Tikhonov regularization-based reconstruction of partial scattering functions obtained from contrast variation small-angle neutron scattering

TL;DR

The paper addresses instability in reconstructing partial scattering functions $S_{ij}$ from contrast variation SANS data due to disparate singular values in the forward matrix. It introduces a Tikhonov regularization framework with a diagonal scaling $L$, transforming to $\boldsymbol{s}=L\boldsymbol{S}$ and solving $\boldsymbol{s}_*=(\alpha^2E+B^TB)^{-1}B^T\boldsymbol{I}^{\delta}$ where $B=AL^{-1}$, using the SVD $B=UDV^T$ with a filter $q(\alpha,\mu)=\mu^2/(\alpha^2+\mu^2)$. The authors demonstrate the approach on numerical tests and experimental CV-SANS data for polyrotaxane, showing stable reconstructions and substantial reductions in fluctuation metrics (e.g., $\sigma$ for $S_{\mathrm{PP}}$ from $0.704$ to $0.197$). They propose a practical reconstruction flow—select $L$, compute $\boldsymbol{s}_*$ for candidate $\alpha$, and use norm–residual plots (L-curve style) to pick $\alpha$—and note the method extends naturally to $p$-component systems. Overall, the work provides a robust, generalizable tool for multicomponent CV-SANS analysis.

Abstract

Contrast variation small-angle neutron scattering (CV-SANS) has been widely employed for nano structural analysis of multicomponent systems. In CV-SANS experiments, scattering intensities of samples with different scattering co\ ntrasts are decomposed into partial scattering functions, corresponding to structure of each component and cross-correlation between different components, by singular value decomposition (SVD). However, the estimation of partial scattering functions with small absolute values often suffers from instability due to the significant differences in the singular values. In this paper, we propose a remedy for this instability by introducing the Tikhonov regularization, which ensures more stable reconstruction of the partial scattering functions.

Figures (12)

• Figure 1: (a) Schematic illustration of partial scattering functions $S_{ij}$ for PR solution. (b) CV-SANS intensities $I(Q)$ of PR solutions with different deuteration levels and partial scattering functions $S_{ij}(Q)$ obtained by a singular value decomposition method Mayumi-etal09. Reprinted from Ref. Mayumi-etal09 with permission from American Chemical Society.
• Figure 2: The regularization matrix is set to $L=E$. The regularization parameter is set to (top) $\alpha=0$ (no regularization) and (bottom) $\alpha=10$. Here, $S_*^{(1)}=S_{{\rm PP},*}$, $S_*^{(2)}=S_{{\rm CC},*}$ and $S_*^{(3)}=S_{{\rm CP},*}$.
• Figure 3: $L=E$. Here, $S_*^{(1)}=S_{{\rm PP},*}$, $S_*^{(2)}=S_{{\rm CC},*}$ and $S_*^{(3)}=S_{{\rm CP},*}$.
• Figure 4: $L_1=1$, $L_2=0.1$, $L_3=1$. Here, $S_*^{(1)}=S_{{\rm PP},*}$, $S_*^{(2)}=S_{{\rm CC},*}$ and $S_*^{(3)}=S_{{\rm CP},*}$.
• Figure 5: The reconstruction for $L_1=1,L_2=0.1,L_3=1$ and $\alpha=10$. Here, $S_*^{(1)}=S_{{\rm PP},*}$, $S_*^{(2)}=S_{{\rm CC},*}$ and $S_*^{(3)}=S_{{\rm CP},*}$.