Novel solutions of the Helmholtz equation and their application to diffraction

Abstract

This paper presents a series of novel representations for the solutions of the Helmholtz equation in a broad class of wedge-like domains including those with curvilinear, non-flat faces. These representations are obtained by an original method which combines ray theory with the probabilistic approach to partial differential equations and uses a specific technique to deal with a need for analytical continuation of the specified boundary function. The main results are reminiscent of the standard Feynman–Kac formula but differ in that the averaging over solutions of stochastic differential equations is replaced by averaging over the trajectories of a new two-scale random motion introduced here. The paper focuses on the development of the solutions and for this reason it includes only a brief outline of numerous applications, consequences, extensions and variations of the method, which include, but are not limited to, problems of diffraction and scattering, problems in three-dimensional domains and problems of wave propagation in non-homogeneous media.

A novel stochastic method for the solution of direct and inverse exterior elliptic problems

“…Although this term does not belong to the principal part of the elliptic operator, it introduces qualitative differences in handling the corresponding boundary value problem via stochastic calculus. We can see [16] that when trying to construct stochastic paths with drift and diffusion pertaining to the Helmholtz equation, then these paths must live in a complex (2n)−dimensional space in contrast to the Laplace equation where the trajectories belong to the real n−dimensional space.…”

“…Even when x comes away from D, the auxiliary attractor ξ, which also has the freedom to pull away 15 , is responsible to keep the ratio |Yi| |x−ξ| in the vicinity of unity. As far as the last assertion according to the remoteness of ξ is concerned, it is worthwhile to mention that in the regime of the mildly conditioned stochastic method, the pair of coaxial cones must be thick 16 and centered at a distant point ξ so that the whole conical structure embodies globally the region D. Then all the points of the surface ∂D have the opportunity to contribute to the mean value (3.38) as the well posedness of the direct b.v.p dictates. Finally the role of Q m (cos(Θ i )) in the denominator of (3.38) must not be underestimated since, for crossings on ∂D close to the cone walls, the function 1 Qm might take large values balancing the smallness of ( |Yi| |x−ξ| ) m+1 .…”

“…Our next goal is to work with the whole Helmholtz operator in order mainly to confront the scattering processes stemmed from acoustics or electromagnetism in the time harmonic regime with arbitrary frequency. We can see [16] that when trying to construct stochastic paths with drift and diffusion pertaining to the Helmholtz operator, then these paths must live in a complex (2n)−dimensional space in contrast to the Laplace equation where the trajectories belong to the real n−dimensional space. This increases the complexity of the problem, which is under current investigation by the authors [23].…”

Sensing and Multiscale Structure

“…This approach to sensing (inverse scattering) is apparently new, and is illustrated here through the example of raydensity fluctuations beyond a subfractal phase-changing screen (SPS). The concept of rays is of great utility in the theory of scattering, most famously in the shortwave limit [4], but also more generally [5]. The ray-density, denoted by R, describes intensity fluctuations induced by a SPS in an incoherent configuration (no interference).…”

A Monte Carlo method for solving the one-dimensional telegraph equations with boundary conditions

“…In the specific field of electromagnetism, several accelerating techniques have been proposed in the literature to solve probabilistically electrostatic problems, such as the floating random walk [6], and walking on spheres [24,25], capable to speed up the computation of the solution by means of a variable time step size. For time-dependent electromagnetic problems, in particular for the time harmonic solution of the wave equation described by the scalar Helmholtz equation, a probabilistic representation was proposed in [5]. This representation is based on a suitable transformation, which allows to transform the original problem into two set of equations, one of them amenable to be solved using the Feynman-Kac formula.…”