Shrink wrapping for Taylor models revisited

Taylor models have been used successfully to calculate verified inclusions of the solutions of initial value problems for ordinary differential equations. In this context, Makino and Berz introduced an accompanying method called shrink wrapping. This method aims to reduce the wrapping effect which occurs during repeated forward integration of Taylor models. We review shrink wrapping as proposed by Makino and Berz, state examples that point to a flaw in their theorem and concept of proof, and present a new, corrected version of shrink wrapping.     

A Chebyshev interval method for nonlinear dynamic systems under uncertainty

This paper proposes a new interval analysis method for the dynamic response of nonlinear systems with uncertain-but-bounded parameters using Chebyshev polynomial series. Interval model can be used to describe nonlinear dynamic systems under uncertainty with low-order Taylor series expansions. However, the Taylor series-based interval method can only suit problems with small uncertain levels. To account for larger uncertain levels, this study introduces Chebyshev series expansions into interval model to develop a new uncertain method for dynamic nonlinear systems. In contrast to the Taylor series, the Chebyshev series can offer a higher numerical accuracy in the approximation of solutions. The Chebyshev inclusion function is developed to control the overestimation in interval computations, based on the truncated Chevbyshev series expansion. The Mehler integral is used to calculate the coefficients of Chebyshev polynomials. With the proposed Chebyshev approximation, the set of ordinary differential equations (ODEs) with interval parameters can be transformed to a new set of ODEs with deterministic parameters, to which many numerical solvers for ODEs can be directly applied. Two numerical examples are applied to demonstrate the effectiveness of the proposed method, in particular its ability to effectively control the overestimation as a non-intrusive method.     

Computing reachable sets for uncertain nonlinear hybrid systems using interval constraint-propagation techniques

We investigate solution techniques for numerical constraint-satisfaction problems and validated numerical set integration methods for computing reachable sets of nonlinear hybrid dynamical systems in the presence of uncertainty. To use interval simulation tools with higher-dimensional hybrid systems, while assuming large domains for either initial continuous state or model parameter vectors, we need to solve the problem of flow/sets intersection in an effective and reliable way. The main idea developed in this paper is first to derive an analytical expression for the boundaries of continuous flows, using interval Taylor methods and techniques for controlling the wrapping effect. Then, the event detection and localization problems underlying flow/sets intersection are expressed as numerical constraint-satisfaction problems, which are solved using global search methods based on branch-and-prune algorithms, interval analysis and consistency techniques. The method is illustrated with hybrid systems with uncertain nonlinear continuous dynamics and nonlinear invariants and guards.     

Taylor series method for dynamical systems with control: Convergence and error estimates

To optimize a complicated function constructed from a solution of a system of ordinary differential equations (ODEs), it is very important to be able to approximate a solution of a system of ODEs very precisely. The precision delivered by the standard Runge-Kutta methods often is insufficient, resulting in a “noisy function” to optimize. We consider an initial-value problem for a system of ordinary differential equations having polynomial right-hand sides with respect to all dependent variables. First we show how to reduce a wide class of ODEs to such polynomial systems. Using the estimates for the Taylor series method, we construct a new “aggregative” Taylor series method and derive guaranteed a priori step-size and error estimates for Runge-Kutta methods of order r. Then we compare the 8,13-Prince-Dormand’s, Taylor series, and aggregative Taylor series methods using seven benchmark systems of equations, including van der Pol’s equations, the “brusselator,” equations of Jacobi’s elliptic functions, and linear and nonlinear stiff systems of equations. The numerical experiments show that the Taylor series method achieves the best precision, while the aggregative Taylor series method achieves the best computational time. The final section of this paper is devoted to a comparative study of the above numerical integration methods for systems of ODEs describing the optimal flight of a spacecraft from the Earth to the Moon. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 24, Dynamical Systems and Optimization, 2005.     

