Regularization of nonlinear ill-posed problems by dynamical system method

Alexandra Smirnova Georgia State University

Let F be a nonlinear twice Fréchet differentiable map in a Hilbert space H and the equation

(1)  F (z) = 0

be solvable (maybe nonuniquely). Assume that y is a solution to (1), and F′(y) is not boundedly invertible. In this case solving (1) is an ill-posed nonlinear problem. A new approach to solving such a problem is presented. The approach is based on a construction of a dynamical system

(2)  u̇ = ϕ(u, t), u(0) = u_{0},

with a unique global solution u(t) for any u_{0} ∈ B(y, r) that tends to an element u(∞) (in the norm of H), and F (u(∞)) = 0.

If y is a unique solution to (1) in the ball B(y, r), then u(∞) = y. Otherwise u(∞) depends on the choice of u_{0}.

The approach is justified, that is, the existence and uniqueness of the global solution to (2) with a suitable choice of ϕ and u_{0} is proved.