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Abstract
The object of this work is the estimate of the global error in the numerical solution of the IVP for a system of ODE's. Given a Runge–Kutta formula of order q, which yields an approximation y
n
to the true value y(x
n
), a general, parallel method is presented, that provides a second value y
n
* of order q+2; the global error e
n
=y
n
–y(x
n
) is then estimated by the difference y
n
–y
n
*. The numerical tests reported, show the very good performance of the procedure proposed. A comparison with the code GEM90 is also appended.
... All such methods utilize prefixed and usually equidistant discretization meshes. These meshes exclude completely an automatic discretization error control, which is presently under extensive investigation because of its theoretical and practical value in applied science and engineering [2,[10][11][12][13][16][17][18][19]21,31,32,[43][44][45]63,65,66,[73][74][75][90][91][92][93][94][95][98][99][100][101]. Moreover, it can be even crucial and necessary for decent state estimations in highly nonlinear continuous-discrete stochastic systems including stiff ones elaborated in [49,53]. ...
This paper further advances the idea of accurate Gaussian filtering towards efficient unscented-type Kalman methods for estimating continuous-time nonlinear stochastic systems with discrete measurements. It implies that the differential equations evolving sigma points utilized in computations of the predicted mean and covariance in time-propagations of the Gaussian distribution are solved accurately, i.e. with negligible error. The latter allows the total error of the unscented Kalman filtering technique to be reduced significantly and gives rise to the novel accurate continuous-discrete unscented Kalman filtering algorithm. At the same time, this algorithm is rather vulnerable to round-off and numerical integration errors committed in each state estimation run because of the need for the Cholesky decomposition of covariance matrices involved. Such a factorization will always fail when the covariance's positivity is lost. This positivity lost issue is commonly resolved with square-root filtering implementations, which propagate not the full covariance matrix but its square root (Cholesky factor) instead. Unfortunately, negative weights encountered in applications of the accurate continuous-discrete unscented Kalman filter to high-dimensional stochastic systems preclude from designing conventional square-root methods. In this paper, we address this problem with low-rank Cholesky factor update procedures or with hyperbolic QR transforms used for yielding J-orthogonal square roots. Our novel square-root algorithms are justified theoretically and examined numerically in an air traffic control scenario.
This chapter presents a brief insight into the theory of Runge–Kutta methods for ordinary differential equations. In particular, it gives precise definitions and explains all basic issues and notions in such numerical integration tools. These include the concept of Runge–Kutta formula, its consistency, convergence and stability. The important families of Runge–Kutta schemes for reversible and Hamiltonian problems are also covered, here. A special emphasis is paid to implementation aspects of four subclasses in the implicit Runge–Kutta method realm, which allow cheap and efficient Newton-type iterations to be applied. This chapter pays its particular attention to variable-stepsize implementation of stepping formulas under consideration and elaborates all technical details of contemporary local and global error control mechanisms used for achieving a desirable accuracy of adaptive numerical integration in automatic mode. The presented theoretical analysis of Runge–Kutta schemes is accompanied by detailed pseudo-codes, which clarify algorithmic particulars of such numerical integration procedures and are situated in Appendix of this chapter. Finally, a proper theoretical analysis and exhaustive numerical examination of five ODE solvers used commonly in nonlinear Kalman filtering is fulfilled in MATLAB.
This chapter presents a brief insight into the theory of numerical integration methods for ordinary differential equations. In particular, it gives precise definitions and explains all basic issues and notions of numerical integration. These include the concepts of stepping formulas, their consistency, convergence and stability. A special emphasis is paid to implementation aspects of implicit numerical schemes, including the accuracy and stability of the fixed-point iteration as well as those of various Newton-type iterations. The important families of reversible and Hamiltonian problems are also covered, here. This chapter pays a particular attention to raising the accuracy of one-step methods via the Richardson extrapolation technique and elaborates all technical details of contemporary local and global error control mechanisms used for achieving a desirable accuracy of numerical integration in automatic mode. The presented theoretical analysis of stepping methods under consideration, which are all summarized in the form of pseudo-codes situated in Appendix of this chapter, is always supported with illustrative calculations performed in MATLAB.
