Fractional exponential sums

This repository contains the code for the numerical experiments in the paper "Low-rank tensor structure preservation in fractional operators by means of exponential sums"; the experiments are briefly described below. Please note that some of them require the use of the Tensor Toolbox, TensorLab, or the TT-Toolbox.

Description of the experiments

Solution of a tensor equation

The example num_exp_dense.m solves a fractional tensor equation in 3 dimensions by exponential sums, and compares the result with the dense solution obtained by diagonalization.

Solution of a low CP-rank equation

The example num_exp_lowrank.m deals with the case above, but assumes that the right hand side in the equation is product of three functions that depend on $x$, $y$, and $z$, respectively. Then, the right hand side is stored in the CP format, and the solution is computed in the same format by exponential sum approximation.

Low-rank approximability properties

The test num_exp_lowrank_approximability.m checks the bounds for the low-rank approximability in CP, TT, and Tucker formats for the solution of fractional differential equations.

High-dimensional TT solver

In the last test, num_exp_tt.m, a high-dimensional equation over $[0, 1]^d$ (up to $d = 20$) is solved by combining a TT-cross approximation for the right hand side, and the exponential sum approximation in the TT-format.