Generalized Intervals and the Dependency Problem

Computing enclosures of the range of functions using interval arithmetic often leads to overestimations due to variables (parameters) appearing more than once within the expression to be evaluated (dependency problem). An experimental implementation of a generalized interval arithmetic which has been proposed by Hansen [2] in 1975 is now available [1, 3]. In many cases generalized intervals are well suited to reduce the dependency problem and/or the so called wrapping effect. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

One-step 9-stage Hermite–Birkhoff–Taylor DAE solver of order 10

The HBT(10)9 method for ODEs is expanded into HBT(10)9DAE for solving nonstiff and moderately stiff systems of fully implicit differential algebraic equations (DAEs) of arbitrarily high fixed index. A scheme to generate first-order derivatives at off-step points is combined with Pryce scheme which generates high order derivatives at step points. The stepsize is controlled by a local error estimator. HBT(10)9DAE uses only the first four derivatives of y instead of the first 10 required by Taylor’s series method T10DAE of order 10. Dormand–Prince’s DP(8,7)13M for ODEs is extended to DP(8,7)DAE for DAEs. HBT(10)9DAE wins over DP(8,7)DAE on several test problems on the basis of CPU time as a function of relative error at the end of the interval of integration. An index-5 problem is equally well solved by HBT(10)9DAE and T10DAE. On this problem, the error in the solution by DP(8,7)DAE increases as time increases.     

An Improved Interval Global Optimization Algorithm Using Higher-order Inclusion Function Forms

We propose an improved algorithm for unconstrained global optimization in the framework of the Moore–Skelboe algorithm of interval analysis (H. Ratschek and J. Rokne, New computer methods for global optimization, Wiley, New York, 1988). The proposed algorithm is an improvement over the one recently proposed in P.S.V. Nataraj and K. Kotecha, (J. Global Optimization, 24 (2002) 417). A novel and powerful feature of the proposed algorithm is that it uses a variety of inclusion function forms for the objective function – the simple natural inclusion, the Taylor model (M. Berz and G. Hoffstatter, Reliable Computing, 4 (1998) 83), and the combined Taylor–Bernstein form (P.S.V. Nataraj and K. Kotecha, Reliable Computing, in press). Several improvements are also proposed for the combined Taylor–Bernstein form. The performance of the proposed algorithm is numerically tested and compared with those of existing algorithms on 11 benchmark examples. The results of the tests show the proposed algorithm to be overall considerably superior to the rest, in terms of the various performance metrics chosen for comparison.     

An Algorithm for Global Optimization using the Taylor–Bernstein Form as Inclusion Function

We investigate the use of higher order inclusion functions in the Moore–Skelboe (MS) algorithm of interval analysis (IA) for unconstrained global optimization. We first propose an improvement of the Taylor–Bernstein (TB) form given in (Lin and Rokne (1996) 101) which has the property of higher order convergence. We make the improvement so that the TB form is more effective in practice. We then use the improved TB form as an inclusion function in a prototype MS algorithm and also modify the cut-off test and termination condition in the algorithm. We test and compare on several examples the performances of the proposed algorithm, the MS algorithm, and the MS algorithm with the Taylor model of Berz and Hoffstatter (1998; 97) as inclusion function. The results of these (preliminary) tests indicate that the proposed algorithm with the improved TB form as inclusion function is quite effective for low to medium dimension problems studied.     

Effective matrix formalism for singularity analysis of differential equations and new intergrable system in nonlinear elasticity

We introduce a simple matrix formalism for Taylor series and generalized Laurent series that can be used for implementing the Taylor method for nonlinear ODEs and singularity analysis of differential equations. Advantages of this approach over conventional techniques are shown on model examples. Surprisingly, the same formalism can be used for proving C-integrability of a 3D model in nonlinear elasticity. An alternative proof is obtained by using similarity between the model in nonlinear elasticity and the classic Pohlmeier-Lund-Regge model from high energy physics. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

B-series analysis of iterated Taylor methods

For stochastic implicit Taylor methods that use an iterative scheme to compute their numerical solution, stochastic B-series and corresponding growth functions are constructed. From these, convergence results based on the order of the underlying Taylor method, the choice of the iteration method, the predictor, and the number of iterations, for Itô and Stratonovich SDEs, and for weak as well as strong convergence are derived. As special case, also the application of Taylor methods to ODEs is considered. The theory is supported by numerical experiments.     

