Tensorisation in the Solution of Smoluchowski Type Equations
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We investigate the structure of the non-linear operator featured in the Smoluchowski-type system of ordinary differential equations, and find a way to express it algebraically in terms of the parameters of the problem and a few auxiliary tensors, describing, in a sense, the “shape” of the system. We find compact representations of these auxiliary tensors in terms of a Tensor Train decomposition. Provided the parameters admit a compact representation in this format as well, this allows us to rather straightforwardly reuse standard numerical algorithms for a wide range of associated problems, obtaining \(O(\log N)\) asymptotic complexity.
The work was supported by the Russian Science Foundation, grant 19-11-00338.
- 6.Timokhin, I.V., Matveev, S.A., Siddharth, N., Tyrtyshnikov, E.E., Smirnov, A.P., Brilliantov, N.V.: Newton method for stationary and quasi-stationary problems for Smoluchowski-type equations. J. Comput. Phys. 382, 124–137 (2019). https://doi.org/10.1016/j.jcp.2019.01.013MathSciNetCrossRefGoogle Scholar