A robust GMRES algorithm in Tensor Train format
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We consider the solution of linear systems with tensor product structure using a GMRES algorithm. In order to cope with the computational complexity in large dimension both in terms of floating point operations and memory requirement, our algorithm is based on low-rank tensor representation, namely the Tensor Train format. In a backward error analysis framework, we show how the tensor approximation affects the accuracy of the computed solution. With the bacwkward perspective, we investigate the situations where the (d+1)-dimensional problem to be solved results from the concatenation of a sequence of d-dimensional problems (like parametric linear operator or parametric right-hand side problems), we provide backward error bounds to relate the accuracy of the (d+1)-dimensional computed solution with the numerical quality of the sequence of d-dimensional solutions that can be extracted form it. This enables to prescribe convergence threshold when solving the (d+1)-dimensional problem that ensures the numerical quality of the d-dimensional solutions that will be extracted from the (d+1)-dimensional computed solution once the solver has converged. The above mentioned features are illustrated on a set of academic examples of varying dimensions and sizes.
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