162 research outputs found
Norm-dependent convergence and stability of the inverse scattering series for diffuse and scalar waves
This work analyzes the forward and inverse scattering series for scalar waves
based on the Helmholtz equation and the diffuse waves from the time-independent
diffusion equation, which are important PDEs in various applications. Different
from previous works, which study the radius of convergence for the forward and
inverse scattering series, the stability, and the approximation error of the
series under the norms, we study these quantities under the Sobolev
norm, which associates with a general class of -based function spaces. The
norm has a natural spectral bias based on its definition in the Fourier
domain: the case biases towards the lower frequencies, while the case
biases towards the higher frequencies. We compare the stability estimates
using different norms for both the parameter and data domains and provide
a theoretical justification for the frequency weighting techniques in practical
inversion procedures. We also provide numerical inversion examples to
demonstrate the differences in the inverse scattering radius of convergence
under different metric spaces.Comment: 31 pages, 6 figure
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Seeing through rock with help from optimal transport
Geophysicists and mathematicians work together to detect geological structures located deep within the earth by measuring and interpreting echoes from manmade earthquakes. This inverse problem naturally involves the mathematics of wave propagation, but we will see that a different mathematical theory – optimal transport – also turns out to be very useful
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