Multiscale Discrete Approximations of Fourier Integral Operators Associated with Canonical Transformations and Caustics

Abstract

Abstract. We develop an algorithm for the computation of general Fourier integral operators associated with canonical graphs. The algorithm is based on dyadic parabolic decomposition using wave packets and enables the discrete approximate evaluation of the action of such operators on data in the presence of caustics. The procedure consists in the construction of a universal operator representation through the introduction of locally singularity-resolving diffeomorphisms, enabling the application of wave packet driven computation, and in the construction of the associated pseudo-differential joint-partition of unity on the canonical graphs. We apply the method to a parametrix of the wave equation in the vicinity of a cusp singularity.1. Introduction. In this paper, we develop an algorithm for applying Fourier integral operators associated with canonical graphs using wave packets. To arrive at such an algorithm, we construct a universal oscillatory integral representation of the kernels of these Fourier integral operators by introducing singularity resolving diffeomorphisms where caustics occur. The universal representation is of the form such that the algorithm based on the dyadic parabolic decomposition of phase space previously developed by the authors applies [2]. We refer to [7,8,10,11] for related computational methods aiming at the evaluation of the action of Fourier integral operators.The algorithm comprises a geometrical component, bringing the local representations in universal form, and a wave packet component which yields the application of the local operators. Here, we develop the geometrical component, which consists of the following steps. First we determine the location of caustics on the canonical relation of the Fourier integral operator. For each point on a caustic we determine the associated specific rank deficiency and construct an appropriate diffeomorphism, resolving the caustic in open neighborhoods of this point. We determine the (local) phase function of the composition of the Fourier integral operator and the inverse of the diffeomorphism in terms of universal coordinates and detect the largest set on which it is defined. We evaluate the preimage of this set on the canonical relation. We continue this procedure until the caustic is covered with overlapping sets, associated with diffeomorphisms for the corresponding rank deficiencies. Then we repeat the steps for each caustic and arrive at a collection of open sets covering the canonical relation.In the special case of Fourier integral operators corresponding to parametrices of evolution equations, for isotropic media, an alternative approach for obtaining solutions in the vicinity of caustics, based on a re-decomposition strategy following a multi-product representation of the propagator, has been proposed previously [2,19,20]. Unlike multi-product representations, our construction does not involve a subdivision of the evolution parameter and yields a single-step computation. Moreover, it is valid for the general class of Fourier integra...

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“…6 for Each block partitioned by discontinuous point sets do 7 Set τ = 2π, since it is not necessary to detect discontinuity here. 8 Apply Algorithm 3 to recover the first row and the first column of each block. 9 Apply Algorithm 3 to recover the second and the third columns of each block.…”

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“…2. This partitioning into time intervals is detected numerically from the points of rank-deficiency of the matrix W t 1 of the Hamiltonian system as detailed in [11] and indicated in Fig. 3 (right).…”