Fast algorithms for Helmholtz Green's functions

Abstract

The formal representation of the quasi-periodic Helmholtz Green's function obtained by the method of images is only conditionally convergent and, thus, requires an appropriate summation convention for its evaluation. Instead of using this formal sum, we derive a candidate Green's function as a sum of two rapidly convergent series, one to be applied in the spatial domain and the other in the Fourier domain (as in Ewald's method). We prove that this representation of Green's function satisfies the Helmholtz equation with the quasi-periodic condition and, furthermore, leads to a fast algorithm for its application as an operator.We approximate the spatial series by a short sum of separable functions given by Gaussians in each variable. For the series in the Fourier domain, we exploit the exponential decay of its terms to truncate it. We use fast and accurate algorithms for convolving functions with this approximation of the quasi-periodic Green's function. The resulting method yields a fast solver for the Helmholtz equation with the quasi-periodic boundary condition. The algorithm is adaptive in the spatial domain and its performance does not significantly deteriorate when Green's function is applied to discontinuous functions or potentials with singularities. We also construct Helmholtz Green's functions with Dirichlet, Neumann or mixed boundary conditions on simple domains and use a modification of the fast algorithm for the quasi-periodic Green's function to apply them.The complexity, in dimension dR2, of these algorithms is O(k d log kCC(log e K1 ) d ), where e is the desired accuracy, k is proportional to the number of wavelengths contained in the computational domain and C is a constant. We illustrate our approach with examples.

Tensor-Structured Preconditioners and Approximate Inverse of Elliptic Operators in ℝ d

“…To the best of our knowledge, low separation rank tensor approximation for oscillating kernels has not yet been addressed in the literature (cf. the asymptotically optimal methods in the volume size based on the fast multipole type representations [5,6]). …”

“…Such representations allow the 3D volume and boundary calculations with sublinear cost O(| log ε| κn), even in the case κ = O(n). Notice that methods presented in [5,6] lead to the linear complexity in the volume O(κ 3 log κ + C(| log ε|) 3 ).…”

“…To the best of our knowledge, the low rank tensor-structured approximation for oscillating kernels has not yet been addressed in the literature (cf. the asymptotically optimal method in the volume size based on the fast multipole type representations [5,6]). In the present paper we construct and prove the rank-O(κ) separable approximation for the oscillating kernel e −iκ x / x , x ∈ R d , κ ∈ R + .…”

Fast convolution with the free space Helmholtz Green’s function

“…The real part is approximated as a sum of two terms which are applied separately, one in the spatial and the other in the Fourier domain. We note that this splitting between domains is the key idea in Ewald's method for evaluating lattice sums [1], and we elaborate further on this connection later (see also [2]). …”

Event Tracking for Real-Time Unaware Sensitivity Analysis (EventTracker)

“…The concept of Green's function stems from the fact that the total output of a linear time-invariant system can be formulated by a summation of terms that adds all outputs of the system for all single points [25]. In other words, each continuous function could be replaced by an infinite sum of delta functions whose distances approach zero.…”