Kenneth R. Jackson

Validated solutions of initial value problems for ordinary differential equations

Compared to standard numerical methods for initial-value problems (IVPs) for ordinary differential equations (ODEs), validated methods for IVPs for ODEs have two important advantages: if they return a solution to a problem, then (1) the problem is guaranteed to have a unique solution, and (2) an enclosure of the true solution is produced. The authors survey Taylor series methods for validated solutions of IVPs for ODEs, describe several such methods in a common framework, and identify areas for future research.

Interpolants for Runge-Kutta formulas

A general procedure for the construction of interpolants for Runge-Kutta (RK) formulas is presented. As illustrations, this approach is used to develop interpolants for three explicit RK formulas, including those employed in the well-known subroutines RKF45 and DVERK. A typical result is that no extra function evaluations are required to obtain an interpolant with <italic>O</italic>(<italic>h</italic><supscrpt>5</supscrpt>) local truncation error for the fifth-order RK formula used in RKF45; two extra function evaluations per step are required to obtain an interpolant with <italic>O</italic>(<italic>h</italic><supscrpt>6</supscrpt>) local truncation error for this RK formula.

On Taylor Model Based Integration of ODEs

Interval methods for verified integration of initial value problems (IVPs) for ODEs have been used for more than 40 years. For many classes of IVPs, these methods are able to compute guaranteed error bounds for the flow of an ODE, where traditional methods provide only approximations to a solution. Overestimation, however, is a potential drawback of verified methods. For some problems, the computed error bounds become overly pessimistic, or the integration even breaks down. The dependency problem and the wrapping effect are particular sources of overestimations in interval computations. Berz and his coworkers have developed Taylor model methods, which extend interval arithmetic with symbolic computations. The latter is an effective tool for reducing both the dependency problem and the wrapping effect. By construction, Taylor model methods appear particularly suitable for integrating nonlinear ODEs. We analyze Taylor model based integration of ODEs and compare Taylor model methods with traditional enclosure methods for IVPs for ODEs.

Nonlinearly Preconditioned Krylov Subspace Methods for Discrete Newton Algorithms

We propose an algorithm for implementing Newtons method for a general nonlinear system

An Effective High-Order Interval Method for Validating Existence and Uniqueness of the Solution of an IVP for an ODE

f(x) = 0

The potential for parallelism in Runge-Kutta methods. Part 1: RK formulas in standard form

where the linear systems that arise at each step of Newtons method are solved by a preconditioned Krylov subspace iterative method. The algorithm requires only function evaluations and does not require the evaluation or storage of the Jacobian matrix. Matrix-vector products involving the Jacobian matrix are approximated by directional differences. We develop a framework for constructing preconditionings for this inner iterative method which do not reference the Jacobian matrix explicitly. We derive a nonlinear SSOR type preconditioning which numerical experiments show to be as effective as the linear SSOR preconditioning that uses the Jacobian explicitly.

An Interval Hermite-Obreschkoff Method for Computing Rigorous Bounds on the Solution of an Initial Value Problem for an Ordinary Differential Equation

Validated methods for initial value problems for ordinary differential equations produce bounds that are guaranteed to contain the true solution of a problem. When computing such bounds, these methods verify that a unique solution to the problem exists in the interval of integration and compute a priori bounds for the solution in this interval. A major difficulty in this verification phase is how to take as large a stepsize as possible, subject to some tolerance requirement. We propose a high-order enclosure method for proving existence and uniqueness of the solution and computing a priori bounds.

Some recent advances in validated methods for IVPs for ODEs

The authors examine the potential for parallelism in Runge–Kutta (RK) methods based on formulas in standard one-step form. Both negative and positive results are presented. Many of the negative results are based on a theorem that bounds the order of an RK formula in terms of the minimal polynomial associated with its coefficient matrix. The positive results are largely examples of prototypical formulas that offer a potential for effective “coarse-grain” parallelism on machines with a few processors.

A comparison of two ode codes: gear and episode☆

To date, the only effective approach for computing guaranteed bounds on the solution of an initial value problem (IVP) for an ordinary differential equation (ODE) has been interval methods based on Taylor series. This paper derives a new approach, an interval Hermite-Obreschkoff (IHO) method, for computing such enclosures. Compared to interval Taylor series (ITS) methods, for the same stepsize and order, our IHO scheme has a smaller truncation error, better stability, and requires fewer Taylor coefficients and high-order Jacobians.The stability properties of the ITS and IHO methods are investigated. We show as an important by-product of this analysis that the stability of an interval method is determined not only by the stability function of the underlying formula, as in a standard method for an IVP for an ODE, but also by the associated formula for the truncation error.

A survey of parallel numerical methods for initial value problems for ordinary differential equations

Compared to standard numerical methods for initial value problems (IVPs) for ordinary differential equations (ODEs), validated methods (often called interval methods) for IVPs for ODEs have two important advantages: if they return a solution to a problem, then (1) the problem is guaranteed to have a unique solution, and (2) an enclosure of the true solution is produced.We present a brief overview of interval Taylor series (ITS) methods for IVPs for ODEs and discuss some recent advances in the theory of validated methods for IVPs for ODEs. In particular, we discuss an interval Hermite-Obreschkoff (IHO) scheme for computing rigorous bounds on the solution of an IVP for an ODE, the stability of ITS and IHO methods, and a new perspective on the wrapping effect, where we interpret the problem of reducing the wrapping effect as one of finding a more stable scheme for advancing the solution.