent any 2x2 shaded cell at the bottom. Once this is done, most everything else falls into place, until completion.