• Corpus ID: 245837595

High-frequency limit of the inverse scattering problem: asymptotic convergence from inverse Helmholtz to inverse Liouville

@article{Chen2022HighfrequencyLO,
  title={High-frequency limit of the inverse scattering problem: asymptotic convergence from inverse Helmholtz to inverse Liouville},
  author={Shi Chen and Zhiyan Ding and Qin Li and Leonardo Zepeda-N'unez},
  journal={ArXiv},
  year={2022},
  volume={abs/2201.03494}
}
. We investigate the asymptotic relation between the inverse problems relying on the Helmholtz equation and the radiative transfer equation (RTE) as physical models, in the high-frequency limit. In particular, we evaluate the asymptotic convergence of a generalized version of inverse scattering problem based on the Helmholtz equation, to the inverse scattering problem of the Liouville equation (a simplified version of RTE). The two inverse problems are connected through the Wigner transform that… 

References

SHOWING 1-10 OF 59 REFERENCES

Inverse scattering via Heisenberg's uncertainty principle

We present a stable method for the fully nonlinear inverse scattering problem of the Helmholtz equation in two dimensions. The new approach is based on the observation that ill-posedness of the

Sensitivity analysis of an inverse problem for the wave equation with caustics

The paper investigates the sensitivity of the inverse problem of recovering the velocity field in a bounded domain from the boundary dynamic Dirichlet-to-Neumann map (DDtN) for the wave equation.

Inverse Problems for the Stationary Transport Equation in the Diffusion Scaling

TLDR
Stability estimates for the inverse problem of reconstructing the optical parameters of the radiative transfer equation from boundary measurements in the diffusion limit are derived and it is shown that the stability is Lipschitz in all regimes, but the coefficient deteriorates as $e^{\frac{1}{\mathsf{Kn}}}$, making the inverse Problem of RTE severely ill-posed when the Knudsen number is small.

Inverse scattering problems with multi-frequencies

This paper is concerned with computational approaches and mathematical analysis for solving inverse scattering problems in the frequency domain. The problems arise in a diverse set of scientific

High frequency limit of the Helmholtz equations.

We derive the high frequency limit of the Helmholtz equations in terms of quadratic observables. We prove that it can be written as a stationary Liouville equation with source terms. Our method is

Semi-classical limit of an inverse problem for the Schrödinger equation

TLDR
This paper shows that using the initial condition and final state of the Schrödinger equation to reconstruct the potential term in the Liouville equation formally bridges an inverse problem in quantum mechanics with an inverse Problem in classical mechanics.

High Resolution Inverse Scattering in Two Dimensions Using Recursive Linearization

TLDR
A fast, stable algorithm for the solution of the inverse acoustic scattering problem in two dimensions, using Chen's method of recursive linearization to reconstruct an unknown sound speed at resolutions of thousands of square wavelengths in a fully nonlinear regime.

Inversion of seismic reflection data in the acoustic approximation

The nonlinear inverse problem for seismic reflection data is solved in the acoustic approximation. The method is based on the generalized least‐squares criterion, and it can handle errors in the data

Semi-classical models for the Schrödinger equation with periodic potentials and band crossings

The Bloch decomposition plays a fundamental role in the study of quantum mechanics and wave propagation in periodic media. Most of the homogenization theory developed for the study of high frequency

Homogenization limits and Wigner transforms

We present a theory for carrying out homogenization limits for quadratic functions (called “energy densities”) of solutions of initial value problems (IVPs) with anti-self-adjoint (spatial)
...