Preconditioned low-rank methods for high-dimensional elliptic PDE eigenvalue problems. We consider elliptic PDE eigenvalue problems on a tensorized domain, discretized such that the resulting matrix eigenvalue problem Ax=λx exhibits Kronecker product structure. In particular, we are concerned with the case of high dimensions, where standard approaches to the solution of matrix eigenvalue problems fail due to the exponentially growing degrees of freedom. Recent work shows that this curse of dimensionality can in many cases be addressed by approximating the desired solution vector x in a low-rank tensor format. In this paper, we use the hierarchical Tucker decomposition to develop a low-rank variant of LOBPCG, a classical preconditioned eigenvalue solver. We also show how the ALS and MALS (DMRG) methods known from computational quantum physics can be adapted to the hierarchical Tucker decomposition. Finally, a combination of ALS and MALS with LOBPCG and with our low-rank variant is proposed. A number of numerical experiments indicate that such combinations represent the methods of choice.

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  1. Bachmayr, Markus; Dahmen, Wolfgang: Adaptive low-rank methods: problems on Sobolev spaces (2016)
  2. Bachmayr, M.; Dahmen, W.: Adaptive low-rank methods for problems on Sobolev spaces with error control in $\mathrmL_2$ (2016)
  3. Etter, Simon: Parallel ALS algorithm for solving linear systems in the hierarchical Tucker representation (2016)
  4. Kressner, Daniel; Steinlechner, Michael; Vandereycken, Bart: Preconditioned low-rank Riemannian optimization for linear systems with tensor product structure (2016)
  5. Kressner, Daniel; Uschmajew, André: On low-rank approximability of solutions to high-dimensional operator equations and eigenvalue problems (2016)
  6. Lee, Namgil; Cichocki, Andrzej: Regularized computation of approximate pseudoinverse of large matrices using low-rank tensor train decompositions (2016)
  7. Bachmayr, Markus; Dahmen, Wolfgang: Adaptive near-optimal rank tensor approximation for high-dimensional operator equations (2015)
  8. Kazeev, Vladimir; Schwab, Christoph: Tensor approximation of stationary distributions of chemical reaction networks (2015)
  9. Savostyanov, Dmitry V.: Quasioptimality of maximum-volume cross interpolation of tensors (2014)
  10. Beckermann, Bernhard; Kressner, Daniel; Tobler, Christine: An error analysis of Galerkin projection methods for linear systems with tensor product structure (2013)
  11. Benner, Peter; Breiten, Tobias: Low rank methods for a class of generalized Lyapunov equations and related issues (2013)
  12. Klinvex, A.; Saied, F.; Sameh, A.: Parallel implementations of the trace minimization scheme tracemin for the sparse symmetric eigenvalue problem (2013)
  13. Lubich, Christian; Rohwedder, Thorsten; Schneider, Reinhold; Vandereycken, Bart: Dynamical approximation by hierarchical Tucker and tensor-train tensors (2013)
  14. Mach, T.: Computing inner eigenvalues of matrices in tensor train matrix format (2013)
  15. Rohwedder, Thorsten; Uschmajew, André: On local convergence of alternating schemes for optimization of convex problems in the tensor train format (2013)
  16. Uschmajew, André; Vandereycken, Bart: The geometry of algorithms using hierarchical tensors (2013)
  17. Kazeev, Vladimir A.; Khoromskij, Boris N.: Low-rank explicit QTT representation of the Laplace operator and its inverse (2012)
  18. Kressner, Daniel; Tobler, Christine: Preconditioned low-rank methods for high-dimensional elliptic PDE eigenvalue problems (2011)
  19. Knyazev, Andrew V.: Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method (2001)