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

Abstract

We consider approximations{xn}\{ {x_n}\}obtained by moment discretization to (i) the minimalL2{\mathcal {L}_2}-norm solution ofKx=y\mathcal {K}x = ywhereK\mathcal {K}is a Hilbert-Schmidt integral operator onL2{\mathcal {L}_2}, and to (ii) the least squares solution of minimalL2{\mathcal {L}_2}-norm of the same equation whenyis not in the rangeR(K)\mathcal {R}(\mathcal {K})ofK\mathcal {K}. In case (i), ify∈R(K)y \in \mathcal {R}(\mathcal {K}), thenxn→K†y{x_n} \to {\mathcal {K}^\dagger }y, whereK†{\mathcal {K}^\dagger }is the generalized inverse ofK\mathcal {K}, and‖xn‖→∞\left \| {{x_n}} \right \| \to \inftyotherwise. Rates of convergence are given in this case if furtherK†y∈K∗(L2){\mathcal {K}^\dagger }y \in {\mathcal {K}^\ast }({\mathcal {L}_2}), whereK∗{\mathcal {K}^\ast }is the adjoint ofK\mathcal {K}, and the Hilbert-Schmidt kernel ofKK∗\mathcal {K}{\mathcal {K}^\ast }satisfies certain smoothness conditions. In case (ii), ify∈R(K)⊕R(K)⊥y \in \mathcal {R}(\mathcal {K}) \oplus \mathcal {R}{(\mathcal {K})^ \bot }, thenxn→K†y{x_n} \to {\mathcal {K}^\dagger }y, and‖xn‖→∞\left \| {{x_n}} \right \| \to \inftyotherwise. If furtherK†y∈K∗K(L2){\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(KK∗)2{(\mathcal {K}{\mathcal {K}^\ast })^2}. Some of these results are generalized to a class of linear operator equations on abstract Hilbert spaces.