Minimal Sparse Sampling for Fourier-polynomial Chaos in Acoustic Scattering

Single frequency acoustic scattering from an uncertain surface (with sinusoidal components) admits an efficient Fourier-polynomial chaos (FPC) expansion of the acoustic field. The expansion coefficients are computed nonintrusively, i.e., by functional sampling from existing acoustic models. The structure of the acoustic decomposition permits sparse selection of FPC orders within the framework of the Smolyak construction. The main result shows a minimal, sparse sampling required to exactly reconstruct FPC expansions of Smolyak form. To this end, this paper defines two concepts: exactly discretizable orthonormal, function systems (EDO); and nested systems created by decimation or “fledging.” An EDO generalizes the Nyquist-Shannon sampling conditions (exact recovery of “bandlimited” functions given sufficient sampling) to multidimensional FPC expansions. EDO criteria replace the concept of polynomially exact quadrature. Fledging parallels the idea of sub-sampling for sub-bands, from higher to lower level. The FPC Smolyak construction is an EDO fledged from a full grid EDO. An EDO results exactly when the sampled FPC expansion can be inverted to find its coefficients. EDO fledging requires that the lower level (1) has grid points and expansion orders nested in the higher level, and (2) derives its map from the samples to the coefficients from the higher level map. The theory begins with a single dimension fledged EDO, since a tensor product of fledged EDOs yields a fledged tensor EDO. A sequence of nested EDO levels fledge recursively from the largest EDO. The Smolyak construction uses telescoping sums of tensor products up to a maximum level to develop nested EDO systems for sparse grids and orders. The Smolyak construction transform gives exactly the inverse of the weighted evaluation map, and that inverse has a condition number that expresses the numerical limitations of the Smolyak construction.