A Projection Method to Solve Linear Systems in Tensor Format

Jonas Ballani and Lars Grasedyck

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Submission date: 03. May. 2010
Pages: 24
published in: Numerical linear algebra with applications, 20 (2013) 1, p. 27-43 
DOI number (of the published article): 10.1002/nla.1818
Bibtex
MSC-Numbers: 15A69, 90C06, 65F10
Keywords and phrases: Low Rank, Tucker, Kronecker-product matrix
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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.