Wave propagation using bases for bandlimited functions

Abstract

We develop a two-dimensional solver for the acoustic wave equation with spatially varying coefficients. In what is a new approach, we use a basis of approximate prolate spheroidal wavefunctions and construct derivative operators that incorporate boundary and interface conditions. Writing the wave equation as a first-order system, we evolve the equation in time using the matrix exponential. Computation of the matrix exponential requires efficient representation of operators in two dimensions and for this purpose we use short sums of one-dimensional operators. We also use a partitioned low-rank representation in one dimension to further speed up the algorithm. We demonstrate that the method significantly reduces numerical dispersion and computational time when compared with a fourth-order finite difference scheme in space and an explicit fourth-order Runge-Kutta solver in time.

Spectral Methods Based on Prolate Spheroidal Wave Functions for Hyperbolic PDEs

“…In [4,5], the author studied the feasibility of using PSWFs as the basis functions in spectral element methods. More recently, in [3] Beylkin and Sandberg developed a two-dimensional solver for the acoustic wave equation by using a basis of approximate PSWFs. However, even basic aspects of approximation and stability theory for methods based on PSWFs remain unknown.…”

Compressed sensing with preconditioning for sparse recovery with subsampled matrices of Slepian prolate functions

“…2 Independent and identically distributed it holds that, with probability at least 1 − N −γ(log s) 3 , the restricted isometry constant δ s of [7]). The stronger restriction m ≥ Cµ 2 s(log N ) 4 where µ is the concentration measure parameter (as studied in e.g.…”

“…The stronger restriction m ≥ Cµ 2 s(log N ) 4 where µ is the concentration measure parameter (as studied in e.g. [11]) yields directly the exact recovery property with higher probability 1 − N −γ(log N ) 3 (which is independent of s).…”

“…• A direct argument hints that one shouldn't expect any uniform bound in the infinity norm to exist for k ∈ N. Indeed, according to Proposition 1 in [3], the following estimate holds: [50] (see formula (11)) with identical normalization:…”

Discrete Symbol Calculus

“…Separation ideas, with an adapted notion of operator calculus, have also been suggested for solving the wave equation; two examples are [6] and [13].…”