There has been a fair amount of work in the last two decades characterizing one step integration methods for ODEs having certain geometric properties when applied to general Hamiltonian systems: symplectic integrators; energy/integral-preserving integrators; conjugate-to-symplectic integrators. A different point of view arises if we restrict the class of problems from the case of general Hamiltonian functions to the case of polynomial Hamiltonian functions and polynomial Hamiltonian vector fields. Then, it becomes often easier to preserve these geometric properties. A remarkable case is the one of the Kahan's method. We show that Kahan's discretization of quadratic vector fields is equivalent to a Runge-Kutta method. When the vector field is Hamiltonian on either a symplectic vector space or a Poisson vector space with constant Poisson structure, the map determined by this discretization has a conserved modied Hamiltonian and an invariant measure, a combination previously unknown amongst Runge-Kutta methods applied to nonlinear vector fields. This produces large classes of integrable rational mappings in two and three dimensions, explaining some of the integrable cases that were previously known.
Joint work with Robert McLachlan, Brynjulf Owren and Reinout Quispel.
Various "non-standard" integration schemes for quadratic vector fields have been proposed by Kahan, Mickens, and Hirota & Kimura. Although it is now understood that the Kahan/Hirota-Kimura method can be viewed as a certain type of Runge-Kutta scheme, the connection of these methods with discrete integrable systems remains somewhat mysterious. Here some results on discretization of bi-Hamiltonian vector fields in three dimensions (with Petrera) are reviewed. Recent results (with Towler) on extending this approach to polynomial vector fields of degree three or more are also explained.
In this talk we review the current state of play with regard to the numerical solution of isospectral flows. Classical numerical methods cannot respect eigenvalues as the solution evolves. The alternative is to regard the flow as an action by the orthogonal group on the linear space of symmetric matrices, thereby reducing the task in hand to the solution of orthogonal flows. This can be accomplished either with symplectic RK methods or by mapping the system into the underlying Lie algebra. An exciting alternative for specific flows (e.g. the double-bracket flow and the Bloch-Iserles equations, but the general idea works for all flows that can be represented in a finite "alphabet") is to expand the solution at the algebra level into (nonlinear) Magnus series. Yet another fascinating connection is between certain isospectral flows, Lis-Poisson flows and Lie-algebra representations. All this will be reviewed in the talk. In addition I plan to discuss the application of isospectral flows to a number of linear algebra calculations and the open mathematical challenges in this context.
The well known theorem of Ge and Marsden states that if a numerical scheme for a Hamiltonian system of ODEs is both symplectic and preserves energy, then the scheme is tracking the exact solution up to a diffeomorphism in the independent variable. Implicit in the statement is that the energy and the solution are those of a smooth system.
It is already known that finite difference (FD) variational problems, which are invariant under translation in a time-like dependent variable, conserve both an FD-energy and an FD-symplectic form. In this talk, we show a similar result for finite element (FE) integration schemes.
Recently the authors demonstrated that standard Finite Element integration methods for variational problems will, a priori, preserve a weak form of the Noether conservation laws, provided the FE-Lagrangian is (weakly) invariant under a Lie group action. For translation and rotation group actions on the base space, standard Lagrangian elements may be used. This implies that there wil be exact preservation of an FE-energy for time-translation invariant Lagrangians using standard FE methods. Further, we show that the FE-energy and the continuum limit energy are close in the XXX-norm. No symmetry conditions are required on the mesh, and the conservation law will be as weak as the Lie group invariance of the Lagrangian in the functional analytic sense. Numerical demonstrations of these laws have been carried out, which show that conservation holds to machine precision.
In the talk, we will also show, (work in progress) that there is an FE-symplectic form. The form may be calculated, as it can in both the smooth and finite difference cases, by calculating the variation of the natural boundary conditions (or a second variation of the Lagrangian).
For systems which are inherently discrete, the continuum limit system is an approximation; its energy and its solutions will only approximate the true discrete energy and the true discrete solutions. In this sense, then, the Ge and Marsden theorem can be understood as saying that you cannot mix and match the provenance of your symplectic forms, your energy, and your solutions.
Joint work with Tristan Pryer
Integrable dynamical systems originally appear in classical mechanics and have rich applications to various branches in mathematics as well as in mathematical physics. Numerical analysis is one of such fields. Recurrence relations of some existing numerical algorithms are equivalent to discrete-time integrable systems themselves having explicit solutions and many conserved quantities. They are classified into three classes.
1) matrix eigenvalue algorithms (e.g. qd algorithm vs. discrete time Toda lattice)
2) convergence acceleration algorithms through continued fractions (e.g. ¥epsilon algorithm vs. discrete time potential KdV)
3) algorithms based on additional formulas (e.g. arithmetic-geometric mean algorithm of Gauss)
Conversely, a time discretization of some integrable system may contribute to design a new effective numerical algorithm. A notion of “integrable algorithms” is then introduced.
In this talk I will speak about the recent developments in integrable algorithms. The Lax representation of integrable system enables us to formulate a numerical algorithm as an isospectral deformation of a structured matrix. The mdLVs algorithm for computing singular values of a bi-diagonal matrix is designed by using a discrete time Lotka-Volterra system in mathematical biology. The resulting singular values have a highly relative accuracy. To speed up the mdLVs a new shift strategy is devised. Then the mdLVs is now compared to the today's standard algorithm. New algorithms for computing eigenvalues of band matrices and totally nonnegative matrices are also designed by using a generalized Toda lattice and a generalized Lotka-Volterra system.
This is a joint work with A. Fukuda, Y. Yamamoto, M. Iwasaki, E. Ishiwata, K. Kimura, T. Yamashita and M. Nagata.
Integrable convergence acceleration and related algorithms of rational type possess certain "cross rules" that give rise to singular rules. Cross rules help avoiding near-singularities and singular rules are useful for avoiding division by zero. Cross rules by themselves constitute integrable systems on the two dimensional integer lattice. The nature of cross rules is going to be discussed with respect to their invariance by the action of the Moebius group.
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Last updated 14th March, 2013