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Tensor-product approach to global time-space-parametric discretization of chemical master equation
Sergey Dolgov and Boris N. Khoromskij
We study the application of the novel tensor formats (TT, QTT, QTT-Tucker) to the solution of $d$-dimensional chemical master equations, applied mostly to gene regulating networks (signaling cascades, toggle switches, phage-$\lambda$). For some important cases, e.g. signaling cascade models, we prove good separability properties of the system operator. The time is treated as an additional variable, with the Quantized tensor representations (QTT, QTT-Tucker) employed, leading to the log-complexity in the system size. This global space-time $(d+1)$-dimensional system, approximated in the QTT or QTT-Tucker formats, is solved in the block-diagonal form by the ALS-type iterations. Another issue considered is the quantification of uncertainty, which means that some model parameters are not known exactly, but only their ranges can be estimated. It occurs frequently in real-life systems. In this case, we introduce the unknown parameters as auxiliary variables discretized on the corresponding grids, and solve the global space-parametric system at once in the tensor formats.