  Component separation is one of the key stages of any modern, cosmic microwave
background (CMB) data analysis pipeline. It is an inherently non-linear
procedure and typically involves a series of sequential solutions of linear
systems with similar, albeit not identical system matrices, derived for
different data models of the same data set. Sequences of this kind arise for
instance in the maximization of the data likelihood with respect to foreground
parameters or sampling of their posterior distribution. However, they are also
common in many other contexts. In this work we consider solving the component
separation problem directly in the measurement (time) domain, which can have a
number of important advantageous over the more standard pixel-based methods, in
particular if non-negligible time-domain noise correlations are present as it
is commonly the case. The time-domain based approach implies, however,
significant computational effort due to the need to manipulate the full volume
of time-domain data set. To address this challenge, we propose and study
efficient solvers adapted to solving time-domain-based, component separation
systems and their sequences and which are capable of capitalizing on
information derived from the previous solutions. This is achieved either via
adapting the initial guess of the subsequent system or through a so-called
subspace recycling, which allows to construct progressively more efficient,
two-level preconditioners. We report an overall speed-up over solving the
systems independently of a factor of nearly 7, or 5, in the worked examples
inspired respectively by the likelihood maximization and likelihood sampling
procedures we consider in this work.
