Hào, Dinh Nho; Duc, Nguyen Van; Lesnic, D. Regularization of parabolic equations backward in time by a non-local boundary value problem method. (English) Zbl 1194.35501 IMA J. Appl. Math. 75, No. 2, 291-315 (2010). 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. Cited in 46 Documents MSC: 35R30 Inverse problems for PDEs Keywords:parabolic equations backward in time; ill-posed problems; regularization; non-local boundary value problems; a priori and a posteriori parameter choice rules PDFBibTeX XMLCite \textit{D. N. Hào} et al., IMA J. Appl. Math. 75, No. 2, 291--315 (2010; Zbl 1194.35501) Full Text: DOI