Gregory J. Reid

Differential Elimination–Completion Algorithms for DAE and PDAE

Differential–algebraic equations (DAE) and partial differential–algebraic equations (PDAE) are systems of ordinary equations and PDAEs with constraints. They occur frequently in such applications as constrained multibody mechanics, spacecraft control, and incompressible fluid dynamics. n n n nA DAE has differential index r if a minimum of r+1 differentiations of it are required before no new constraints are obtained. Although DAE of low differential index (0 or 1) are generally easier to solve numerically, higher index DAE present severe difficulties. n n n nReich et al. have presented a geometric theory and an algorithm for reducing DAE of high differential index to DAE of low differential index. Rabier and Rheinboldt also provided an existence and uniqueness theorem for DAE of low differential index. We show that for analytic autonomous first-order DAE, this algorithm is equivalent to the Cartan–Kuranishi algorithm for completing a system of differential equations to involutive form. The Cartan–Kuranishi algorithm has the advantage that it also applies to PDAE and delivers an existence and uniqueness theorem for systems in involutive form. We present an effective algorithm for computing the differential index of polynomially nonlinear DAE. A framework for the algorithmic analysis of perturbed systems of PDAE is introduced and related to the perturbation index of DAE. Examples including singular solutions, the Pendulum, and the Navier–Stokes equations are given. Discussion of computer algebra implementations is also provided.

A complete symbolic-numeric linear method for camera pose determination

Camera pose estimation is the problem of determining the position and orientation of an internally calibrated camera from known 3D reference points and their images. We briefly survey several existing methods for pose estimation, then introduce our new complete linear method, which is based on a symbolic-numeric method from the geometric (Jet) theory of partial differential equations. The method is stable and robust. In particular, it can deal with the points near critical configurations. Numerical experiments are given to show the performance of the new method.

Fast differential elimination in C: The CDiffElim environment

We introduce the CDiffElim environment, written in C, and an algorithm developed in this environment for simplifying systems of overdetermined partial differential equations by using differentiation and elimination. n nThis environment has strategies for addressing difficulties encountered in differential elimination algorithms, such as exhaustion of computer memory due to intermediate expression swell, and failure to complete due to the massive number of calculations involved. These strategies include low-level memory management strategies and data representations that are tailored for efficient differential elimination algorithms. These strategies, which are coded in a low-level C implementation, seem much more difficult to implement in high-level general purpose computer algebra systems. n nA differential elimination algorithm written in this environment is applied to the determination of symmetry properties of classes of (n+1)-dimensional coupled nonlinear partial differential equations of form iut+∇2u+(a(t)|x|2+b(t)·x+c(t)+d|u|4/n)u=0, where u is an m-component vector-valued function. The resulting systems of differential equations for the symmetries have been made available on the web, to be used as benchmark systems for other researchers. The new differential elimination algorithm in C, runs on the test suite an average of 400 times faster than our RifSimp algorithm in Maple. n nNew algorithms, including an enhanced GCD algorithm, and a hybrid symbolic-numeric differential elimination algorithm, are also described.

Symmetry Classification Using Noncommutative Invariant Differential Operators

Given a class F of differential equations, the symmetry classification problem is to determine for each member f ∈ F the structure of its Lie symmetry group Gf or, equivalently, of its Lie symmetry algebra. The components of the symmetry vector fields of the Lie algebra are solutions of an associated overdetermined defining system of differential equations. The usual computer classification method which applies a sequence of total derivative operators and eliminations to this associated system often fails on problems of interest due to the excessive size of expressions generated in intermediate computations. We provide an alternative classification method which exploits the knowledge of an equivalence group G preserving the class. A noncommutative differential elimination procedure due to Lemaire, Reid, and Zhang, where each step of the procedure is invariant under G, can be applied and an existence and uniqueness theorem for the output used to classify the structure of symmetry groups for each f ∈ F. The method is applied to a class of nonlinear diffusion convection equations vx = u, vt = B(u) ux - K(u) which is invariant under a large but easily determined equivalence group G. In this example the complexity of the calculations is much reduced by the use of G-invariant differential operators.

