On Optimal Algorithms in an Asymptotic Model with Gaussian Measure

Abstract

We study the approximate solution of linear problems in separable Hilbert spaces equipped with a Gaussian measure. We find information and algorithms with the best possible rate of convergence. Although adaptive information and nonlinear algorithms are permitted, we prove that nonadaptive information and linear algorithms are optimal. An algorithm is optimal if it converges with a rate of convergence that is no worse than the rate of any other algorithm except on sets of measure zero. We prove that algorithms and information that minimize the average errors lead to the best possible rate of convergence. This exhibits a close relation between the asymptotic and average case models.