Error Estimates for Adaptive Spectral Decompositions
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Adaptive spectral (AS) decompositions associated with a piecewise constant function, u, yield small subspaces where the characteristic functions comprising u 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 L^2-error estimates for the AS decomposition of u, truncated after K terms, when u is piecewise constant and consists of K 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.
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