Linda R. Petzold

Numerical solution of initial-value problems in differential-algebraic equations

Preface 1. Introduction: why DAEs? Basic types of DAEs applications Overview 2. Theory of DAEs Iintroduction solvability and the index Linear constant coefficient DAEs Linear time varying DAEs Nonlinear systems 3. Multistep methods Introduction DBF convergence BDF methods, DAEs and stiff problems General linear multistep methods 4. One-step methods Introduction Linear constant coefficient systems Nonlinear index one systems Semi-Explicit Nonlinear Index Two systems Order reduction and stiffness Extrapolation Methods 5. Software and DAEs Introduction Algorithms and Strategies in Dassl Obtaining numerical solutions Solving higher index systems 6. Applications. Introduction Systems of rigid bodies Trajectory prescribed path control Electrical networks DAEs arising from the method of lines Bibliography 7. The DAE home page Introduction theoretical advances Numerical analysis advancements DAE software DASSL Supplementary bibliography Index.

Automatic Selection of Methods for Solving Stiff and Nonstiff Systems of Ordinary Differential Equations

This paper describes a scheme for automatically determining whether a problem can be solved more efficiently using a class of methods suited for nonstiff problems or a class of methods designed for stiff problems. The technique uses information that is available at the end of each step in the integration for making the decision between the two types of methods. If a problem changes character in the interval of integration, the solver automatically switches to the class of methods which is likely to be most efficient for that part of the problem. Test results, using a modified version of the LSODE package, indicate that many problems can be solved more efficiently using this scheme than with a single class of methods, and that the overhead of choosing the most efficient methods is relatively small.

Stiffness in stochastic chemically reacting systems: The implicit tau-leaping method

We show how stiffness manifests itself in the simulation of chemical reactions at both the continuous-deterministic level and the discrete-stochastic level. Existing discrete stochastic simulation methods, such as the stochastic simulation algorithm and the (explicit) tau-leaping method, are both exceedingly slow for such systems. We propose an implicit tau-leaping method that can take much larger time steps for many of these problems.

Improved leap-size selection for accelerated stochastic simulation

In numerically simulating the time evolution of a well-stirred chemically reacting system, the recently introduced “tau-leaping” procedure attempts to accelerate the exact stochastic simulation algorithm by using a special Poisson approximation to leap over sequences of noncritical reaction events. Presented here is an improved procedure for determining the maximum leap size for a specified degree of accuracy.

Using Krylov methods in the solution of large-scale differential-algebraic systems

In this paper, a new algorithm for the solution of large-scale systems of differential-algebraic equations is described. It is based on the integration methods in the solver DASSL, but instead of a direct method for the associated linear systems which arise at each time step, we apply the preconditioned GMRES iteration in combination with an Inexact Newton Method. The algorithm, along with those in DASSL, is implemented in a new solver called DASPK. We outline the algorithms and strategies used, and discuss the use of the solver. We develop and analyze some preconditioners for a certain class of DAE stems, and finally demonstrate the application of DASPK on two example problems.

Efficient formulation of the stochastic simulation algorithm for chemically reacting systems

In this paper we examine the different formulations of Gillespies 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 Brucks 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.

A New Look at Proper Orthogonal Decomposition

We investigate some basic properties of the proper orthogonal decomposition (POD) method as it is applied to data compression and model reduction of finite dimensional nonlinear systems. First we provide an analysis of the errors involved in solving a nonlinear ODE initial value problem using a POD reduced order model. Then we study the effects of small perturbations in the ensemble of data from which the POD reduced order model is constructed on the reduced order model. We explain why in some applications this sensitivity is a concern while in others it is not. We also provide an analysis of computational complexity of solving an ODE initial value problem and study the computational savings obtained by using a POD reduced order model. We provide several examples to illustrate our theoretical results.

Numerical methods and software for sensitivity analysis of differential-algebraic systems

Abstract A modification to explicit Runge-Kutta (RK) methods is proposed. Schemes are constructed which require less derivative-evaluations to achieve a certain order than the classical RK methods do. As an example, we give a second-order method requiring one evaluation, two third-order methods using one and two evaluations, respectively and finally a fourth-order method which requires two evaluations. Numerical examples illustrate the behaviour of these schemes.

Adjoint Sensitivity Analysis for Differential-Algebraic Equations: The Adjoint DAE System and Its Numerical Solution

An adjoint sensitivity method is presented for parameter-dependent differential-algebraic equation systems (DAEs). The adjoint system is derived, along with conditions for its consistent initialization, for DAEs of index up to two (Hessenberg). For stable linear DAEs, stability of the adjoint system (for semi-explicit DAEs) or of an augmented adjoint system (for fully implicit DAEs) is shown. In addition, it is shown for these systems that numerical stability is maintained for the adjoint system or for the augmented adjoint system.

Avoiding negative populations in explicit Poisson tau-leaping.

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.