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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Authors: D. A. Kouba and John Locker
Journal: Trans. Amer. Math. Soc. 293 (1986), 229-255
MSC: Primary 47E05; Secondary 34B05, 47A50
MathSciNet review: 814921
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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\Vert {Lx - y} \right\Vert^2} + {\alpha ^2}{\left\Vert {Tx} \right\Vert^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.

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Article copyright: © Copyright 1986 American Mathematical Society

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