Low-rank tensor approximation of singularly perturbed boundary value problems in one dimension
- Marcati, Carlo
- Rakhuba, Maxim
- Ulander, Johan E.M.
DOIAbstract
We derive rank bounds on the quantized tensor train (QTT) compressed approximation of singularly perturbed reaction diffusion boundary value problems in one dimension. Specifically, we show that, independently of the scale of the singular perturbation parameter, a numerical solution with accuracy 0<ε<1 can be represented in the QTT format with a number of parameters that depends only polylogarithmically on ε. In other words, QTT-compressed solutions converge exponentially fast to the exact solution, with respect to a root of the number of parameters. We also verify the rank bound estimates numerically and overcome known stability issues of the QTT-based solution of partial differential equations (PDEs) by adapting a preconditioning strategy...
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