This paper aims at constructing the first variable-stepsize doubly quasi-consistent singly diagonally implicit two-step peer pairs equipped with local error control for accurate and efficient numerical integration of stiff ordinary differential equations (ODEs). The main novelty of the numerical integration tools under consideration is that these accommodate the property of double quasi-consistency to variable meshes. This property means that the principal terms of the local and global errors coincide. It improves greatly the accuracy of conventional local error control facility and even translates it into a global one when the influence of the remaining terms in the mentioned error expansions is negligible. Our study proves theoretical conditions, which ensure the double quasi-consistency in the family of variable-stepsize singly diagonally implicit two-step peer formulas, and presents a feasible procedure for constructing such numerical schemes in practice. In particular, two embedded doubly quasi-consistent pairs of convergence order 3 with local error control are built in this way. Our numerical examples confirm their high efficiency for treating stiff ODEs, including large-scale systems obtained by semidiscretization of partial differential equations (PDEs). Moreover, the numerical integration tools designed here are shown to outperform the well-optimized Matlab code ODE23s with local error control, which is also a stiff ODE solver of order 3. An exhaustive comparison to the other built-in Matlab code ODE15s with local error control, which is considered to be a benchmark means for solving stiff ODEs by many practitioners, is also fulfilled and commented.
The problem of accurate and efficient numerical integration of ordinary differential equations (ODEs) is a key issue of computation science, which has a valued practical impact in various topics of research. In this respect, effective global error estimation and control strategies are of particular theoretical and practical significance. They allow many applied dynamic systems to be treated numerically for user-supplied accuracy conditions in automatic mode. The property of double quasi-consistency plays an outstanding role in the mentioned topic because the global (or true) error of a doubly quasi-consistent method is asymptotically equal to its local error. Therefore, this entails quite a nontrivial implication that the true error of numerical integration can be cheaply evaluated and regulated within a single solution run. The latter was considered to be impossible for decades. The property of double quasi-consistency goes back to 2009, when Kulikov studied it in the first time but failed to find doubly quasi-consistent numerical methods among Nordsieck formulas. Later on, Kulikov and Weiner proved the existence of doubly quasi-consistent numerical schemes among superconvergent explicit two-step parallel peer methods and examined their practical efficiency in comparison to explicit Matlab ODE solvers in 2010. The focus of the present research is on accurate numerical integration formulas for treating stiff ODEs, which often arise in practice and for which explicit methods are shown to be ineffective. Here, we make the first step towards an accurate and efficient numerical solution of stiff ODEs and prove existence of implicit doubly quasi-consistent formulas. We fulfill our investigation of double quasi-consistency within the family of fixed-stepsize implicit two-step peer schemes and construct two methods of convergence orders 3 and 4, which possess excellent stability properties. Our theoretical results are confirmed with numerical integration of both nonstiff and stiff test problems in this paper.
This paper elaborates a new numerical method for treating stiff ordinary differential equations (ODEs) which constitute a basic simulation tool in many areas of study. For its efficiency, this technique is grounded in singly diagonally implicit two-step peer schemes designed recently. The latter means that linear systems with the same coefficient matrix are solved to advance a step of the method and to evaluate its global error as well. Moreover, in contrast to many other numerical methods with global error control, we monitor and regulate the scaled global error, which corresponds better with practitioners' needs. In other words, we present and study one peer triple which is able to solve stiff problems and to ensure that the true error of the derived numerical solution does not exceed the user-supplied bound at the same time. This implies that local and global error estimation and control mechanisms are also discussed here. Finally, efficiency of our numerical tool is checked in comparison to efficiency of stiff built-in MATLAB ODE solvers on various test problems. © 2015 Society for Industrial and Applied Mathematics. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
This paper elaborates a global error estimation and control mechanism in explicit two-step peer methods. These recently designed methods exhibit their high efficiency even in comparison to the best explicit Runge-Kutta pairs. More precisely, we form here triples of the so-called superconvergent explicit peer schemes and show that they are cheap and able to achieve preassigned accuracy conditions in automatic mode. For comparison, we present also numerical data derived by built-in explicit Matlab ODE solvers implemented with only local error control. Especially, we point out that a scaled global error is computed and regulated in this paper in contrast to the earlier published results where the absolute values of the global error have been utilized.
