Related papers: Sensitivity analysis for active control of the Hel…
Most problems in electrodynamics do not have an analytical solution so much effort has been put in the development of numerical schemes, such as the finite-difference method, volume element methods, boundary element methods, and related…
We consider a plane wave, a radiation solution, and the sum of these solutions (total solution) for the Helmholtz equation in an exterior region in $\R^3$. We consider a ray in this region, such that its direction is different from the…
We introduce a novel multi-resolution Localized Orthogonal Decomposition (LOD) for time-harmonic acoustic scattering problems that can be modeled by the Helmholtz equation. The method merges the concepts of LOD and operator-adapted wavelets…
A convergence theory for the $hp$-FEM applied to a variety of constant-coefficient Helmholtz problems was pioneered in the papers [Melenk-Sauter, 2010], [Melenk-Sauter, 2011], [Esterhazy-Melenk, 2012], [Melenk-Parsania-Sauter, 2013]. This…
This paper considers the Helmholtz problem in the exterior of a ball with Dirichlet boundary conditions and radiation conditions imposed at infinity. The differential Helmholtz operator depends on the complex wavenumber with non-negative…
We present analytic solutions to the Teukolsky equation for massless perturbations of any spin in the 4-dimensional de Sitter background. The angular part of the equation fixes the separation constant to a discrete set and its solution is…
For strongly continous semigroups on Hilbert spaces, we investigate admissibility properties of control and observation operators shifted along continuous scales of spaces built by means of either interpolation and extrapolation or…
This paper presents an adaptive numerical framework for solving exterior "sound-soft" scattering problems governed by the Helmholtz equation. By interpreting the Method of Fundamental Solutions through the lens of rational approximation, we…
This paper tackles the data completion problem related to the Helmholtz equation. The goal is to identify unknown boundary conditions on parts of the boundary that cannot be accessed directly, by making use of measurements collected from…
We consider the Dirichlet problem of the indefinite Helmholtz equation in 1D, $u''+k^2u=f$ in $(0,1)$, $u(0)=g_0$, $u(1)=g_1$, with a constant wavenumber $k\in(0,\infty)\backslash\pi\mathbb{N}$ and a source term $f\in H^p_0(0,1)$, $p\ge 4$.…
A rigorous mathematical theory is developed to explain the super-resolution phenomenon observed in the experiment by F.Lemoult, M.Fink and G.Lerosey (Acoustic resonators for far-field control of sound on a subwavelength scale, Phys. Rev.…
We investigate the statistical recovery of solutions to first-kind Fredholm integral equations with discrete, scattered, and noisy pointwise measurements. Assuming the forward operator's range belongs to the Sobolev space of order $m$,…
In this paper we propose a {\it discontinuous} plane wave neural network (DPWNN) method with $hp-$refinement for approximately solving Helmholtz equation and time-harmonic Maxwell equations. In this method, we define a quadratic functional…
We establish boundary observability and control for the fractional heat equation over arbitrary time horizons $T > 0$, within the optimal range of fractional exponents $s \in (1/2, 1)$. Our approach introduces a novel synthesis of…
Exact solutions describing the Rayleigh-Bloch waves for the two-dimensional Helmholtz equation are constructed in the case when the refractive index is a sum of a constant and a small amplitude function which is periodic in one direction…
We develop efficient and high-order accurate solvers for the Helmholtz equation on complex geometry. The schemes are based on the WaveHoltz algorithm which computes solutions of the Helmholtz equation by time-filtering solutions of the wave…
In this work we study the increasing resolution of linear inverse scattering problems at a large fixed frequency. We consider the problem of recovering the density of a Herglotz wave function, and the linearized inverse scattering problem…
Recovering a low-complexity signal from its noisy observations by regularization methods is a cornerstone of inverse problems and compressed sensing. Stable recovery ensures that the original signal can be approximated linearly by optimal…
We construct a unique global solution to the Cauchy problem of the 3D Boltzmann equation for initial data around the Maxwellian in the spatially critical homogeneous Besov space…
We develop an unconditionally energy-stable tensor-product space-time discretization framework for the solution of a linear kinetic transport equation in one space dimension. The kinetic equation is a simplified model of radiative transfer…