求解微分方程初值问题的一种弧长法

对于连续介质力学问题中导出的微分方程初值问题,常常具有解奇异性,如不连续、Ｓｔｉｆ性质或激波间断·本文通过在相应空间,引入一个或数个弧长参数变量,克服解的奇异性·对于常微分方程组引入弧长参数变量后,奇异性得以消除和削弱,应用一般的解常微分方程组的方法（如Ｒｕｎｇｅ＿Ｋｕｔａ法）求解·对于偏微分方程引入弧长参数变量后,在相应的空间离散成常微分方程组,用解奇异性常微分方程组相同的方法即可求解·本文给出了两个算例     

A verified inexact implicit Runge–Kutta method for nonsmooth ODEs

Structural pounding and oscillations have been extensively investigated by using ordinary differential equations (ODEs). In many applications, force functions are defined by piecewise continuously differentiable functions and the ODEs are nonsmooth. Implicit Runge–Kutta (IRK) methods for solving the nonsmooth ODEs are numerically stable, but involve systems of nonsmooth equations that cannot be solved exactly in practice. In this paper, we propose a verified inexact IRK method for nonsmooth ODEs which gives a global error bound for the inexact solution. We use the slanting Newton method to solve the systems of nonsmooth equations, and interval method to compute the set of matrices of slopes for the enclosure of solution of the systems. Numerical experiments show that the algorithm is efficient for verification of solution of systems of nonsmooth equations in the inexact IRK method. We report numerical results of nonsmooth ODEs arising from simulation of the collapse of the Tacoma Narrows suspension bridge, steel to steel impact experiment, and pounding between two adjacent structures in 27 ground motion records for 12 different earthquakes. This work is partly supported by a Grant-in-Aid from Japan Society for the Promotion of Science and a scholarship from Egyptian Government.     

Category-measure duality: convexity,midpoint convexity and Berz sublinearity

Category-measure duality concerns applications of Baire-category methods that have measure-theoretic analogues. The set-theoretic axiom needed in connection with the Baire category theorem is the Axiom of Dependent Choice, DC rather than the Axiom of Choice, AC. Berz used the Hahn–Banach theorem over \({\mathbb {Q}}\) to prove that the graph of a measurable sublinear function that is \({\mathbb {Q}}_{+}\)-homogeneous consists of two half-lines through the origin. We give a category form of the Berz theorem. Our proof is simpler than that of the classical measure-theoretic Berz theorem, our result contains Berz’s theorem rather than simply being an analogue of it, and we use only DC rather than AC. Furthermore, the category form easily generalizes: the graph of a Baire sublinear function defined on a Banach space is a cone. The results are seen to be of automatic-continuity type. We use Christensen Haar null sets to extend the category approach beyond the locally compact setting where Haar measure exists. We extend Berz’s result from Euclidean to Banach spaces, and beyond. Passing from sublinearity to convexity, we extend the Bernstein–Doetsch theorem and related continuity results, allowing our conditions to be ‘local’—holding off some exceptional set.     

Breaking the limits: The Taylor series method

This paper discusses several examples of ordinary differential equation (ODE) applications that are difficult to solve numerically using conventional techniques, but which can be solved successfully using the Taylor series method. These results are hard to obtain using other methods such as Runge-Kutta or similar schemes; indeed, in some cases these other schemes are not able to solve such systems at all. In particular, we explore the use of the high-precision arithmetic in the Taylor series method for numerically integrating ODEs. We show how to compute the partial derivatives, how to propagate sets of initial conditions, and, finally, how to achieve the Brouwer’s Law limit in the propagation of errors in long-time simulations. The TIDES software that we use for this work is freely available from a website.     