The global error estimation and control problem is studied in embedded Runge–Kutta methods applied to ordinary differential equations (ODEs). We consider both nonstiff equations and stiff ones. Such problems arise in many areas of application and their accurate numerical solution is an important issue of applied mathematics. Usually, global error estimation and control techniques implemented in ODE solvers considerably extend execution time and they are rarely used in practical computations, especially, in implicit methods applied to stiff ODEs. In this paper we show that useful information can be obtained via simple summation of local error estimates calculated cheaply in the course of integration by some embedded Runge–Kutta pairs. Thus, since the embedded method local error estimation is a standard option of any practical solver for differential equations, our technique allows the global error estimation to be implemented naturally and without extra computations. Numerical tests presented here confirm this conclusion in practice. A comparison with explicit and implicit ODE solvers implemented with only local error control in MATLAB is also conducted.
Although the issue of mental illness among offender populations has received attention in the last number of years, there are a number of issues related to mental illness among such groups that require more study. One such topic relates to the association between mental illness, actuarially assessed risk of recidivism, and observed rates of reoffending. In the present investigation, file information was reviewed to determine the presence of a variety of mental health conditions. Actuarially based risk assessment data were also collected for participants as well as information regarding suspension, new charges, and convictions. A sample of 136 offenders housed in a halfway house operated by Correctional Service of Canada was included in the present investigation. Results indicated very high rates of serious mental illness in this high-risk population. Offenders with borderline personality disorder and attention deficit hyperactivity disorder were significantly more likely to recidivate or be suspended. Suspensions refer to administrative decisions to place an offender in jail due to problematic behaviour (typically involving a breach of his release conditions or new charges/convictions). Offenders with a diagnosis of paraphilic disorder were significantly less likely to recidivate or be suspended. Results are discussed in light of the available literature.
The user of a code for solving the initial value problem for ordinary differential equations is normally interested in the global error, i. e. the difference between the solution of the problem posed and the numerical result returned by the code. This paper describes a way of estimating the global error reliably while still solving the problem with acceptable efficiency. Global extrapolation procedures are applied to parallel solutions obtained by a Runge-Kutta-Fehlberg method. These ideas are implemented in a Fortran program called GERK, which is ACM Algorithm 504.
How can small-scale parallelism best be exploited in the solution of nonstiff initial value problems? It is generally accepted that only modest gains inefficiency are possible, and it is often the case that fast parallel algorithms have quite crude error control and stepsize selection components. In this paper we consider the possibility of using parallelism to improvereliability andfunctionality rather than efficiency. We present an algorithm that can be used with any explicit Runge-Kutta formula. The basic idea is to take several smaller substeps in parallel with the main step. The substeps provide an interpolation facility that is essentially free, and the error control strategy can then be based on a defect (residual) sample. If the number of processors exceeds (p – 1)/2, wherep is the order of the Runge-Kutta formula, then the interpolant and the error control scheme satisfy very strong reliability conditions. Further, for a given orderp, the asymptotically optimal values for the substep lengths are independent of the problem and formula and hence can be computed a priori. Theoretical comparisons between the parallel algorithm and optimal sequential algorithms at various orders are given. We also report on numerical tests of the reliability and efficiency of the new algorithm, and give some parallel timing statistics from a 4-processor machine.
