Solution of the Fokker-Planck Equation by Cross Approximation Method in the Tensor Train Format.

We propose the novel numerical scheme for solution of the multidimensional Fokker-Planck equation, which is based on the Chebyshev interpolation and the spectral differentiation techniques as well as low rank tensor approximations, namely, the tensor train decomposition and the multidimensional cross approximation method, which in combination makes it possible to drastically reduce the number of degrees of freedom required to maintain accuracy as dimensionality increases. We demonstrate the effectiveness of the proposed approach on a number of multidimensional problems, including Ornstein-Uhlenbeck process and the dumbbell model. The developed computationally efficient solver can be used in a wide range of practically significant problems, including density estimation in machine learning applications.

Method of Surgical Treatment of Spine Injury with the Use of Extracorporeal Navigation System Based on Additive Technologies.

We present a method of minimally invasive transcutaneous insertion of screws using a prefabricated extracorporeal navigation system using additive technologies (based on primary data obtained from the DICOM package in multi-detector computed tomography of the affected spine segment) according to the principle of personalized medicine. The method was tested on 10 dogs of different breeds with generally similar mechanism of trauma and typical consequences that led to fracture and dislocation of one of the lumbar vertebrae. In all animals, a positive treatment outcome of different degrees was achieved.