Regularization for 𝑛th-order linear boundary value problems using 𝑚th-order differential operators

Abstract

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