Various techniques that have been proposed for estimating the accumulated discretization error in the numerical solution of differential equations, particularly ordinary differential equations, are classified, described, and compared. For most of the schemes either an outline of an error analysis is given which explains why the scheme works or a weakness of the scheme is illustrated.
In order to calculate a Runge-Kutta method of order 10, one has to solve a non-linear algebraic system of 1205 equations.
We give here a special class of methods that needs only 17 function evaluations per step.
The user of a code for solving the initial value problem for ordinary differential systems is normally left with the difficult task of assessing the accuracy of the numerical result returned by the code. Even when the code reports an estimate of the global error, the question may remain whether this estimate is correct, i. e. whether the user can rely on the estimate. This paper proposes a simple idea of measuring the reliability of the global error estimate with the aim of assisting the user in the validation of the numerical result. The idea is put into practice with the existing code GERK (ACM Algorithm 504) developed by Shampine and Watts. This code uses global Richardson extrapolation for the error estimation, which in many cases can also be applied to Runge-Kutta codes for delay equations.
Numerical methods for systems of first order ordinary differential equations are tested on a variety of initial value problems. The methods are compared primarily as to how well they can handle relatively routine integration steps under a vareity of accuracy requirements, rather than how well they handle difficulties caused by discontinuities, stiffness, roundoff or getting started. The problems, methods and comparison criteria are specified very carefully. One objective in doing so is to provide a rigorous conceptual basis for comparing methods. Another is to provide a useful standard for such comparisons. A program called DETEST is used to obtain the relevant statistics for any particular method.
It is shown how to exploit the powerful codes based on extrapolation of the midpoint rule so as to solve the initial value problem for a system of ordinary differential equations (ODEs) and to provide estimates of the global (true) error in the solution values. Particular attention is given to the reliability of the estimates and to the overall efficiency of the integration.
Interpolation of high-order Runge-Kutta formulas is always theoretically possible, but in practice it is still unsatisfactory for its expensiveness. In this paper, rather than trying to improve the efficiency, we concentrate our attention to the possibility of using parallelism to improvereliability andfunctionality. Nevertheless, as we shall see, some modest speedup can also be gained. As an illustration, our approach is applied to the well-known RK8(7) pair of Prince and Dormand [10], and itsspeedup andefficiency examined. Numerical experimentation using the nonstiff package of test problems by Enright and Pryce [4] shows the very good performance of the technique proposed.Theoretisch ist eine Interpolation fr Runge-Kutta-Verfahren hoher Ordnung immer mglich, in der Praxis ist sie jedoch wegen ihrer Kosten nicht befriedigend. Anstatt zu versuchen, die Effizienz des Vorgehens zu verbessern, konzentrieren wir uns in dieser Arbeit auf die Verbesserung der Zuverlssigkeit und der Funktionalitt mit Hilfe der Parallelisierung. Tatschlich erreichen wir, wie sich zeigen wird, auch eine bescheidene Beschleunigung. Zur Demonstration wenden wir unsere Vorgangsweise auf das bekannte RK8(7)-Paar von Prince und Dormand [10] an und untersuchen seine Beschleunigung und Effizienz. Numerische Experimente mit dem Testpaket von Enright und Pryce [4] fr nichtsteife Probleme zeigen das ausgezeichnete Verhalten der vorgeschlagenen Technik.
A personal view of the historical development of software based on explicit Runge-Kutta formulas is presented, and some future developments are suggested.
Efficient and asymptotically (h --, 0) correct global error estimation procedures for initial value problem ODE-codes may be based on a suitable version of defect correction. An implementation of such a procedure for a variable order, variable step Adams PECE-code (typically STEP of Shampme) is described; the global error vector is produced simultaneously with the solution vector at each output point which need not be a grid point. The results of extensive experimentation are presented and d~scussed; m these tests, the qualitative and quantitative behavior of the true error was correctly reflected by the numerical estnnates throughout.
Thesis--University of Illinois at Urbana-Champaign. Bibliography: p. 67-68.