We derive the high frequency limit of the Helmholtz equation with source term when the source is the sum of two point sources. We study it in terms of Wigner measures (quadratic observables). We prove that the Wigner measure associated with the solution satisfies a Liouville equation with, as source term, the sum of the source terms that would be created by each of the two point sources taken separately. The first step, and main difficulty, in our study is the obtention of uniform estimates on the solution. Then, from these bounds, we derive the source term in the Liouville equation together with the radiation condition at infinity satisfied by the Wigner measure.
In this article, we discuss quantitative Runge approximation properties for the acoustic Helmholtz equation and prove stability improvement results in the high frequency limit for an associated partial data inverse problem modelled on \cite{AU04, KU19}. The results rely on quantitative unique continuation estimates in suitable function spaces with explicit frequency dependence. We contrast the frequency dependence of interior Runge approximation results from non-convex and convex sets.
This paper studies for large frequency number $k>0$ the existence and multiplicity of solutions of the semilinear problem $$ -Δu -k^2 u=Q(x)|u|^{p-2}u\quad\text{ in }\mathbb{R}^N, \quad N\geq 2. $$ The exponent $p$ is subcritical and the coefficient $Q$ is continuous, nonnegative and satisfies the condition $$ \limsup_{|x|\to\infty}Q(x)<\sup_{x\in\mathbb{R}^N}Q(x). $$ In the limit $k\to\infty$, sequences of solutions associated to ground states of a dual equation are shown to concentrate, after rescaling, at global maximum points of the function $Q$.
Consider the one-dimensional stochastic Helmholtz equation where the source is assumed to be driven by the white noise. This paper concerns the stability analysis of the inverse random source problem which is to reconstruct the statistical properties of the source such as the mean and variance. Our results show that increasing stability can be obtained for the inverse problem by using suitable boundary data with multi-frequencies.
We consider the high frequency Helmholtz equation with a variable refraction index $n^2(x)$ ($x \in \R^d$), supplemented with a given high frequency source term supported near the origin $x=0$. A small absorption parameter $α_{\varepsilon}>0$ is added, which somehow prescribes a radiation condition at infinity for the considered Helmholtz equation. The semi-classical parameter is $\varepsilon>0$. We let $\eps$ and $\a_\eps$ go to zero {\em simultaneaously}. We study the question whether the indirectly prescribed radiation condition at infinity is satisfied {\em uniformly} along the asymptotic …
The paper is concerned with an inverse point source problem for the Helmholtz equation. It consists of recovering the locations and amplitudes of a finite number of radiative point sources inside a given inhomogeneous medium from the knowledge of a single boundary measurement. The main result of the paper is a new Hölder type stability estimate for the inversion under the assumption that the point sources are well separated. The proof of the stability is based on a combination of Carleman estimates and a technique for proving uniqueness of the Cauchy problem.
In this paper, a new model is proposed for the inverse random source scattering problem of the Helmholtz equation with attenuation. The source is assumed to be driven by a fractional Gaussian field whose covariance is represented by a classical pseudo-differential operator. The work contains three contributions. First, the connection is established between fractional Gaussian fields and rough sources characterized by their principal symbols. Second, the direct source scattering problem is shown to be well-posed in the distribution sense. Third, we demonstrate that the micro-correlation strengt…
In this paper, we introduce quadrature domains for the Helmholtz equation. We show existence results for such domains and implement the so-called partial balayage procedure. We also give an application to inverse scattering problems, and show that there are non-scattering domains for the Helmholtz equation at any positive frequency that have inward cusps.
Over the last ten years, results from [Melenk-Sauter, 2010], [Melenk-Sauter, 2011], [Esterhazy-Melenk, 2012], and [Melenk-Parsania-Sauter, 2013] decomposing high-frequency Helmholtz solutions into "low"- and "high"-frequency components have had a large impact in the numerical analysis of the Helmholtz equation. These results have been proved for the constant-coefficient Helmholtz equation in either the exterior of a Dirichlet obstacle or an interior domain with an impedance boundary condition. Using the Helffer-Sjöstrand functional calculus, this paper proves analogous decompositions for scatt…
Adapting Mourre's commutator method to the dissipative setting, we prove a limiting absorption principle for a class of abstract dissipative operators. A consequence is the resolvent estimates for the high frequency Helmholtz equation when trapped trajectories meet the set where the imaginary part of the potential is non-zero. We also give the resolvent estimates in Besov spaces.
