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 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 $$L^2$$
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. 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.
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 lowdimensional 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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.