Simultaneous deconvolution and system identification. If you have more than one kernel type, then you have a problem of dictionary learning.
Deconvolving a signal without knowing what it was convolved with (how blurry it is or what kind of blur it was). Say, reconstructing the pure sound of an instrument, and the sound of the echo in a church, from a recording made in a reverberant church, without knowing which church it was.
If you can find some way of making your problem linear-ish, and your signal is “sparse”, this turns out, amazingly, to be sometimes tractable.
TODO: cite Vetterli's grad student, name TBC.