Abstract

Motivated by application to Tikhonov regularization in convex minimization where the objective functions are strongly convex, we study the rate of convergence of the first- and second-order evolution equations associated with a maximal strongly monotone operator on a real Hilbert space. We show that the convergence rate of solutions to the second-order evolution equation of monotone type to a zero of the monotone operator (or a minimum point of a convex function) is faster than the first-order one when the maximal monotone operator is strongly monotone. Since the bounded solutions of the second-order evolution equation on the half line are not directly computable, we have used  he computation of solutions to the corresponding second-order boundary value problem of monotone type. By using the numerical methods and approximation of solutions to the second-order boundary value problem associated with a gradient of a convex function with at least a minimum point, we approximate a minimum of the convex function. Finally, with a simple concrete example a comparison between this method and the steepest descent method is presented.