Convergence rates of a dual gradient method for constrained linear ill-posed problems
Qinian Jin
Abstract
Abstract In this paper we consider a dual gradient method for solving linear ill-posed problems $$Ax = y$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mi>y</mml:mi> </mml:mrow> </mml:math> , where $$A : X \rightarrow Y$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>:</mml:mo> <mml:mi>X</mml:mi> <mml:mo>→</mml:mo> <mml:mi>Y</mml:mi> </mml:mrow> </mml:math> is a bounded linear operator from a Banach space X to a Hilbert space Y . A strongly convex penalty function is used in the method to select a solution with desired feature. Under variational source conditions on the sought solution, convergence rates are derived when the method is terminated by either an a priori stopping rule or the discrepancy principle. We also consider an acceleration of the method as well as its various applications.