Sobolev and Max Norm Error Estimates for Gaussian Beam Superpositions
Provides rigorous error bounds for Gaussian beam methods, which are important for high-frequency wave propagation problems in domains like seismology and optics, but the results are incremental as they extend known theory to more general norms and caustic scenarios.
This work derives Sobolev and max norm error estimates for Gaussian beam superpositions approximating solutions to linear hyperbolic PDEs and the Schrödinger equation, showing convergence rates in terms of the short wavelength ε. The results demonstrate that k-th order beams converge as O(ε^{k/2-s}) in Sobolev norms and O(ε^{⌈k/2⌉}) in max norm away from caustics, with slower rates near caustics.
This work is concerned with the accuracy of Gaussian beam superpositions, which are asymptotically valid high frequency solutions to linear hyperbolic partial differential equations and the Schrödinger equation. We derive Sobolev and max norms estimates for the difference between an exact solution and the corresponding Gaussian beam approximation, in terms of the short wavelength $\varepsilon$. The estimates are performed for the scalar wave equation and the Schrödinger equation. Our result demonstrates that a Gaussian beam superposition with $k$-th order beams converges to the exact solution as $O(\varepsilon^{k/2-s})$ in order $s$ Sobolev norms. This result is valid in any number of spatial dimensions and it is unaffected by the presence of caustics in the solution. In max norm, we show that away from caustics the convergence rate is $O(\varepsilon^{\lceil k/2\rceil})$ and away from the essential support of the solution, the convergence is spectral in $\varepsilon$. However, in the neighborhood of a caustic point we are only able to show the slower, and dimensional dependent, rate $O(\varepsilon^{(k-n)/2})$ in $n$ spatial dimensions.