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Discretizing Helmholtz problems via finite elements yields linear systems whose efficient solution remains a major challenge for classical computation. In this paper, we investigate how variational quantum algorithms could address this…
The multigrid algorithm is a multilevel approach to accelerate the numerical solution of discretized differential equations in physical problems involving long-range interactions. Multiresolution analysis of wavelet theory provides an…
The Helmholtz equation arises when modeling wave propagation in the frequency domain. The equation is discretized as an indefinite linear system, which is difficult to solve at high wave numbers. In many applications, the solution of the…
We present a preconditioning method for the multi-dimensional Helmholtz equation with smoothly varying coefficient. The method is based on a frame of functions, that approximately separates components associated with different singular…
The use of operator methods of algebraic nature is shown to be a very powerful tool to deal with different forms of relativistic wave equations. The methods provide either exact or approximate solutions for various forms of differential…
We consider the application of the WaveHoltz iteration to time-harmonic elastic wave equations with energy conserving boundary conditions. The original WaveHoltz iteration for acoustic Helmholtz problems is a fixed-point iteration that…
The multigrid algorithm is an efficient numerical method for solving a variety of elliptic partial differential equations (PDEs). The method damps errors at progressively finer grid scales, resulting in faster convergence compared to…
This paper analyzes the error estimates of the hybridizable discontinuous Galerkin (HDG) method for the Helmholtz equation with high wave number in two and three dimensions. The approximation piecewise polynomial spaces we deal with are of…
Pseudodifferential parabolic equations with an operator square root arise in wave propagation problems as a one-way counterpart of the Helmholtz equation. The expression under the square root usually involves a differential operator and a…
Multiscale problems are computationally costly to solve by direct simulation because the smallest scales must be represented over a domain determined by the largest scales of the problem. We have developed and analyzed new numerical methods…
In this paper we present and analyse a high accuracy method for computing wave directions defined in the geometrical optics ansatz of Helmholtz equation with variable wave number. Then we define an "adaptive" plane wave space with small…
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…
Numerical simulations of seismic wave propagation in heterogeneous 3D media are central to investigating subsurface structures and understanding earthquake processes, yet are computationally expensive for large problems. This is…
The Helmholtz equation with variable wavenumbers is challenging to solve numerically due to the pollution effect, which often results in a huge ill-conditioned linear system. In this paper, we present a high-order wavelet Galerkin method to…
Consider the elastic scattering of a time-harmonic wave by multiple well separated rigid particles in two dimensions. To avoid using the complex Green's tensor of the elastic wave equation, we utilize the Helmholtz decomposition to convert…
In several geophysical applications, such as full waveform inversion and data modelling, we are facing the solution of inhomogeneous Helmholtz equation. The difficulties of solving the Helmholtz equa- tion are two fold. Firstly, in the case…
We solve the one-dimensional Helmholtz equation in several scenarios using the quantum annealer provided by the D-Wave systems within a pseudospectral scheme, where its solution is encoded into certain set of suitable basis functions. We…
In this article, discrete variants of several results from vector calculus are studied for classical finite difference summation by parts operators in two and three space dimensions. It is shown that existence theorems for scalar/vector…
We develop a new dispersion minimizing compact finite difference scheme for the Helmholtz equation in 2 and 3 dimensions. The scheme is based on a newly developed ray theory for difference equations. A discrete Helmholtz operator and a…
In this paper we present a new highly efficient calculation method for the far field amplitude pattern that arises from scattering problems governed by the d-dimensional Helmholtz equation and, by extension, Schr\"odinger's equation. The…