Local regularization methods for inverse Volterra equations applicable to the structure of solid surfaces

Abstract

Deconvolution of appearance potential spectra is an old strategy commonly used to investigate electronic properties of solids in the surface region. Recently, this strategy was found to be effective in the study of nanostructures. In this context, the density of unoccupied states in the surface region of a solid is recovered from the measured AP-spectrum data from the governing equation $k*x*x=g$, where $k$ is a Lorentzian type function, $g$ is a measured APS-signal and $x$ is the density function to be recovered. As an important step in solving for $x$, we need to solve the autoconvolution problem $x*x=f$, which is a nonlinear ill-posed Volterra problem. In this paper, we first improve upon the existing local regularization theory developed in [{\bf9}] for solving the autoconvolution problem, allowing for $L_p$ data, where $1 \le p \le \infty$. We prove the solutions of the regularized equation $x_\alpha^\delta \in L_\infty(0,1)$ (smoother than $x_\alpha^\delta \in L_2(0,1)$ as in [{\bf9}]) converge to the true solution $\overline{x}$ of the autoconvolution equation in $L_\infty$ norm (stronger than $L_2$ norm as in [{\bf9}]) when the noise level in the data shrinks to 0. It is worth noting that we obtain the improved convergence theory while imposing less restrictions on the true solution $\overline{x}$; namely $\overline{x} \in C^1(0,1)$ in contrast to $\overline{x}\in W^{2,\infty}(0,1)$. Further, for the stable deconvolution of appearance potential spectra, we apply the local regularization methods to solve a combination of two ill-posed Volterra problems: the linear problem of determining $f$ from $f*k=g$ and then the nonlinear autoconvolution problem of determining $x$ from $x*x=f$. The results include a convergence theory and a fast sequential numerical method which essentially preserves the causal nature of the combined deconvolution problem. Numerical examples are included to show the effectiveness and efficiency of the methods.