8 research outputs found
Error Estimates for Adaptive Spectral Decompositions
Adaptive spectral (AS) decompositions associated with a piecewise constant function, , yield small subspaces where the characteristic functions comprising are well approximated. When combined with Newton-like optimization methods, AS decompositions have proved remarkably efficient in providing at each nonlinear iteration a low-dimensional search space for the solution of inverse medium problems. Here, we derive -error estimates for the AS decomposition of , truncated after terms, when is piecewise constant and consists of characteristic functions over Lipschitz domains and a background. Numerical examples illustrate the accuracy of the AS decomposition for media that either do, or do not, satisfy the assumptions of the theory
Adaptive spectral decompositions for inverse medium problems
Inverse medium problems involve the reconstruction of a spatially varying
unknown medium from available observations by exploring a restricted search
space of possible solutions. Standard grid-based representations are very
general but all too often computationally prohibitive due to the high dimension
of the search space. Adaptive spectral (AS) decompositions instead expand the
unknown medium in a basis of eigenfunctions of a judicious elliptic operator,
which depends itself on the medium. Here the AS decomposition is combined with
a standard inexact Newton-type method for the solution of time-harmonic
scattering problems governed by the Helmholtz equation. By repeatedly adapting
both the eigenfunction basis and its dimension, the resulting adaptive spectral
inversion (ASI) method substantially reduces the dimension of the search space
during the nonlinear optimization. Rigorous estimates of the AS decomposition
are proved for a general piecewise constant medium. Numerical results
illustrate the accuracy and efficiency of the ASI method for time-harmonic
inverse scattering problems, including a salt dome model from geophysics
Error Estimates for Adaptive Spectral Decompositions
Adaptive spectral (AS) decompositions associated with a piecewise constant
function yield small subspaces where the characteristic functions
comprising are well approximated. When combined with Newton-like
optimization methods for the solution of inverse medium problems, AS
decompositions have proved remarkably efficient in providing at each nonlinear
iteration a low-dimensional search space. Here, we derive -error estimates
for the AS decomposition of , truncated after terms, when is
piecewise constant and consists of characteristic functions over Lipschitz
domains and a background. Our estimates apply both to the continuous and the
discrete Galerkin finite element setting. Numerical examples illustrate the
accuracy of the AS decomposition for media that either do, or do not, satisfy
the assumptions of the theory
Energy Decay and Stability of a Perfectly Matched Layer For the Wave Equation
In Grote and Sim (Efficient PML for the wave equation. Preprint,arXiv:1001.0319[math:NA],2010; in: Proceedings of the ninth international conference on numerical aspectsof wave propagation (WAVES 2009, held in Pau, France,2009), pp 370–371), a PML formulation was proposed for the wave equation in its standard second-order form. Here, energydecay andL2stability bounds in two and three space dimensions are rigorously proved bothfor continuous and discrete formulations with constant damping coefficients. Numericalresults validate the theory