Institute for Numerical Simulation
Rheinische Friedrich-Wilhelms-Universität Bonn



The programming components are as follows: The special features are the following items: Brief description:

The high dimensional approximation is valid for all applications but we are more interested in quantum data. We consider two forms of data inputs. First, the observations of the function to be approximated are directly in disposition. Second, only some functions of the function values are available while the local data are unobserved. An illustration of this later is an electronic structure computation which provides only the global potential energy of an atomic system but the local energy per atom is needed. Our high-dimensional implementation works for both input structures which could include some noisy imperfection. As a motivation, we can consider a molecular dynamics program necessitates the energy per atom in order to be able to compute the force applied at each atom. That needs to be calculated at each molecular dynamic step. In contrast, an electronic structure packet such as DFT does not provide energy per atom but only the total energy of the complete system. In the same research frame, we approximate also DFT by Tight-Binding by using a correction term in form of a two-body repulsive potential which is represented as a multilevel B-spline.



After applying the high-dimensional program to several atomic configurations, we collect the major results on the next table. It mainly summarizes the elapsed time for the preparation and the performance of the new method. In the tabulated outcomes, we consider first Body Centered Cubic of germanium. In addition, we use composites which are labeled Gex Si1-x where x in [0,1] controls the amounts of germanium and silicon. More precisely, we consider the composites which are Ge0.75 Si0.25, Ge0.50 Si0.50 and Ge0.25 Si0.75. All those composites are structured by using Face Centered Cubic as space group symmetry. The appropriate parameter values are obtained from the American Mineralogist Crystal Structure Database.

The preparation step consists of a geometry optimization and the determination of the kernel approximation. The duration of the Gaussian kernel is dominated by the DFT computations related to point samples. The numbers of point samples are 70, 105, 200 and 250 for isotropic, 2D anisotropic, 3D anisotropic and Voigt tensor respectively. The stochastic computation is very fast. In fact, the application of stochastic simulation is at most 2 percent of the whole preparation. All the computations were performed with the DFT basis unpolarized single ζ which is the least intensive basis available in the implementation. If other bases were used, the time for the DFT computation would last even much longer. In the case of DFT double ζ polarized, the scaling of the computation intensity might be doubled. In all dimensions, the preparation expense has long durations. Depending on the tensor transformation and the number of sample points, the preparation overhead can last a few minutes till several days. But the output of those preparations can be stored so that they need only be computed once for all. The ratio between the direct DFT-evaluation and the evaluation using kernel approximation is displayed in the last column of the Table. It exhibits that in average, the orders of acceleration are respectively 2.62e-06, 9.16e-06, 3.00e-05 and 4.22e-05 for the isotropic, 2D anisotropic, 3D anisotropic and 6D tensor cases. Since the acceleration advantage is very good, investing on the preparation process is worth calculating as the results can be stored and subsequently post-processed.