This paper is concerned with resolvent estimates on the real axis for the Helmholtz equation posed in the exterior of a bounded obstacle with Dirichlet boundary conditions when the obstacle is trapping. There are two resolvent estimates for this situation currently in the literature: (i) in the case of elliptic trapping the general "worst case" bound of exponential growth applies, and examples show that this growth can be realised through some sequence of wavenumbers, (ii) in the prototypical case of hyperbolic trapping where the Helmholtz equation is posed in the exterior of two strictly conv…
We consider GMRES applied to discretisations of the high-frequency Helmholtz equation with strong trapping; recall that in this situation the problem is exponentially ill-conditioned through an increasing sequence of frequencies. Under certain assumptions about the distribution of the eigenvalues, we prove upper bounds on how the number of GMRES iterations grows with the frequency. Our main focus is on boundary-integral-equation formulations of the exterior Dirichlet and Neumann obstacle problems in 2- and 3-d; for these problems, we investigate numerically the sharpness (in terms of dependenc…
This paper concerns the inverse source problem for the time-harmonic wave equation in a one dimensional domain. The goal is to determine the source function from the boundary measurements. The problem is challenging due to complexity of the Green's function for the Helmholtz equation. Our main result is a logarithmic estimate consists of two parts: the data discrepancy and the high frequency tail.
Multiple standing waves of Helmholtz equation with mixed dispersion concentrating in the high frequency limit
Yunnan Normal University（云南师范大学）；Yunnan Research Center Of Modern Analysis And Partial Differential Equations（云南现代分析与偏微分方程研究中心）；Universidade de Brasília（巴西利亚大学）
It is well known that when the geometry and/or coefficients allow stable trapped rays, the outgoing solution operator of the Helmholtz equation (a.k.a. the resolvent of the Laplacian) grows exponentially through a sequence of real frequencies tending to infinity. In this paper we show that, even in the presence of the strongest-possible trapping, if a set of frequencies of arbitrarily small measure is excluded, the Helmholtz solution operator grows at most polynomially as the frequency tends to infinity. One significant application of this result is in the convergence analysis of several numer…
The aim of this paper is to extend the method of improving cloaking structures in the conductivity to scattering problems. We construct very effective near-cloaking structures for the scattering problem at a fixed frequency. These new structures are, before using the transformation optics, layered structures and are designed so that their first scattering coefficients vanish. Inside the cloaking region, any target has near-zero scattering cross section for a band of frequencies. We analytically show that our new construction significantly enhances the cloaking effect for the Helmholtz equation…
We design sources for the two-dimensional Helmholtz equation that can cloak an object by cancelling out the incident field in a region, without the sources completely surrounding the object to hide. As in previous work for real positive wavenumbers, the sources are also determined by the Green identities. The novelty is that we prove that the same approach works for complex wavenumbers which makes it applicable to a variety of media, including media with dispersion, loss and gain. Furthermore, by deriving bounds on Graf's addition formulas with complex arguments, we obtain new estimates that a…
We consider the scalar Helmholtz equation with variable, discontinuous coefficients, modelling transmission of acoustic waves through an anisotropic penetrable obstacle. We first prove a well-posedness result and a frequency-explicit bound on the solution operator, with both valid for sufficiently-large frequency and for a class of coefficients that satisfy certain monotonicity conditions in one spatial direction, and are only assumed to be bounded (i.e., $L^\infty$) in the other spatial directions. This class of coefficients therefore includes coefficients modelling transmission by penetrable…
The results in \cite{O2} (see \cite{O1} for the quasistatics regime) consider the Helmholtz equation with fixed frequency $k$ and, in particular imply that, for $k$ outside a discrete set of resonant frequencies and given a source region $D_a\subset \mathbb{R}^{d}$ ($d=\overline{2,3}$) and $u_0$, a solution of the homogeneous scalar Helmholtz equation in a set containing the control region $D_c\subset \mathbb{R}^{d}$, there exists an infinite class of boundary data on $\partial D_a$ so that the radiating solution to the corresponding exterior scalar Helmholtz problem in $\mathbb{R}^{d} \setmin…
We study a commonly-used second-kind boundary-integral equation for solving the Helmholtz exterior Neumann problem at high frequency, where, writing $Γ$ for the boundary of the obstacle, the relevant integral operators map $L^2(Γ)$ to itself. We prove new frequency-explicit bounds on the norms of both the integral operator and its inverse. The bounds on the norm are valid for piecewise-smooth $Γ$ and are sharp up to factors of $\log k$ (where $k$ is the wavenumber), and the bounds on the norm of the inverse are valid for smooth $Γ$ and are observed to be sharp at least when $Γ$ is smooth with …
This paper is concerned with the asymptotic expansions of the amplitude of the solution of the Helmholtz equation. The original expansions were obtained using a pseudo-differential decomposition of the Dirichlet to Neumann operator. This work uses first and second order approximations of this operator to derive new asymptotic expressions of the normal derivative of the total field. The resulting expansions can be used to appropriately choose the ansatz in the design of high-frequency numerical solvers, such as those based on integral equations, in order to produce more accurate approximation o…
We are concerned with the field concentration between two nearly-touching inclusions with high-contrast material parameters, which is a central topic in the theory of composite materials. The degree of concentration is characterised by the blowup rate of the gradient of the underlying field. In this paper, we derive optimal gradient estimates for the wave filed of the 3-D Helmholtz system in the quasi-static regime. There are two salient features of our results that are new to the literature. First, we cover all the possible scenarios that the size of the inclusions are in different scales in …
We study the 2-D Helmholtz equation in perturbed stratified media, allowing the existence of guided waves. Our assumptions on the perturbing and source terms are not too restrictive. We prove two results. Firstly, we introduce a Sommerfeld-Rellich radiation condition and prove the uniqueness of the solution for the studied equation. Then, by careful asymptotic estimates, we prove the existence of a bounded solution satisfying our radiation condition.
