Jens Hoefkens

Computing Validated Solutions of Implicit Differential Equations

Ordinary differential equations (ODEs), including high-order implicit equations, describe important problems in mechanical and chemical engineering. However, the use of self-validated methods providing rigorous enclosures of the solution has mostly been limited to explicit and weakly nonlinear problems, and no general-purpose algorithm for the validated integration of general ODE initial value problems has been developed. Since most integration techniques for Differential Algebraic Equations (DAEs) are based on transformation to implicit ODEs, the integration of DAE initial value problems has traditionally been restricted to few hand-picked problems from the relatively small class of low-index systems. The recently developed Taylor model method combines high-order differential algebraic descriptions of functional dependencies with intervals for verification. It has proven its power in several applications, including verified integration of ODEs under avoidance of the wrapping effect. Recognizing antiderivation (integration) as a natural operation on Taylor models yields methods that treat DEs within a fully differential algebraic context as implicit equations made of conventional functions and antiderivation. This method has the potential to be applied to high-index DAE problems and allows the computation of guaranteed enclosures of final conditions from large initial regions for large classes of initial value problems. In the framework of this method, a Taylor model represents the highest derivative of the solution function occurring in the DE and all lower derivatives are treated as antiderivatives of this Taylor model. Consequently, one obtains a set of implicit equations involving only the highest derivative. Utilizing methods of verified inversion of functional dependencies described by Taylor models allows the computation of a guaranteed enclosure of the solution in the form of a Taylor model. The performance of the method is illustrated by detailed examples.

Efficient high-order methods for ODEs and DAEs

We present methods for the high-order differentiation through ordinary differential equations (ODEs), and more importantly, differential algebraic equations (DAEs). First, methods are developed that assert that the requested derivatives are really those of the solution of the ODE, and not those of the algorithm used to solve the ODE. Next, high-order solvers for DAEs are developed that in a fully automatic way turn an n-th order solution step of the DAEs into a corresponding step for an ODE initial value problem. In particular, this requires the automatic high-order solution of implicit relations, which is achieved using an iterative algorithm that converges to the exact result in at most n+1 steps. We give examples of the performance of the method.

Controlling the Wrapping Effect in the Solution of ODEs for Asteroids

During the last decade, substantial progress has been made in fighting the wrapping effect in self-validated integrations of linear systems. However, it is still the main problem limiting the applicability of such methods to the long-term integration of non-linear systems. Here we show how high-order self-validated methods can successfully overcome this obstacle.We study and compare the validated integration of a Kepler problem with conventional and high-order methods represented by AWA and Taylor models, respectively. We show that this simple model problem exhibits significant wrapping that is particularly difficult to control for conventional first-order methods. It will become clear that utilizing high-order methods with shrink wrapping allows the system to be analyzed in a fully validated context over large integration times. By comparing high-order Taylor model integrations with Taylor model methods subjected to an artificial wrapping effect, we show that utilizing high-order methods to propagate initial conditions is indeed the foremost reason for the successful suppression of the wrapping effect.To further demonstrate that high-order Taylor model methods can be used for the integration of complicated non-linear systems, we summarize results obtained from a fully verified and self-validated orbit integration of the near earth asteroid 1997 XF11. Since this asteroid will have several close encounters with Earth, its analysis is an important application of reliable computations.

Verified High-Order Integration of DAEs and Higher-Order ODEs

Within the framework of Taylor models, no fundamental difference exists between the antiderivation and the more standard elementary operations. Indeed, a Taylor model for the antiderivative of another Taylor model is straightforward to compute and trivially satisfies inclusion monotonicity.

Verification of invertibility of complicated functions over large domains

A new method to decide the invertibility of a given high-dimensional function over a domain is presented. The problem arises in the field of verified solution of differential algebraic equations (DAEs) related to the need to perform projections of certain constraint manifolds over large domains. The question of invertibility is reduced to a verified linear algebra problem involving first partials of the function under consideration. Different from conventional approaches, the elements of the resulting matrices are Taylor models for the derivatives of the functions.The linear algebra problem is solved based on Taylor model methods, and it will be shown the method is able to decide invertibility with a conciseness that often goes substantially beyond what can be obtained with other interval methods. The theory of the approach is presented. Comparisons with three other interval-based methods are performed for practical examples, illustrating the applicability of the new method.

Verified High-Order Inversion of Functional Depedencies and Interval Newton Methods

A new method for computing verified enclosures of the inverses of given functions over large domains is presented. The approach is based on Taylor Model methods, and the sharpness of the enclosures scales with a high order of the domain. These methods have applications in the solution of implicit equations and the Taylor Model based integration of Differential Algebraic Equations (DAE) as well as other tasks where obtaining verified high-order models of inverse functions is required.The accuracy of Taylor model methods has been shown to scale with the (n + 1)-st order of the underlying domain, and as a consequence, they are particularly well suited to model functions over relatively large domains. Moreover, since Taylor models can control the cancellation and dependency problems (see Makino, K. and Berz, M.: Efficient Control of the Dependency Problem Based on Taylor Model Methods, Reliable Computing5(1) (1999)) that often affect regular interval techniques, the new method can successfully deal with complicated multidimensional problems. As an application of these new methods, a high-order extension of the standard Interval Newton method that converges approximately with the (n + 1)-st order of the underlying domain is developed.Several examples showing various aspects of the practical behavior of the methods are given.

Verified Control of Near-Earth Asteroid Orbits

Abstract Orbit calculations of near-earth asteroids pose several difficult challenges. First of all, due to the fact that the impact of such an object is potentially serious, the problem requires as precise an answer as possible, for a relatively large range of initial conditions determined by measurement inaccuracies. Secondly, while the primary force to govern the equation of motion is the gravity of the sun, there are many other factors to influence the details of asteroid motion, making the resulting ODE quite complicated. Thirdly, based on the guaranteed prediction of the near-earth passage, a control of the asteroid orbit has to be provided if necessary, which requires verified global optimizations to determine the timing and the direction and intensity of the momentum transfer. In this paper, we outline an approach to exclude collisions or determine the minimum amount of momentum transfer necessary for their avoidance. The approach is based on verified ODE integration and verified global optimization based on the Taylor Model approach.

Differential Algebraic Methods in Feedforward Control Theory1

Abstract Feedforward methods have often been met with concern in the field of control theory for a variety of reasons. One of them has been the lack of efficient and accurate methods for the computation of the inverse dynamics. In this paper we will take a new look at the problem of feedforward control and show how results from the field of differential algebraic methods can be used to obtain very accurate high-order inverses of functional relations. Together with sophisticated DA based ODE integration tools that allow differentiation of the solution with respect to parameters, this opens new opportunities for the application of feedforward methods in many control problems. The applicability of the method will be studied in detail with examples from autonomous feedforward control and time-dependent prescribed path problems. All presented methods have been implemented using the code COSY Infinity (Berz, 1997).