Regularization of parabolic equations backward in time by a non-local boundary value problem method

Let H be a Hilbert space with norm |·|, A: D(A) ⊂ H → H a positive definite, self-adjoint operator with compact inverse on H, and T and ϵ are given positive numbers. The ill-posed parabolic equation backward in time $$\cases {u_t + Au =0, \qquad 0 < t < T, \\ \parallel u(T) - f \parallel \leq \epsilon }$$ is regularized by the well-posed non-local boundary value problem $$\cases {\upsilon_{\alpha t} + A\upsilon _\alpha =0, \qquad 0 < t < aT, \\ \alpha \upsilon _\alpha (0) + \upsilon _\alpha (aT) = f}$$ with a > 1 being given and α > 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.