We study the high frequency limit for the dissipative Helmholtz equation when the source term concentrates on a submanifold of R^n. We prove that the solution has a unique semi-classical measure, which is precisely described in terms of the classical properties of the problem. This result is already known when the micro-support of the source is bounded, we now consider the general case.
We consider the high-frequency Helmholtz equation with a given source term, and a small absorption parameter $\a>0$. The high-frequency (or: semi-classical) parameter is $\eps>0$. We let $\eps$ and $\a$ go to zero simultaneously. We assume that the zero energy is non-trapping for the underlying classical flow. We also assume that the classical trajectories starting from the origin satisfy a transversality condition, a generic assumption. Under these assumptions, we prove that the solution $u^\eps$ radiates in the outgoing direction, {\bf uniformly} in $\eps$. In particular, the function $u^\ep…
We couple the mixed variational problem for the generalized Hodge-Helmholtz or Hodge-Laplace equation posed on a bounded three-dimensional Lipschitz domain with the first-kind boundary integral equation arising from the latter when constant coefficients are assumed in the unbounded complement. Recently developed Calderon projectors for the relevant boundary integral operators are used to perform a symmetric coupling. We prove stability of the coupled problem away from resonant frequencies by establishing a generalized Garding inequality (T-coercivity). The resulting system of equations describ…
We prove in this paper an existence result for frequency modes coupling seismic waves and vibrating tall buildings. The derivation from physical principles of a set of equations modeling this phenomenon was done in previous studies. In this model all vibrations are assumed to be anti plane and time harmonic so the two dimensional Helmholtz equation can be used. A coupling frequency mode is obtained once we can determine a wavenumber such that the solution of the corresponding Helmholtz equation in the lower half plane with relevant Neumann and Dirichlet at the interface satisfies a specific in…
In this paper, we study real solutions of the nonlinear Helmholtz equation $$ - Δu - k^2 u = f(x,u),\qquad x\in \R^N $$ satisfying the asymptotic conditions $$ u(x)=O(|x|^{\frac{1-N}{2}}) \quad \text{and} \quad \frac{\partial^2 u}{\partial r^2}(x)+k^2 u(x)) =o(|x|^{\frac{1-N}{2}}) \qquad \text{as $r=|x| \to \infty$.} $$ We develop the variational framework to prove the existence of nontrivial solutions for compactly supported nonlinearities without any symmetry assumptions. In addition, we consider the radial case in which, for a larger class of nonlinearities, infinitely many solutions are sh…
We are concerned with an inverse problem associated with the fractional Helmholtz system that arises from the study of viscoacoustics in geophysics and thermoviscous modelling of lossy media. We are particularly interested in the case that both the medium parameter and the internal source of the wave equation are unknown. Moreover, we consider a general class of source functions which can be frequency-dependent. We establish several general uniqueness results in simultaneously recovering both the medium parameter and the internal source by the corresponding exterior measurements. In sharp cont…
This work extends monotonicity-based methods in inverse problems to the case of the Helmholtz (or stationary Schrödinger) equation $(Δ+ k^2 q) u = 0$ in a bounded domain for fixed non-resonance frequency $k>0$ and real-valued scattering coefficient function $q$. We show a monotonicity relation between the scattering coefficient $q$ and the local Neumann-Dirichlet operator that holds up to finitely many eigenvalues. Combining this with the method of localized potentials, or Runge approximation, adapted to the case where finitely many constraints are present, we derive a constructive monotonicit…
Numerically solving the 2D Helmholtz equation is widely known to be very difficult largely due to its highly oscillatory solution, which brings about the pollution effect. A very fine mesh size is necessary to deal with a large wavenumber leading to a severely ill-conditioned huge coefficient matrix. To understand and tackle such challenges, it is crucial to analyze how the solution of the 2D Helmholtz equation depends on (perturbed) boundary and source data for large wavenumbers. In fact, this stability analysis is critical in the error analysis and development of effective numerical schemes.…
This paper is concerned with an inverse source problem for the three-dimensional Helmholtz equation by a single boundary measurement at a fixed frequency. We show the Lipschitz stability under the assumption that the source function is piecewise constant on a domain which is made of a union of disjoint convex polyhedral subdomains..
