Regularization for $n$th-order linear boundary value problems using $m$th-order differential operators

Abstract:

Let $X$ and $Y$ denote real Hilbert spaces, and let $L: X \to Y$ be a closed densely-defined linear operator having closed range. Given an element $y \in Y$, we determine least squares solutions of the linear equation $Lx = y$ by using the method of regularization. Let $Z$ be a third Hilbert space, and let $T: X \to Z$ be a linear operator with $\mathcal {D}(L) \subseteq \mathcal {D}(T)$. Under suitable conditions on $L$ and $T$ and for each $\alpha \ne 0$, we show that there exists a unique element ${x_\alpha } \in \mathcal {D}(L)$ which minimizes the functional ${G_\alpha }(x) = {\left \| {Lx - y} \right \|^2} + {\alpha ^2}{\left \| {Tx} \right \|^2}$, and the ${x_\alpha }$ converge to a least squares solution ${x_0}$ of $Lx = y$ as $\alpha \to 0$. We apply our results to the special case where $L$ is an $n$th-order differential operator in $X = {L^2}[a,b]$, and we regularize using for $T$ an $m$th-order differential operator in ${L^2}[a,b]$ with $m \le n$. Using an approximating space of Hermite splines, we construct numerical solutions to $Lx = y$ by the method of continuous least squares and the method of discrete least squares.