Finite-dimensional approximation of constrained Tikhonov-regularized solutions of ill-posed linear operator equations

Neubauer, A.

doi:10.1090/S0025-5718-1987-0878691-2

Finite-dimensional approximation of constrained Tikhonov-regularized solutions of ill-posed linear operator equations
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Math. Comp. 48 (1987), 565-583 Request permission

Abstract:

In this paper we derive conditions under which the finite-dimensional constrained Tikhonov-regularized solutions ${x_{\alpha ,{C_n}}}$ of an ill-posed linear operator equation $Tx = y$ (i.e., ${x_{\alpha ,{C_n}}}$ is the minimizing element of the functional ${\left \| {Tx - y} \right \|^2} + \alpha {\left \| x \right \|^2}$, $\alpha > 0$ in the closed convex set ${C_n}$, which is a finite-dimensional approximation of a closed convex set C) converge to the best approximate solution of the equation in C. Moreover, we develop an estimate for the approximation error, which is optimal for certain sets C and ${C_n}$. We present numerical results that verify the theoretical results.

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