Pressure-induced isostructural phase transition of metal-doped silicon clathrates

We propose an atomistic model for the pressure-induced isostructural phase transition of metal-doped silicon clathrates, Ba8Si46 and K8Si46, that has been observed at 14 GPa and 23 GPa, respectively. The model explains successfully the equation of state, transition pressure, change of Raman spectra and dependence on the doped cations as well as the effects of substituting Si(6c) atoms with noble metals.Comment: 5 pages, two coumn, 5 figures. See http://www.iitaka.org/down.html for more informatio

Zooming from Global to Local: A Multiscale RBF Approach

Because physical phenomena on Earth's surface occur on many different length scales, it makes sense when seeking an efficient approximation to start with a crude global approximation, and then make a sequence of corrections on finer and finer scales. It also makes sense eventually to seek fine scale features locally, rather than globally. In the present work, we start with a global multiscale radial basis function (RBF) approximation, based on a sequence of point sets with decreasing mesh norm, and a sequence of (spherical) radial basis functions with proportionally decreasing scale centered at the points. We then prove that we can "zoom in" on a region of particular interest, by carrying out further stages of multiscale refinement on a local region. The proof combines multiscale techniques for the sphere from Le Gia, Sloan and Wendland, SIAM J. Numer. Anal. 48 (2010) and Applied Comp. Harm. Anal. 32 (2012), with those for a bounded region in $\mathbb{R}^d$ from Wendland, Numer. Math. 116 (2012). The zooming in process can be continued indefinitely, since the condition numbers of matrices at the different scales remain bounded. A numerical example illustrates the process