  We propose a preconditioner that can accelerate the rate of convergence of
the Multiple Shooting Shadowing (MSS) method. This recently proposed method can
be used to compute derivatives of time-averaged objectives (also known as
sensitivities) to system parameter(s) for chaotic systems. We propose a block
diagonal preconditioner, which is based on a partial singular value
decomposition of the MSS constraint matrix. The preconditioner can be computed
using matrix-vector products only (i.e. it is matrix-free) and is fully
parallelised in the time domain. Two chaotic systems are considered, the Lorenz
system and the 1D Kuramoto Sivashinsky equation. Combination of the
preconditioner with a regularisation method leads to tight bracketing of the
eigenvalues to a narrow range. This combination results in a significant
reduction in the number of iterations, and renders the convergence rate almost
independent of the number of degrees of freedom of the system, and the length
of the trajectory that is used to compute the time-averaged objective. This can
potentially allow the method to be used for large chaotic systems (such as
turbulent flows) and optimal control applications. The singular value
decomposition of the constraint matrix can also be used to quantify the effect
of the system condition on the accuracy of the sensitivities. In fact,
neglecting the singular modes affected by noise, we recover the correct values
of sensitivity that match closely with those obtained with finite differences
for the Kuramoto Sivashinsky equation in the light turbulent regime.
