This paper proposes a new method for estimating the error in the solution of matrix equations. The estimate is based on the adjoint method in combination with small sample statistical theory. It can be implemented simply and is inexpensive to compute. Numerical examples are presented which illustrate the power and effectiveness of the new method. (C) 2004 IMACS. Published by Elsevier B.V. All rights reserved.

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## Scholarly Works (9 results)

In this paper we propose a general method for a posteriori error estimation in the solution of initial value problems in ordinary differential equations (ODEs). With the help of adjoint sensitivity software, this method can be implemented efficiently. It provides a condition estimate for the ODE system. We also propose an algorithm for global error control, based on the condition of the system and the perturbation due to the numerical approximation.

In this paper we examine the different formulations of Gillespie's stochastic simulation algorithm (SSA) [D. Gillespie, J. Phys. Chem. 81, 2340 (1977)] with respect to computational efficiency, and propose an optimization to improve the efficiency of the direct method. Based on careful timing studies and an analysis of the time-consuming operations, we conclude that for most practical problems the optimized direct method is the most efficient formulation of SSA. This is in contrast to the widely held belief that Gibson and Bruck's next reaction method [M. Gibson and J. Bruck, J. Phys. Chem. A 104, 1876 (2000)] is the best way to implement the SSA for large systems. Our analysis explains the source of the discrepancy. (C) 2004 American Institute of Physics.

In this paper, we introduce a multiscale stochastic simulation algorithm (MSSA) which makes use of Gillespie's stochastic simulation algorithm (SSA) together with a new stochastic formulation of the partial equilibrium assumption (PEA). This method is much more efficient than SSA alone. It works even with a very small population of fast species, Implementation details are discussed, and an application to the modeling of the heat shock response of E. Coli is presented which demonstrates the excellent efficiency and accuracy obtained with the new method. (c) 2005 Elsevier Inc. All rights reserved.

Tau-leaping methods have recently been proposed for the acceleration of discrete stochastic simulation of chemically reacting systems. This paper considers the numerical stability of these methods. The concept of stochastic absolute stability is defined, discussed, and applied to the following leaping methods: the explicit tau, implicit tau, and trapezoidal tau.(C) 2004 American Institute of Physics.

The explicit tau-leaping procedure attempts to speed up the stochastic simulation of a chemically reacting system by approximating the number of firings of each reaction channel during a chosen time increment tau as a Poisson random variable. Since the Poisson random variable can have arbitrarily large sample values, there is always the possibility that this procedure will cause one or more reaction channels to fire so many times during tau that the population of some reactant species will be driven negative. Two recent papers have shown how that unacceptable occurrence can be avoided by replacing the Poisson random variables with binomial random variables, whose values are naturally bounded. This paper describes a modified Poisson tau-leaping procedure that also avoids negative populations, but is easier to implement than the binomial procedure. The new Poisson procedure also introduces a second control parameter, whose value essentially dials the procedure from the original Poisson tau-leaping at one extreme to the exact stochastic simulation algorithm at the other; therefore, the modified Poisson procedure will generally be more accurate than the original Poisson procedure. (C) 2005 American Institute of Physics.

Reactions in real chemical systems often take place on vastly different time scales, with "fast" reaction channels firing very much more frequently than "slow" ones. These firings will be interdependent if, as is usually the case, the fast and slow reactions involve some of the same species. An exact stochastic simulation of such a system will necessarily spend most of its time simulating the more numerous fast reaction events. This is a frustratingly inefficient allocation of computational effort when dynamical stiffness is present, since in that case a fast reaction event will be of much less importance to the system's evolution than will a slow reaction event. For such situations, this paper develops a systematic approximate theory that allows one to stochastically advance the system in time by simulating the firings of only the slow reaction events. Developing an effective strategy to implement this theory poses some challenges, but as is illustrated here for two simple systems, when those challenges can be overcome, very substantial increases in simulation speed can be realized. (C) 2005 American Institute of Physics.