Recent Developments in Inverse Problems - Abstract
Bürger, Steven
We consider the autoconvolution operator for real-valued, square-integrable functions defined on $[0,1]$: [F:L^2(0,1)to L^2(0,1),quad [F(x)](s) = int_0^s x(s-t)x(t)textdt] The aim of deautoconvolution is to approximately reconstruct a function $x_0$ from noisy data $y^delta$ for $F(x_0)$. Since this problem is ill-posed, one needs to regularize it and one possible way to do this is Lavrentiev-regularization. This means to solve the equation [alpha(x^*-x) + y^delta - F(x) = 0tag$*$] for a regularization parameter $alpha>0$ and a reference element $x^*$. In the paper J. Janno, newblock em Lavrent'ev regularization of ill-posed problems containing nonlinear near to monotone operators with application to autoconvolution equation, newblock Inverse Problems bf 16 (2000), it was shown that, under certain conditions on the exact solution, the solution of $(*)$ converges to the exact solution as the noise tends to zero. We consider a discretized version of $(*)$, namely [Q(alpha(x^*-x) + y^delta - F(Qx)) = 0] with a certain projection operator $Q$. For solutions of this equation we present a convergence result, discuss how to solve the equation and give numerical examples.