  For a large class of orthogonal basis functions, there has been a recent
identification of expansion methods for computing accurate, stable
approximations of a quantity of interest. This paper presents, within the
context of uncertainty quantification, a practical implementation using basis
adaptation, and coherence motivated sampling, which under assumptions has
satisfying guarantees. This implementation is referred to as Basis Adaptive
Sample Efficient Polynomial Chaos (BASE-PC). A key component of this is the use
of anisotropic polynomial order which admits evolving global bases for
approximation in an efficient manner, leading to consistently stable
approximation for a practical class of smooth functionals. This fully adaptive,
non-intrusive method, requires no a priori information of the solution, and has
satisfying theoretical guarantees of recovery. A key contribution to stability
is the use of a presented correction sampling for coherence-optimal sampling in
order to improve stability and accuracy within the adaptive basis scheme.
Theoretically, the method may dramatically reduce the impact of dimensionality
in function approximation, and numerically the method is demonstrated to
perform well on problems with dimension up to 1000.
