Adaptive Near-Optimal Rank Tensor Approximation for High-Dimensional Operator Equations

Abstract

We consider a framework for the construction of iterative schemes for operator equations that combine low-rank approximation in tensor formats and adaptive approximation in a basis. Under fairly general assumptions, we conduct a rigorous convergence analysis, where all parameters required for the execution of the methods depend only on the underlying infinite-dimensional problem, but not on a concrete discretization. Under certain assumptions on the rates for the involved low-rank approximations and basis expansions, we can also give bounds on the computational complexity of the iteration as a function of the prescribed target error. Our theoretical findings are illustrated and supported by computational experiments. These demonstrate that problems in very high dimensions can be treated with controlled solution accuracy.

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Acknowledgments

This work was funded in part by the Excellence Initiative of the German Federal and State Governments, DFG Grant GSC 111 (Graduate School AICES), the DFG Special Priority Program 1324, and NSF Grant #1222390.

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Correspondence to Wolfgang Dahmen.

Additional information

Communicated by Albert Cohen.

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Bachmayr, M., Dahmen, W. Adaptive Near-Optimal Rank Tensor Approximation for High-Dimensional Operator Equations. Found Comput Math 15, 839–898 (2015). https://doi.org/10.1007/s10208-013-9187-3

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Keywords

  • Low-rank tensor approximation
  • Adaptive methods
  • High-dimensional operator equations
  • Computational complexity

Mathematics Subject Classification

  • 41A46
  • 41A63
  • 65D99
  • 65J10
  • 65N12
  • 65N15