Random walks are developed to approximate the solutions of multiscaling, fractional-order, anomalous diffusion equations. The essential elements of the diffusion are described by the matrix-order scaling indexes and the mixing measure, which describes the diffusion coefficient in every direction. Two forms of the governing equation (also called the multiscaling fractional diffusion equation), b...

BACKGROUND Diffusion imaging tractography is increasingly used to trace critical fiber tracts in brain tumor patients to reduce the risk of post-operative neurological deficit. However, the effects of peritumoral edema pose a challenge to conventional tractography using the standard diffusion tensor model. The aim of this study was to present a novel technique using a two-tensor unscented Kalma...

This report aims to present my research updates on distance function wavelets (DFW) based on the fundamental solutions and general solutions of the Helmholtz, modified Helmholtz, and convection-diffusion equations, which include the isotropic HelmholtzFourier (HF) transform and series, the Helmholtz-Laplace (HL) transform, and the anisotropic convection-diffusion wavelets and ridgelets. The lat...

Diffusion tensor imaging provides information about structure and location of white matter tracts within the human brain which is of particular interest for neurosurgery. The reconstruction of neuronal structures from diffusion tensor data is commonly solved by tracking algorithms based on streamline propagation. These approaches generate streamline bundles that approximate the course of neuron...

— We consider the discretization of linear, nonstationary, convection-dominated, convection-diffusion Systems by the streamline diffusion finite element method and give local error estimâtes in the energy norm for both linear scalar équations in arbitrary dimensions and for Systems in one space dimension. For piecewise linear shape functions in time-space that are continuons in space and discon...

We investigate the optimal accuracy of the streamline diffusion finite element method applied to convection–dominated problems. For linear/bilinear elements the theoretical order of convergence given in the literature is either O(h3/2) for quasi–uniform meshes or O(h2) for some uniform meshes. The determination of the optimal order in general was an open problem. By studying a special type of m...