Pseudo-differential representation of the metaplectic transform and its application to fast algorithms

Abstract

The metaplectic transform (MT) is a unitary integral mapping which is widely used in signal processing and can be viewed as a generalization of the Fourier transform. For a given function ψ on an N -dimensional continuous space q, the MT of ψ is parameterized by a rotation (or more generally, a linear symplectic transformation) of the 2N -dimensional phase space (q, p), where p is the wavevector space dual to q. Here, we derive a pseudo-differential form of the MT. For smallangle rotations, it readily yields asymptotic differential representations of the MT, which are easy to compute numerically. Rotations by larger angles are implemented as successive applications of K ≫ 1 small-angle MTs. The algorithm complexity scales as O(KN 3 Np), where Np is the number of grid points. We present a numerical implementation of this algorithm and discuss how to mitigate the two associated numerical instabilities.

Metaplectic geometrical optics for modeling caustics in uniform and non-uniform media

“…= ±1. The sign ambiguity in σ is of fundamental importance, as it contributes to the well-known phase shifts that a wavefield acquires upon touching a caustic [38,[54][55][56][57], the π/2 phase shift following reflection being a famous example. Note that when det B = 0, the right-hand side of Eq.…”

“…( 79) is no longer unitary. A detailed analysis [38] showed that this loss of unitarity results in the unbounded growth of high-wavenumber oscillations, which can be partially mitigated by using low-pass smoothing methods.…”

“…( 70)]. The corresponding algorithms are: (i) a curvature-dependent adaptive ray-tracing scheme [39], (ii) a Gram-Schmidt orthogonalization scheme for constructing the rotation matrices S t [39], (iii) a fast lineartime algorithm for performing the small-angle NIMTs from one tangent plane to the next [38,41], and (iv) a numerical steepest-descent quadrature rule for the efficient computation of Υ t [42]. We shall now briefly describe each algorithm separately.…”

“…As an attempt to resolve these remaining shortcomings, we have developed a new ray-tracing framework called metaplectic geometrical optics (MGO) [38][39][40][41][42]. In essence, MGO introduces the metaplectic rotation scheme into the general operator formalism of XGO to yield a formalism that (i) avoids caustics by construction, (ii) can be applied to any linear wave equation, including integro-differential wave equations that arise in kinetic treatments of plasma waves, (iii) includes an envelope equation along each ray that can readily diffraction and eventually mode conversion too, and (iv) is practical for implementation in a ray-tracing code, as exemplified by the several efficient algorithms for MGO that have already been developed.…”

Steepest-descent algorithm for simulating plasma-wave caustics via metaplectic geometrical optics

“…As part of ongoing work on MGO algorithms [31,32,34,35], here we present a quadrature rule for calculating MGO integrals based on numerical steepest-descent integration [36]. This algorithm emerges naturally from the MGO framework in that MGO integrals always contain saddlepoints that correspond to the ray contributions to the wavefield.…”