Conference on Applied Inverse Problems, July 20-24, 2009, Vienna, Austria
Poster Presentation
Daniel Lesnic: Non-local methods for some inverse problems

Mon, 20 July, 2009, 17:15-18:15, Foyer

The ill-posed parabolic equation backward in time
$$u_t+ Au=0, \; 0 < t < T,\\
\|u(T)-f\| \leq \epsilon$$ with the positive self-adjoint unbounded
operator $A$ and $\epsilon > 0$ being given is regularized by the
well-posed non-local boundary value problem
%
$$v_{\alpha t}+ Av_\alpha=0, \quad 0<t<aT,\\
\alpha v_\alpha(0)+v_\alpha(aT)=f$$
%
with $a \geq 1$ being given and
$\alpha> 0$, the regularization parameter.

Similarly, the ill-posed Cauchy problem for elliptic equations
%
$$u_{tt}- Au=0, \; 0 < t < T,\\
\|u(0)-\varphi\| \leq \epsilon,\\
u_{t}(0)=0$$
%
is regularized by the well-posed non-local boundary
value problem
%
$$u_{tt}- Au=0, \; 0 < t < aT,\\
u(0)+\alpha u(aT)=\varphi,\\
u_t(0)=0.$$
%
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.

This is joint work with Dinh Nho Hao (Hanoi Institute of
Mathematics, Vietnam) and Nguyen Van Duc (Vinh University, Vietnam).

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