Fast recursive filters for simulating nonlinear dynamic systems

A fast and accurate computational scheme for simulating nonlinear dynamic systems is presented. The scheme assumes that the system can be represented by a combination of components of only two different types: first-order low-pass filters and static nonlinearities. The parameters of these filters and nonlinearities may depend on system variables, and the topology of the system may be complex, including feedback. Several examples taken from neuroscience are given: phototransduction, photopigment bleaching, and spike generation according to the Hodgkin-Huxley equations. The scheme uses two slightly different forms of autoregressive filters, with an implicit delay of zero for feedforward control and an implicit delay of half a sample distance for feedback control. On a fairly complex model of the macaque retinal horizontal cell it computes, for a given level of accuracy, 1-2 orders of magnitude faster than 4th-order Runge-Kutta. The computational scheme has minimal memory requirements, and is also suited for computation on a stream processor, such as a GPU (Graphical Processing Unit).Comment: 20 pages, 8 figures, 1 table. A comparison with 4th-order Runge-Kutta integration shows that the new algorithm is 1-2 orders of magnitude faster. The paper is in press now at Neural Computatio

Dynamic Estimation of Rigid Motion from Perspective Views via Recursive Identification of Exterior Differential Systems with Parameters on a Topological Manifold

We formulate the problem of estimating the motion of a rigid object viewed under perspective projection as the identification of a dynamic model in Exterior Differential form with parameters on a topological manifold. We first describe a general method for recursive identification of nonlinear implicit systems using prediction error criteria. The parameters are allowed to move slowly on some topological (not necessarily smooth) manifold. The basic recursion is solved in two different ways: one is based on a simple extension of the traditional Kalman Filter to nonlinear and implicit measurement constraints, the other may be regarded as a generalized "Gauss-Newton" iteration, akin to traditional Recursive Prediction Error Method techniques in linear identification. A derivation of the "Implicit Extended Kalman Filter" (IEKF) is reported in the appendix. The ID framework is then applied to solving the visual motion problem: it indeed is possible to characterize it in terms of identification of an Exterior Differential System with parameters living on a C0 topological manifold, called the "essential manifold". We consider two alternative estimation paradigms. The first is in the local coordinates of the essential manifold: we estimate the state of a nonlinear implicit model on a linear space. The second is obtained by a linear update on the (linear) embedding space followed by a projection onto the essential manifold. These schemes proved successful in performing the motion estimation task, as we show in experiments on real and noisy synthetic image sequences

Implicit complexity for coinductive data: a characterization of corecurrence

We propose a framework for reasoning about programs that manipulate coinductive data as well as inductive data. Our approach is based on using equational programs, which support a seamless combination of computation and reasoning, and using productivity (fairness) as the fundamental assertion, rather than bi-simulation. The latter is expressible in terms of the former. As an application to this framework, we give an implicit characterization of corecurrence: a function is definable using corecurrence iff its productivity is provable using coinduction for formulas in which data-predicates do not occur negatively. This is an analog, albeit in weaker form, of a characterization of recurrence (i.e. primitive recursion) in [Leivant, Unipolar induction, TCS 318, 2004].Comment: In Proceedings DICE 2011, arXiv:1201.034

Asymptotic integration algorithms for nonhomogeneous, nonlinear, first order, ordinary differential equations

New methods for integrating systems of stiff, nonlinear, first order, ordinary differential equations are developed by casting the differential equations into integral form. Nonlinear recursive relations are obtained that allow the solution to a system of equations at time t plus delta t to be obtained in terms of the solution at time t in explicit and implicit forms. Examples of accuracy obtained with the new technique are given by considering systems of nonlinear, first order equations which arise in the study of unified models of viscoplastic behaviors, the spread of the AIDS virus, and predator-prey populations. In general, the new implicit algorithm is unconditionally stable, and has a Jacobian of smaller dimension than that which is acquired by current implicit methods, such as the Euler backward difference algorithm; yet, it gives superior accuracy. The asymptotic explicit and implicit algorithms are suitable for solutions that are of the growing and decaying exponential kinds, respectively, whilst the implicit Euler-Maclaurin algorithm is superior when the solution oscillates, i.e., when there are regions in which both growing and decaying exponential solutions exist