This work concerns the analysis of wave propagation in random media. Our medium of interest is sea ice, which is a composite of a pure ice background and randomly located inclusions of brine and air. From a pulse emitted by a source above the sea ice layer, the main objective of this work is to derive a model for the backscattered signal measured at the source/detector location. The problem is difficult in that, in the practical configuration we consider, the wave impinges on the layer with a non-normal incidence. Since the sea ice is seen by the pulse as an effective (homogenized) medium, the…
We study the high frequency limit for a non-dissipative Helmholtz equation. We first prove the absence of eigenvalue on the upper half-plane and close to an energy which satisfies a weak damping assumption on trapped trajectories. Then we generalize to this setting the resolvent estimates of Robert-Tamura and prove the limiting absorption principle. We finally study the semiclassical measures of the solution when the source term concentrates on a bounded submanifold of R^n.
We study the semiclassical measures for the solution of a dissipative Helmholtz equation with a source term concentrated on a bounded submanifold. The potential is not assumed to be non-trapping, but trapped trajectories have to go through the region where the absorption coefficient is positive. In that case, the solution is microlocally written around any point away from the source as a sum (finite or infinite) of lagragian distributions. Moreover we prove and use the fact that the outgoing solution of the dissipative Helmholtz equation is microlocally zero in the incoming region.
Cet article est consacre a l'etude des mesures limites associees a la solution de l'equation de Helmholtz avec un terme source se concentrant en un point. Le potentiel est suppose regulier et l'operateur non-captif. La solution de l'equation de Schrodinger semi-classique s'ecrit alors micro-localement comme somme finie de distributions lagrangiennes. Sous une hypothese geometrique, qui generalise l'hypothese du viriel, on en deduit que la mesure limite existe et qu'elle verifie des proprietes standard. Enfin, on donne un exemple d'operateur qui ne verifie pas l'hypothese geometrique et pour le…
The general solution of the modified Helmholtz equation, q_{xx}(x,y)+q_{yy}(x,y)-4b^2q(x,y)=0, in the wedge 0 < x < y < infinity, is presented. This solution is used to find the explicit steady-state of diffusion-limited coalescence, A+A<-->A, on the half-line, with a trap-source at the origin.
Comparison of substructured non-overlapping domain decomposition and overlapping additive Schwarz methods for large-scale Helmholtz problems with multiple sources
Assigning boundary conditions, such as acoustic impedance, to the frequency domain thermoviscous wave equations (TWE), derived from the linearized Navier-Stokes equations (LNSE) poses a Helmholtz problem, solution to which yields a discrete set of complex eigenfunctions and eigenvalue pairs. The proposed method -- the inverse Helmholtz solver (iHS) -- reverses such procedure by returning the value of acoustic impedance at one or more unknown impedance boundaries (IBs) of a given domain, via spatial integration of the TWE for a given real-valued frequency with assigned conditions on other bound…
This paper introduces a novel method to extend the Helmholtz Decomposition to n-dimensional sufficiently smooth and fast decaying vector fields. The rotation is described by a superposition of n(n-1)/2 rotations within the coordinate planes. The source potential and the rotation potential are obtained by convolving the source and rotation densities with the fundamental solutions of the Laplace equation. The rotation-free gradient of the source potential and the divergence-free rotation of the rotation potential sum to the original vector field. The approach relies on partial derivatives and Ne…
For the Helmholtz equation posed in the exterior of a Dirichlet obstacle, we prove that if there exists a family of quasimodes (as is the case when the exterior of the obstacle has stable trapped rays), then there exist near-zero eigenvalues of the standard variational formulation of the exterior Dirichlet problem (recall that this formulation involves truncating the exterior domain and applying the exterior Dirichlet-to-Neumann map on the truncation boundary). Our motivation for proving this result is that a) the finite-element method for computing approximations to solutions of the Helmholtz…
