  Compressive sampling has been widely used for sparse polynomial chaos (PC)
approximation of stochastic functions. The recovery accuracy of compressive
sampling highly depends on the incoherence properties of the measurement
matrix. In this paper, we consider preconditioning the underdetermined system
of equations that is to be solved. Premultiplying a linear equation system by a
non-singular matrix results in an equivalent equation system, but it can
potentially improve the incoherence properties of the resulting preconditioned
measurement matrix and lead to a better recovery accuracy. When measurements
are noisy, however, preconditioning can also potentially result in a worse
signal-to-noise ratio, thereby deteriorating recovery accuracy. In this work,
we propose a preconditioning scheme that improves the incoherence properties of
measurement matrix and at the same time prevents undesirable deterioration of
signal-to-noise ratio. We provide theoretical motivations and numerical
examples that demonstrate the promise of the proposed approach in improving the
accuracy of estimated polynomial chaos expansions.
