Convergence rates for regularized solutions

Abstract:

Given a first-kind integral equation \[ \mathcal {K}u(x) = \int _0^1 {K(x,t)u(t) dt = f(x)} \] with discrete noisy data ${d_i} = f({x_i}) + {\varepsilon _i}$, $i = 1,2, \ldots ,n$, let ${u_{n\alpha }}$ be the minimizer in a Hilbert space W of the regularization functional $(1/n)\sum {(\mathcal {K}} u({x_i}) - {d_i}{)^2} + \alpha \left \| u \right \|_W^2$. It is shown that in any one of a wide class of norms, which includes ${\left \| \cdot \right \|_W}$, if $\alpha \to 0$ in a certain way as $n \to \infty$, then ${u_{n\alpha }}$ converges to the true solution ${u_0}$. Convergence rates are obtained and are used to estimate the optimal regularization parameter $\alpha$.