Parallel Algorithms for Balanced Truncation Model Reduction of Sparse Systems

Badía, Jordi; Benner, Peter; Mayo, Rafael; Quintana–Ort́ı, Enrique S.

doi:10.1007/11558958_32

Cited by 14 publications

(12 citation statements)

References 23 publications

(33 reference statements)

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“…To exemplify the concepts above, we depict in Figure 3 a comparison of (A) the solution of the CME of (27) with propensities shown in Table I ; (B) the solution of the CME of the stochastic Michaelis-Menten in (28) with the nonlinear propensity in (31); and (C) the solution of the reduced model described in Section III for the last state of the Markov chain, which represents total conversion of the substrate to product. The parameters used are…”

Section: B Stochastic Michaelis-mentenmentioning

confidence: 99%

Order Reduction of the Chemical Master Equation via Balanced Realisation

López‐Caamal

Marquez‐Lago

2014

PLoS ONE

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We consider a Markov process in continuous time with a finite number of discrete states. The time-dependent probabilities of being in any state of the Markov chain are governed by a set of ordinary differential equations, whose dimension might be large even for trivial systems. Here, we derive a reduced ODE set that accurately approximates the probabilities of subspaces of interest with a known error bound. Our methodology is based on model reduction by balanced truncation and can be considerably more computationally efficient than solving the chemical master equation directly. We show the applicability of our method by analysing stochastic chemical reactions. First, we obtain a reduced order model for the infinitesimal generator of a Markov chain that models a reversible, monomolecular reaction. Later, we obtain a reduced order model for a catalytic conversion of substrate to a product (a so-called Michaelis-Menten mechanism), and compare its dynamics with a rapid equilibrium approximation method. For this example, we highlight the savings on the computational load obtained by means of the reduced-order model. Furthermore, we revisit the substrate catalytic conversion by obtaining a lower-order model that approximates the probability of having predefined ranges of product molecules. In such an example, we obtain an approximation of the output of a model with 5151 states by a reduced model with 16 states. Finally, we obtain a reduced-order model of the Brusselator.

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Section: B Stochastic Michaelis-mentenmentioning

confidence: 99%

Order Reduction of the Chemical Master Equation via Balanced Realisation

López‐Caamal

Marquez‐Lago

2014

PLoS ONE

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“…In these numerical experiments, we have used MATLAB's balanced truncation code balancmr, which does not take advantage of the extreme sparsity of the FSP formulation. With parallel algorithms for the balanced truncation of sparse systems, such as those in [22], much of this computational cost may be recovered.…”

Section: Q4mentioning

confidence: 99%

Transient analysis of stochastic switches and trajectories with applications to gene regulatory networks

Munsky

Khammash

2008

IET Syst. Biol.

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Many gene regulatory networks are modeled at the mesoscopic scale, where chemical populations change according to a discrete state (jump) Markov process. The chemical master equation (CME) for such a process is typically infinite dimensional and unlikely to be computationally tractable without reduction. The recently proposed Finite State Projection (FSP) technique allows for a bulk reduction of the CME while explicitly keeping track of its own approximation error. In previous work, this error has been reduced in order to obtain more accurate CME solutions for many biological examples. In this paper, we show that this "error" has far more significance than simply the distance between the approximate and exact solutions of the CME. In particular, we show that this error term serves as an exact measure of the rate of first transition from one system region to another. We demonstrate how this term may 1 be used to (i) directly determine the statistical distributions for stochastic switch rates, escape times, trajectory periods, and trajectory bifurcations, and (ii) evaluate how likely it is that a system will express certain behaviors during certain intervals of time. We also present two systems-theory based FSP model reduction approaches that are particularly useful in such studies. We illustrate the benefits of this approach to analyze the switching behavior of a stochastic model of Gardner's genetic toggle switch.

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“…Lyapunov equations are key mathematical objects in systems theory, analysis and design of (control) systems, and in many applications. Solving these equations is an essential step in balanced realization algorithms [1,2], in procedures for reduced order models for systems or controllers [3][4][5][6][7], in Newton methods for algebraic Riccati equations (AREs) [8][9][10][11][12][13][14], or in stabilization algorithms [12,15,16]. Stability analyses for dynamical systems may also resort to Lyapunov equations.…”

Section: Introductionmentioning

confidence: 99%

“…where x(t) ∈ IR n , and δ(x(t)) is either dx(t)/dt-the differential operator, or x(t + 1)-the advance difference operator, respectively. A necessary and sufficient condition for asymptotic stability of system (3) is that for any symmetric positive definite matrix Y, denoted Y > 0, there is a unique solution X > 0 of the Lyapunov Equation (1), or (2). Several other facts deserve to be mentioned.…”

Section: Introductionmentioning

confidence: 99%

Comparative Performance Evaluation of an Accuracy-Enhancing Lyapunov Solver

Sima

2019

Information

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Lyapunov equations are key mathematical objects in systems theory, analysis and design of control systems, and in many applications, including balanced realization algorithms, procedures for reduced order models, Newton methods for algebraic Riccati equations, or stabilization algorithms. A new iterative accuracy-enhancing solver for both standard and generalized continuous- and discrete-time Lyapunov equations is proposed and investigated in this paper. The underlying algorithm and some technical details are summarized. At each iteration, the computed solution of a reduced Lyapunov equation serves as a correction term to refine the current solution of the initial equation. The best available algorithms for solving Lyapunov equations with dense matrices, employing the real Schur(-triangular) form of the coefficient matrices, are used. The reduction to Schur(-triangular) form has to be done only once, before starting the iterative process. The algorithm converges in very few iterations. The results obtained by solving series of numerically difficult examples derived from the SLICOT benchmark collections for Lyapunov equations are compared to the solutions returned by the MATLAB and SLICOT solvers. The new solver can be more accurate than these state-of-the-art solvers and requires little additional computational effort.

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Parallel Algorithms for Balanced Truncation Model Reduction of Sparse Systems

Cited by 14 publications

References 23 publications

Order Reduction of the Chemical Master Equation via Balanced Realisation

Order Reduction of the Chemical Master Equation via Balanced Realisation

Transient analysis of stochastic switches and trajectories with applications to gene regulatory networks

Comparative Performance Evaluation of an Accuracy-Enhancing Lyapunov Solver

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