I plan to upload photos Sunday night. Thanks.

Well, I took photos today and I'm embarrassed to say it probably isn't Ivy. There are (at least) two different plants growing on the property walls. The super thick one is probably some sort of shrub. This is from an angle where it was trimmed back near a gate opening so you can see how deep it is from the wall.

Some closeups of the leaves for ID purposes:

This is a brown patch which might have been under a tree that was cut down. (There is some of the other plant trying to grow there too.)

The other plant (on a different wall) is not as thick and has much smaller leaves, but needs attention.

Right - I'm afraid I don't know what those plants for certain is but it is definitely not a Hedera species ie it is not Ivy. I believe the bottom one is Creeping Fig. I suggest you post on the Name That Plant Forum to get a definite id.

Name that Plant ID'd it as creeping fig.

If creeping fig, then to get back to original question, and please, I'm no creeping fig expert, but I do believe genus Ficus to be one of vigorous species, the plant could probably be cut back very heavily and assuming good growing conditions, ie rainfall at some point, should grow back abundantly.

Thanks. We are having a drought in Southern California, very little rain. But I can water it.

Yes, looks like some pretty good rains coming up for you in the near-term. But yes, all plantings need water on occasion. Especially where you have trimmed a plant way back-such as this case may end up-it is advisable to get the recovery going right with moist soil. You'll succeed.