# A multi-scale Gaussian beam parametrix for the wave equation: the Dirichlet boundary value problem

## TL;DR

This paper develops a multi-scale Gaussian beam method for solving the wave equation with Dirichlet boundary conditions, analyzing its convergence in highly oscillatory scenarios, building on wave-atom and Gaussian beam techniques.

## Contribution

It introduces a novel multi-scale Gaussian beam parametrix for the wave equation boundary problem, extending existing wave-atom and Gaussian beam methods.

## Key findings

- Convergence rate of the parametrix to the true solution
- Extension of Gaussian beam propagation techniques to multi-scale settings
- Application to highly oscillatory wave regimes

## Abstract

We present a construction of a multi-scale Gaussian beam parametrix for the Dirichlet boundary value problem associated with the wave equation, and study its convergence rate to the true solution in the highly oscillatory regime. The construction elaborates on the wave-atom parametrix of Bao, Qian, Ying, and Zhang and extends to a multi-scale setting the technique of Gaussian beam propagation from a boundary of Katchalov, Kurylev and Lassas.