309 IGPM309.pdf        April 2010
TITLE A Projection Method to Solve Linear Systems in Tensor Format
AUTHORS Jonas Ballani, Lars Grasedyck
ABSTRACT In this paper we propose a method for the numerical solution of linear systems of equations in low rank tensor format. Such systems may arise from the discretisation of PDEs in high dimensions but our method is not limited to this type of application. We present an iterative scheme which is based on the projection of the residual to a low dimensional subspace. The subspace is spanned by vectors in low rank tensor format which - similarly to Krylov subspace methods - stem from the subsequent (approximate) application of the given matrix to the residual. All calculations are performed in hierarchical Tucker format which allows for applications in high dimensions. The mode size dependency is treated by a multilevel method. We present numerical examples that include high-dimensional convection-diffusion equations and shift-invert eigenvalue solvers.
KEYWORDS Low rank, Tucker, hierarchical Tucker, Kronecker-product matrix
DOI 10.1002/nla.1818
PUBLICATION Numerical linear algebra with applications 20(1),
27-43 (2013)