we introduce a rbfs mesheless method of lines that decomposes the interior and boundary centers to obtain the numerical solution of the time dependent pdes. then, the method is applied with an adaptive algorithm to obtain the numerical solution of one dimensional problem. we show that in the problems in which the solutions contain region with rapid variation, the adaptive rbfs methods are succe...

In various applications, data in multi-dimensional space are normalized to unit length. This paper considers the problem of best fitting given points on the m-dimensional unit sphere S by k-dimensional great circles with k much less than m. The task is cast as an algebraically constrained low-rank matrix approximation problem. Using the fidelity of the low-rank approximation to the original dat...

The tetrahedron equation is a three-dimensional generalization of the Yang-Baxter equation. Its solutions define integrable three-dimensional lattice models of statistical mechanics and quantum field theory. Their integrability is not related to the size of the lattice, therefore the same solution of the tetrahedron equation defines different integrable models for different finite periodic cubi...

In this paper, we introduce and analyze a new low-rank multilevel strategy for the solution of random diffusion problems. Using a standard stochastic collocation scheme, we first approximate the infinite dimensional random problem by a deterministic parameter-dependent problem on a high-dimensional parameter domain. Given a hierarchy of finite element discretizations for the spatial approximati...

This paper is concerned with the development and analysis of an iterative solver for high-dimensional second-order elliptic problems based on subspace-based low-rank tensor formats. Both the subspaces giving rise to low-rank approximations and corresponding sparse approximations of lower-dimensional tensor components are determined adaptively. A principal obstruction to a simultaneous control o...