A dynamical low-rank approach to solve the chemical master equation for biological reaction networks

Prugger, Martina; Einkemmer, Lukas; López, Carlos F.

doi:10.1016/j.jcp.2023.112250

Journal of Computational Physics

2023

DOI: 10.1016/j.jcp.2023.112250

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A dynamical low-rank approach to solve the chemical master equation for biological reaction networks

Martina Prugger

Lukas Einkemmer

Carlos F. López

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Cited by 5 publications

(1 citation statement)

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“…Though dynamical low-rank approximation (DLRA) [33] offers a significant reduction of computational costs and memory consumption when solving tensor differential equations [26,10,54], the use of DLRA to solve matrix differential equations has sparked immense interest in several communities. Research fields in which DLRA for matrix differential equations has a considerable impact include plasma physics [21,23,16,5,18,12,24,13,20,55], radiation transport [47,14,45,39,46,56,17,3,38], chemical kinetics [29,48,22], wave propagation [28,57], uncertainty quantification [49,2,25,41,42,43,36,30,15,1], and machine learning [52,58,50,51]. These application fields commonly require memory-intensive and computational costly numerical simulations due to the solution's prohibitively large phase space.…”

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confidence: 99%

On the Stability of Robust Dynamical Low-Rank Approximations for Hyperbolic Problems

Kusch

Einkemmer

Ceruti

2023

SIAM J. Sci. Comput.

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The dynamical low-rank approximation (DLRA) is used to treat high-dimensional problems that arise in such diverse fields as kinetic transport and uncertainty quantification. Even though it is well known that certain spatial and temporal discretizations when combined with the DLRA approach can result in numerical instability, this phenomenon is poorly understood. In this paper we perform a L 2 stability analysis for the corresponding nonlinear equations of motion. This reveals the source of the instability for the projector splitting integrator when first discretizing the equations and then applying the DLRA. Based on this we propose a projector splitting integrator, based on applying DLRA to the continuous system before performing the discretization, that recovers the classic CFL condition. We also show that the unconventional integrator has more favorable stability properties and explain why the projector splitting integrator performs better when approximating higher moments, while the unconventional integrator is generally superior for first order moments. Furthermore, an efficient and stable dynamical low-rank update for the scattering term in kinetic transport is proposed. Numerical experiments for kinetic transport and uncertainty quantification, which confirm the results of the stability analysis, are presented.

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mentioning

confidence: 99%

On the Stability of Robust Dynamical Low-Rank Approximations for Hyperbolic Problems

Kusch

Einkemmer

Ceruti

2023

SIAM J. Sci. Comput.

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A robust second-order low-rank BUG integrator based on the midpoint rule

Ceruti,

Einkemmer,

Kusch

et al. 2024

Bit Numer Math

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Dynamical low-rank approximation has become a valuable tool to perform an on-the-fly model order reduction for prohibitively large matrix differential equations. A core ingredient is the construction of integrators that are robust to the presence of small singular values and the resulting large time derivatives of the orthogonal factors in the low-rank matrix representation. Recently, the robust basis-update & Galerkin (BUG) class of integrators has been introduced. These methods require no steps that evolve the solution backward in time, often have favourable structure-preserving properties, and allow for parallel time-updates of the low-rank factors. The BUG framework is flexible enough to allow for adaptations to these and further requirements. However, the BUG methods presented so far have only first-order robust error bounds. This work proposes a second-order BUG integrator for dynamical low-rank approximation based on the midpoint quadrature rule. The integrator first performs a half-step with a first-order BUG integrator, followed by a Galerkin update with a suitably augmented basis. We prove a robust second-order error bound which in addition shows an improved dependence on the normal component of the vector field. These rigorous results are illustrated and complemented by a number of numerical experiments.

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Full-rank and low-rank splitting methods for the Swift–Hohenberg equation

Zhao,

2023

Communications in Nonlinear Science and Numerical Simulation

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A dynamical low-rank approach to solve the chemical master equation for biological reaction networks

Cited by 5 publications

References 50 publications

On the Stability of Robust Dynamical Low-Rank Approximations for Hyperbolic Problems

On the Stability of Robust Dynamical Low-Rank Approximations for Hyperbolic Problems

A robust second-order low-rank BUG integrator based on the midpoint rule

Full-rank and low-rank splitting methods for the Swift–Hohenberg equation

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