The taut string approach to statistical inverse problems: Theory and applications

Sangjin Kim, Michael Pokojovy, Xiang Wan

Abstract: A novel solution approach to a class of nonlinear statistical inverse problems with finitely many observations collected over a compact interval on the real line blurred by Gaussian white noise of arbitrary intensity is presented. Exploiting the nonparametric taut string estimator, we prove the state recovery strategy is convergent to a solution of the unnoisy problem at the rate of n−1∕2<math><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mo is="true">−</mo><mn is="true">1</mn><mo is="true">∕</mo><mn is="true">2</mn></mrow></msup></math> as the number of observations n<math><mi is="true">n</mi></math> grows to infinity. Illustrations of the method’s application to real-world examples from hydrology, civil & electrical engineering are given and an empirical study on the robustness of our approach is presented.