Convergence rates of approximate least squares solutions of linear integral and operator equations of the first kind

Abstract:

We consider approximations $\{ {x_n}\}$ obtained by moment discretization to (i) the minimal ${\mathcal {L}_2}$-norm solution of $\mathcal {K}x = y$ where $\mathcal {K}$ is a Hilbert-Schmidt integral operator on ${\mathcal {L}_2}$, and to (ii) the least squares solution of minimal ${\mathcal {L}_2}$-norm of the same equation when y is not in the range $\mathcal {R}(\mathcal {K})$ of $\mathcal {K}$. In case (i), if $y \in \mathcal {R}(\mathcal {K})$, then ${x_n} \to {\mathcal {K}^\dagger }y$, where ${\mathcal {K}^\dagger }$ is the generalized inverse of $\mathcal {K}$, and $\left \| {{x_n}} \right \| \to \infty$ otherwise. Rates of convergence are given in this case if further ${\mathcal {K}^\dagger }y \in {\mathcal {K}^\ast }({\mathcal {L}_2})$, where ${\mathcal {K}^\ast }$ is the adjoint of $\mathcal {K}$, and the Hilbert-Schmidt kernel of $\mathcal {K}{\mathcal {K}^\ast }$ satisfies certain smoothness conditions. In case (ii), if $y \in \mathcal {R}(\mathcal {K}) \oplus \mathcal {R}{(\mathcal {K})^ \bot }$, then ${x_n} \to {\mathcal {K}^\dagger }y$, and $\left \| {{x_n}} \right \| \to \infty$ otherwise. If further ${\mathcal {K}^\dagger }y \in {\mathcal {K}^\ast }\mathcal {K}({\mathcal {L}_2})$, then rates of convergence are given in terms of the smoothness properties of the Hilbert-Schmidt kernel of ${(\mathcal {K}{\mathcal {K}^\ast })^2}$. Some of these results are generalized to a class of linear operator equations on abstract Hilbert spaces.