The Metrics of Prokhorov and Ky Fan for Assessing Uncertainty in Inverse Problems
نویسنده
چکیده
To assess the quality of solutions in stochastic inverse problems, a proper measure for the distance of random variables is essential. The aim of this note is the comparison of the metrics of Ky Fan and Prokhorov with other concepts such as expected values, probability estimates and almost sure convergence. In ill-posed problems one aims to find an appropriate solution x† to an equation of the form
منابع مشابه
Convergence rates for the Bayesian approach to linear inverse problems∗
Recently, the metrics of Ky Fan and Prokhorov were introduced as a tool for studying convergence in stochastic ill-posed problems. In this work, we show that the Bayesian approach to linear inverse problems can be examined in the new framework as well. We consider the finitedimensional case where the measurements are disturbed by an additive normal noise and the prior distribution is normal. Co...
متن کاملAssessing Uncertainty in Nonlinear Inverse Problems with the Metric of Ky Fan
Recently, the metric of Ky Fan was successfully applied to assess uncertainty in linear inverse problems. In this work, we present an extension of these results to nonlinear problems. More precisely, we derive results on convergence and convergence rates for Tikhonov regularization applied to nonlinear stochastic ill-posed problems, and discuss strategies to extend these local convergence resul...
متن کاملKy Fan Inequality for Set-Valued Maps
In this paper, employing the Ky Fan section theorem,we give a class of Ky Fan inequality for set-valued maps,usually it takes Ky Fan inequality ,vector equilibrium problems as its special cases.
متن کاملUsing the Taxonomy and the Metrics: What to Study When and Why; Comment on “Metrics and Evaluation Tools for Patient Engagement in Healthcare Organization- and System-Level Decision-Making: A Systematic Review”
Dukhanin and colleagues’ taxonomy of metrics for patient engagement at the organizational and system levels has great potential for supporting more careful and useful evaluations of this ever-growing phenomenon. This commentary highlights the central importance to the taxonomy of metrics assessing the extent of meaningful participation in decision-making by patients, consumers and community mem...
متن کاملConvergence results for the Bayesian inversion theory∗
In this paper we derive convergence results for regularized solutions of linear inverse problems obtained by the Bayesian approach in the Ky Fan metric. We show that the convergence rate is order optimal in finite dimensional spaces. Moreover, we prove that order optimal rates can be obtained for weighted Bayesian solutions when the dimension goes to infinity.
متن کامل