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It is increasingly realized that taking stochastic effects into account is important in order to study biological cells. However, the corresponding mathematical formulation, the chemical master equation (CME), suffers from the curse of dimensionality and thus solving it directly is not feasible for most realistic problems. In this paper we propose a dynamical low-rank algorithm for the CME that reduces the dimensionality of the problem by dividing the reaction network into partitions. Only reactions that cross partitions are subject to an approximation error (everything else is computed exactly). This approach, compared to the commonly used stochastic simulation algorithm (SSA, a Monte Carlo method), has the advantage that it is completely noise-free. This is particularly important if one is interested in resolving the tails of the probability distribution. We show that in some cases (e.g. for the lambda phage) the proposed method can drastically reduce memory consumption and run time and provide better accuracy than SSA. • We propose a low-rank algorithm that does not rely on single orbital basis functions. • The proposed approach is better suited for biological/chemical reaction networks. • We demonstrate better accuracy and improved efficiency compared to SSA for some problems. • The proposed complexity reduction algorithm is noise free (in contrast to SSA). • We demonstrate that our algorithm accurately resolves the tails of the distribution (in contrast to SSA). [ABSTRACT FROM AUTHOR]

In this paper, a new descent approximate modified residual algorithm is developed to solve a large scale system of nonlinear symmetric equations, where the basic strategy to improve its numerical performance is to approximately compute the gradients and the difference of gradients. The error bounds of this approximation are presented, and in virtue of this approximation, a conjugate gradient algorithm for solving large-scale optimization problems in the literature is extended to solve the system of nonlinear symmetric equations without needs of computing and storing the Jacobian matrices or their approximate matrices. It is proved that the obtained search directions in our developed algorithm are sufficiently decent with respect to the so-called approximate modified residues. Under mild assumptions, global and local convergence results of the developed algorithm are proved. Numerical tests indicate that the developed algorithm outperforms the other similar ones available in the literature. [ABSTRACT FROM AUTHOR]