A New Integrator for Special Third Order Differential Equations With Application to Thin Film Flow Problem

In recent time, Runge-Kutta methods that integrate special third order ordinary differential equations (ODEs) directly are proposed to address efficiency issues associated with classical Runge-Kutta methods. Albeit, the methods require evaluation of three set of equations to proceed with the numerical integration. In this paper, we propose a class of multistep-like Runge-Kutta methods (hybrid methods), which integrates special third order ODEs directly. The method is completely derivative-free. Algebraic order conditions of the method are derived. Using the order conditions, a four-stage method is presented. Numerical experiment is conducted on some test problems. The method is also applied to a practical problem in Physics and engineering to ascertain its validity. Results from the experiment show that the new method is more accurate and efficient than the classical Runge-Kutta methods and a class of direct Runge-Kutta methods recently designed for special third order ODEs.     

Stress analysis of cross-ply composite laminates with transverse cracks

A boundary-discontinuous Fourier expansion method to analyze the displacements and stresses in cross-ply composite laminates with transverse cracks is presented. The governing equations of the problem are derived on the basis of the generalized plane strain assumption and two-dimensional equations of elasticity. By employing the boundary-discontinuous Fourier expansion method, the governing equations in the form of coupled high-order ODEs are transformed to a set of systems of linear algebraic equations. The method is used to obtain solutions for which published results can be found for comparisons. Compared with the conventional numerical methods for solving coupled high-order ODEs, the method presented is more efficient. Further parametric studies are carried out for cracked laminates with various geometric and material properties. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 6, pp. 795–826, November–December, 2008.     

On interval predictor-corrector methods

One can approximate numerically the solution of the initial value problem using single or multistep methods. Linear multistep methods are used very often, especially combinations of explicit and implicit methods. In floating-point arithmetic from an explicit method (a predictor), we can get the first approximation to the solution obtained from an implicit method (a corrector). We can do the same with interval multistep methods. Realizing such interval methods in floating-point interval arithmetic, we compute solutions in the form of intervals which contain all possible errors. In this paper, we propose interval predictor-corrector methods based on conventional Adams-Bashforth-Moulton and Nyström-Milne-Simpson methods. In numerical examples, these methods are compared with interval methods of Runge-Kutta type and methods based on high-order Taylor series. It appears that the presented methods yield comparable approximations to the solutions.     

Computational bounds on polynomial differential equations

In this paper we study from a computational perspective some properties of the solutions of polynomial ordinary differential equations.We consider elementary (in the sense of Analysis) discrete-time dynamical systems satisfying certain criteria of robustness. We show that those systems can be simulated with elementary and robust continuous-time dynamical systems which can be expanded into fully polynomial ordinary differential equations in Q[π]. This sets a computational lower bound on polynomial ODEs since the former class is large enough to include the dynamics of arbitrary Turing machines.We also apply the previous methods to show that the problem of determining whether the maximal interval of definition of an initial-value problem defined with polynomial ODEs is bounded or not is in general undecidable, even if the parameters of the system are computable and comparable and if the degree of the corresponding polynomial is at most 56.Combined with earlier results on the computability of solutions of polynomial ODEs, one can conclude that there is from a computational point of view a close connection between these systems and Turing machines.     

Factorization of the Loop Algebra and Integrable Toplike Systems

With any Lie algebra of Laurent series with coefficients in a semisimple Lie algebra and its decomposition into a sum of the subalgebra consisting of the Taylor series and a complementary subalgebra, we associate a hierarchy of integrable Hamiltonian nonlinear ODEs. In the case of the so(3) Lie algebra, our scheme covers all classical integrable cases in the Kirchhoff problem of the motion of a rigid body in an ideal fluid. Moreover, the construction allows generating integrable deformations for known integrable models.