Two Tikhonov-type regularization methods for inverse source problem on the Poisson equation (Q2846211)

The goal of the paper is to formulate conditional stability estimates and two versions of a Tikhonov-type regularization approach for the linear inverse problem of identifying the source term \(f \in L^2(\mathbb{R})\) from observations (noisy data) of the function \(g \in L^2(\mathbb{R})\) in the Poisson equation problem NEWLINE\[NEWLINE \begin{cases} -\Delta u(x,y)=f(x),\, x \in \mathbb{R}, \, y>0,\\ u(x,0)=0,\, x \in \mathbb{R}, \\ u(x,y)_{y \to \infty} \,\text{bounded}, \, x \in \mathbb{R},\\ u(x,1)=g(x), \, x \in \mathbb{R}. \end{cases} NEWLINE\]NEWLINE Because the model considers infinite domains of the functions \(f\) and \(g\), the Fourier transform is applicable and yields an explicit solution formula for \(f\) depending linearly on the Fourier transform of \(g\). This allows the authors to formulate and prove the Hölder-type conditional stability estimate NEWLINE\[NEWLINE\|f\| \leq e\|g\|+\left(\frac{1}{1-e^{-1}} \right)^\frac{p}{p+2}\|f\|_p^\frac{2}{p+2}\|g\|^\frac{p}{p+2}\leq C(M)\,\|g\|^\frac{p}{p+2}NEWLINE\]NEWLINE for \(L^2\)-norms \(\|\cdot\|\) and \(H^p\)-norms \(\|\cdot\|_p,\;p>0,\) and under the source (smoothness) condition \(\|f\|_p\leq M\), where the constant \(C\) depends on the \(H^p\)-bound \(M\) of \(f\).NEWLINENEWLINEMoreover, a classical form of Tikhonov regularization with quadratic misfit term and quadratic \(H^p\)-penalty is formulated and analyzed. Furthermore, the authors also suggest a modification simplifying the multiplier in the corresponding formula in the frequency space. A numerical case study completes the paper. The references list appears quite modest, given the fact that the inverse source problem under consideration is quite a standard problem.