We describe a model where an independent component problem and a related linear inverse problem are modelled simultaneously, and construct an algorithm which in some circumstances produces demixing matrices of better quality than the basic ICA algorithms. The effect is achieved by adding a penalty term, motivated by the inverse problem, to the ICA objective function. Our method is related to the idea, which has received some attention in the brain imaging context, that solutions of independent component problems can be used as a basis for inverse methods.