New Error Bounds for Approximations from Projected Linear Equations

Joint Technical Report of U.H. and M.I.T. Technical Report C-2008-43 Dept. Computer Science University of Helsinki and LIDS Report 2797 Dept. EECS M.I.T. July 2008; revised July 2009We consider linear fixed point equations and their approximations by projection on a low dimensional subspace. We derive new bounds on the approximation error of the solution, which are expressed in terms of low dimensional matrices and can be computed by simulation. When the fixed point mapping is a contraction, as is typically the case in Markov decision processes (MDP), one of our bounds is always sharper than the standard contraction-based bounds, and another one is often sharper. The former bound is also tight in a worst-case sense. Our bounds also apply to the non-contraction case, including policy evaluation in MDP with nonstandard projections that enhance exploration. There are no error bounds currently available for this case to our knowledge

Measures for pathway analysis in brain white matter using diffusion tensor images

In this paper we discuss new measures for connectivity analysis of brain white matter, using MR diffusion tensor imaging. Our approach is based on Riemannian geometry, the viability of which has been demonstrated by various researchers in foregoing work. In the Riemannian framework bundles of axons are represented by geodesies on the manifold. Here we do not discuss methods to compute these geodesies, nor do we rely on the availability of geodesies. Instead we propose local measures which are directly computable from the local DTI data, and which enable us to preselect viable or exclude uninteresting seed points for the potentially time consuming extraction of geodesies. If geodesies are available, our measures can be readily applied to these as well. We consider two types of geodesic measures. One pertains to the connectivity saliency of a geodesic, the second to its stability with respect to local spatial perturbations. For the first type of measure we consider both differential as well as integral measures for characterizing a geodesic's saliency either locally or globally. (In the latter case one needs to be in possession of the geodesic curve, in the former case a single tangent vector suffices.) The second type of measure is intrinsically local, and turns out to be related to a well known tensor in Riemannian geometry.</p

Finite-Element Discretization of Static Hamilton-Jacobi Equations Based on a Local Variational Principle

We propose a linear finite-element discretization of Dirichlet problems for static Hamilton-Jacobi equations on unstructured triangulations. The discretization is based on simplified localized Dirichlet problems that are solved by a local variational principle. It generalizes several approaches known in the literature and allows for a simple and transparent convergence theory. In this paper the resulting system of nonlinear equations is solved by an adaptive Gauss-Seidel iteration that is easily implemented and quite effective as a couple of numerical experiments show.Comment: 19 page

Recursive computation of limited lookahead supervisory controls for discrete event systems

We continue the study of limited lookahead policies in supervisory control of discrete event systems undertaken in a previous paper. On-line control of discrete event systems using limited lookahead policies requires, after the execution of each event, the calculation of the supremal controllable sublanguage of a given language with respect to another larger language. These two languages are finite and represented by their tree generators, where one tree is a subtree of the other. These trees change dynamically from step to step, where one step is the execution of one event by the system. We show in this paper how to perform this calculation in a recursive manner, in the sense that the calculation for a new pair of trees can make use of the calculation for the preceding pair, thus substantially reducing the amount of computation that has to be done on-line. In order to make such a recursive procedure possible from step to step, we show how the calculation for a single step (i.e., for a given pair of trees) can itself be performed recursively by means of a backward dynamic programming algorithm on the vertices of the larger tree. These two nested recursive procedures are also extended to the limited lookahead version of the “supervisory control problem with tolerance.”Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45118/1/10626_2005_Article_BF01439177.pd

Efficient Algorithms for Image and High Dimensional Data Processing Using Eikonal Equation on Graphs

International audienceIn this paper we propose an adaptation of the static eikonal equation over weighted graphs of arbitrary structure using a framework of discrete operators. Based on this formulation, we provide explicit solu- tions for the L1,L2 and L∞ norms. Efficient algorithms to compute the explicit solution of the eikonal equation on graphs are also described. We then present several applications of our methodology for image processing such as superpixels decomposition, region based segmentation or patch- based segmentation using non-local configurations. By working on graphs, our formulation provides an unified approach for the processing of any data that can be represented by a graph such as high-dimensional data

Fast Marching Method for Generic Shape from Shading

International audienceWe develop a fast numerical method to approximate the solutions of a wide class of equations associated to the Shape From Shading problem. Our method, which is based on the control theory and the interfaces propagation, is an extension of the ?Fast Marching Method? (FMM) [30,25]. In particular our method extends the FMM to some equations for which the solution is not systematically decreasing along the optimal trajectories. We apply with success our one-pass method to the Shape From Shading equations which are involved by the most relevant and recent modelings [22,21] and which cannot be handled by the most recent extensions of the FMM [26,8]

Coordination in multiagent systems and Laplacian spectra of digraphs

Constructing and studying distributed control systems requires the analysis of the Laplacian spectra and the forest structure of directed graphs. In this paper, we present some basic results of this analysis partially obtained by the present authors. We also discuss the application of these results to decentralized control and touch upon some problems of spectral graph theory.Comment: 15 pages, 2 figures, 40 references. To appear in Automation and Remote Control, Vol.70, No.3, 200