Determination of maximal symmetry groups of classes of differential equations

A symmetry of a differential equation is a transformation which leaves invariant its family of solutions. As the functional form of a member of a class of differential equations changes, its symmetry group can also change.nWe give an algorithm for determining the structure and dimension of the symmetry group(s) of maximal dimension for classes of partial differential equations. It is based on the application of differential elimination algorithms to the linearized equations for the unknown symmetries. Existence and Uniqueness theorems are applied to the output of these algorithms to give the dimension of the maximal symmetry group.nClasses of differential equations considered include ODE of form <italic>u<subscrpt>xx</subscrpt></italic> = ƒ(<italic>x, u, u<subscrpt>x</subscrpt></italic>), Reaction-Diffusion Systems of form <italic>u<subscrpt>t</subscrpt></italic> - <italic>u<subscrpt>xx</subscrpt></italic> = ƒ(<italic>u, v</italic>), <italic>v<subscrpt>t</subscrpt></italic> - <italic>v<subscrpt>xx</subscrpt></italic> = <italic>g</italic>(<italic>u, v</italic>), and Nonlinear Telegraph Systems of form <italic>v<subscrpt>t</subscrpt></italic> = <italic>u<subscrpt>x</subscrpt></italic>, <italic>v<subscrpt>x</subscrpt></italic> = <italic>C</italic>(<italic>u, x</italic>)<italic>u<subscrpt>x</subscrpt></italic> + <italic>B</italic>(<italic>u, x</italic>).

AN EXPLORATION OF HOMOTOPY SOLVING IN MAPLE

Homotopy continuation methods find approximate solutions of a given system by a continuous deformation of the solutions of a related exactly solvable system. There has been much recent progress in the theory and implementation of such path following methods for polynomial systems. In particular, exactly solvable related systems can be given which enable the computation of all isolated roots of a given polynomial system. Extension of such methods to determine manifolds of solutions has also been recently achieved. This progress, and our own research on extending continuation methods to identifying missing constraints for systems of dierential equations, motivated us to implement higher order continuation methods in the computer algebra language Maple. By higher order, we refer to the iterative scheme used to solve for the roots of the homotopy equation at each step. We provide examples for which the higher order iterative scheme achieves a speed up when compared with the standard second order scheme. We also demonstrate how existing Maple numerical ODE solvers can be used to give a predictor only continuation method for solving polynomial systems. We apply homotopy continuation to determine the missing constraints in a system of nonlinear PDE, which is to our knowledge, the first published instance of such a calculation.

Application of numerical algebraic geometry and numerical linear algebra to PDE

The computational difficulty of completing nonlinear pde to involutive form by differential elimination algorithms is a significant obstacle in applications. We apply numerical methods to this problem which, unlike existing symbolic methods for exact systems, can be applied to approximate systems arising in applications.We use Numerical Algebraic Geometry to process the lower order leading nonlinear parts of such pde systems. The irreducible components of such systems are represented by certain generic points lying on each component and are computed by numerically following paths from exactly given points on components of a related system. To check the conditions for involutivity Numerical Linear Algebra techniques are applied to constant matrices which are the leading linear parts of such systems evaluated at the generic points. Representations for the constraints result from applying a method based on Polynomial Matrix Theory.Examples to illustrate the new approach are given. The scope of the method, which applies to complexified problems, is discussed. Approximate ideal and differential ideal membership testing are also discussed.

An algebraic method for analyzing open-loop dynamic systems

This paper reports on the results of combining the Maple packages Dynaflex and RifSimp. The Dynaflex package has been developed to generate the governing dynamical equations for mechanical systems; the RifSimp package has been developed for the symbolic analysis of differential equations. We show that the output equations from Dynaflex can be converted into a form which can be analyzed by RifSimp. Of particular interest is the ability of RifSimp to split a set of differential equations into different cases; each case corresponds to a different set of assumptions, and under some sets of assumptions there are significant simplifications. In order to allow RifSimp to conduct its analysis, the governing equations must be converted from trigonometric form into a polynomial form. After this is done, RifSimp can analyze the system and present its results either graphically, or in list form. The mechanical systems considered are restricted to open-loop systems, because at present, closed-loop systems require too much computation by RifSimp to permit analysis.

Hybrid method for solving new pose estimation equation system

Camera pose estimation is the problem of determining the position and orientation of an internally calibrated camera from known 3D reference points and their images. We introduce a new polynomial equation system for 4-point pose estimation and apply our symbolic-numeric method to solve it stably and efficiently. In particular, our algorithm can also recognize the points near critical configurations and deal with these near critical cases carefully. Numerical experiments are given to show the performance of the hybrid algorithm.

Structure of symmetry of PDE: exploiting partially integrated systems

This work is part of a sequence in which we develop and refine algorithms for computer symmetry analysis of differential equations. We show how to exploit partially integrated forms of symmetry defining systems to assist the differential elimination algorithms that uncover structure of the Lie symmetry algebras. We thus incorporate a key advantage of heuristic integration methods, that of exploiting easy integrals of simple (e.g. one term) PDE that frequently occur in such analyses. A single unified method is given that computes structure constants whether the defining system is unsolved, or has been partially or completely integrated.n We also give a symbolic-numeric algorithm which for the first time can determine the structure of Lie symmetry algebras specified by defining systems that contain floating point coefficients. This algorithm incorporates a numerical version of the Cartan-Kuranishi prolongation projection algorithm from the geometry of differential equations.