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Regularization of parabolic equations backward in time by a non-local boundary value problem method. (English) Zbl 1194.35501

Summary: Let \(H\) be a Hilbert space with norm \(|\cdot|\), \(A: D(A)\subset H\to H\) a positive definite, self-adjoint operator with compact inverse on \(H\), and \(T\) and \(\varepsilon\) are given positive numbers. The ill-posed parabolic equation backward in time
\[ \begin{cases} u_t+Au=0,\quad & 0<t<T,\\ \|u(T)-f\|\leq \varepsilon\end{cases} \]
is regularized by the well-posed non-local boundary value problem
\[ \begin{cases} v_{\alpha t}+Av_\alpha=0,\quad & 0<t<aT,\\ av_\alpha(0)+v_\alpha(aT)=f\end{cases} \]
with \( a>1\) being given and \(\alpha > 0\), the regularization parameter. A priori and a posteriori parameter choice rules are suggested which yield order optimal regularization methods. Numerical results based on the boundary element method are presented and discussed to confirm the theory.

MSC:

35R30 Inverse problems for PDEs
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