Incremental Sparse GP Regression for Continuous-time Trajectory Estimation & Mapping

Recent work on simultaneous trajectory estimation and mapping (STEAM) for mobile robots has found success by representing the trajectory as a Gaussian process. Gaussian processes can represent a continuous-time trajectory, elegantly handle asynchronous and sparse measurements, and allow the robot to query the trajectory to recover its estimated position at any time of interest. A major drawback of this approach is that STEAM is formulated as a batch estimation problem. In this paper we provide the critical extensions necessary to transform the existing batch algorithm into an extremely efficient incremental algorithm. In particular, we are able to vastly speed up the solution time through efficient variable reordering and incremental sparse updates, which we believe will greatly increase the practicality of Gaussian process methods for robot mapping and localization. Finally, we demonstrate the approach and its advantages on both synthetic and real datasets.Comment: 10 pages, 10 figure

Geometrically-exact time-integration mesh-free schemes for advection-diffusion problems derived from optimal transportation theory and their connection with particle methods

We develop an Optimal Transportation Meshfree (OTM) particle method for advection-diffusion in which the concentration or density of the diffusive species is approximated by Dirac measures. We resort to an incremental variational principle for purposes of time discretization of the diffusive step. This principle characterizes the evolution of the density as a competition between the Wasserstein distance between two consecutive densities and entropy. Exploiting the structure of the Euler-Lagrange equations, we approximate the density as a collection of Diracs. The interpolation of the incremental transport map is effected through mesh-free max-ent interpolation. Remarkably, the resulting update is geometrically exact with respect to advection and volume. We present three-dimensional examples of application that illustrate the scope and robustness of the method.Comment: 19 pages, 8 figure

Sliding joints in 3D beams: conserving algorithms using the master-slave approach

This paper proposes two time-integration algorithms for motion of geometrically exact 3D beams under sliding contact conditions. The algorithms are derived using the socalled master–slave approach, in which constraint equations and the related time-integration of a system of differential and algebraic equations are eliminated by design. Specifically, we study conservation of energy and momenta when the sliding conditions on beams are imposed and discuss their algorithmic viability. Situations where the contact jumps to adjacent finite elements are analysed in detail and the results are tested on two representative numerical examples. It is concluded that an algorithmic preservation of kinematic constraint conditions is of utmost importance.Peer ReviewedPostprint (author's final draft

Enumeration of the Monomials of a Polynomial and Related Complexity Classes

We study the problem of generating monomials of a polynomial in the context of enumeration complexity. In this setting, the complexity measure is the delay between two solutions and the total time. We present two new algorithms for restricted classes of polynomials, which have a good delay and the same global running time as the classical ones. Moreover they are simple to describe, use little evaluation points and one of them is parallelizable. We introduce three new complexity classes, TotalPP, IncPP and DelayPP, which are probabilistic counterparts of the most common classes for enumeration problems, hoping that randomization will be a tool as strong for enumeration as it is for decision. Our interpolation algorithms proves that a lot of interesting problems are in these classes like the enumeration of the spanning hypertrees of a 3-uniform hypergraph. Finally we give a method to interpolate a degree 2 polynomials with an acceptable (incremental) delay. We also prove that finding a specified monomial in a degree 2 polynomial is hard unless RP = NP. It suggests that there is no algorithm with a delay as good (polynomial) as the one we achieve for multilinear polynomials

Polynomial Interpretations over the Natural, Rational and Real Numbers Revisited

Polynomial interpretations are a useful technique for proving termination of term rewrite systems. They come in various flavors: polynomial interpretations with real, rational and integer coefficients. As to their relationship with respect to termination proving power, Lucas managed to prove in 2006 that there are rewrite systems that can be shown polynomially terminating by polynomial interpretations with real (algebraic) coefficients, but cannot be shown polynomially terminating using polynomials with rational coefficients only. He also proved the corresponding statement regarding the use of rational coefficients versus integer coefficients. In this article we extend these results, thereby giving the full picture of the relationship between the aforementioned variants of polynomial interpretations. In particular, we show that polynomial interpretations with real or rational coefficients do not subsume polynomial interpretations with integer coefficients. Our results hold also for incremental termination proofs with polynomial interpretations.Comment: 28 pages; special issue of RTA 201

Localized direct boundary-domain integro-differential formulations for incremental elasto-plasticity of inhomogeneous body

A quasi-static mixed boundary value problem of incremental elasto-plasticity for a continuously inhomogeneous body is considered. Using the two-operator Green–Betti formula and the fundamental solution of a reference homogeneous linear elasticity problem, with frozen initial or tangent elastic coefficients, a boundary-domain integro-differential formulation of the elasto-plastic problem is presented, with respect to the displacement rates and their gradients. Using a cut-off function approach, the corresponding localized parametrix of the reference problem is constructed to reduce the elasto-plastic problem to a nonlinear localized boundary-domain integro-differential equation. Algorithms of mesh-based and mesh-less discretizations are presented resulting in sparsely populated systems of nonlinear algebraic equations for the displacement increments

Distributed-memory large deformation diffeomorphic 3D image registration

We present a parallel distributed-memory algorithm for large deformation diffeomorphic registration of volumetric images that produces large isochoric deformations (locally volume preserving). Image registration is a key technology in medical image analysis. Our algorithm uses a partial differential equation constrained optimal control formulation. Finding the optimal deformation map requires the solution of a highly nonlinear problem that involves pseudo-differential operators, biharmonic operators, and pure advection operators both forward and back- ward in time. A key issue is the time to solution, which poses the demand for efficient optimization methods as well as an effective utilization of high performance computing resources. To address this problem we use a preconditioned, inexact, Gauss-Newton- Krylov solver. Our algorithm integrates several components: a spectral discretization in space, a semi-Lagrangian formulation in time, analytic adjoints, different regularization functionals (including volume-preserving ones), a spectral preconditioner, a highly optimized distributed Fast Fourier Transform, and a cubic interpolation scheme for the semi-Lagrangian time-stepping. We demonstrate the scalability of our algorithm on images with resolution of up to $1024^3$ on the "Maverick" and "Stampede" systems at the Texas Advanced Computing Center (TACC). The critical problem in the medical imaging application domain is strong scaling, that is, solving registration problems of a moderate size of $256^3$---a typical resolution for medical images. We are able to solve the registration problem for images of this size in less than five seconds on 64 x86 nodes of TACC's "Maverick" system.Comment: accepted for publication at SC16 in Salt Lake City, Utah, USA; November 201

Incremental localized boundary-domain integro-differential equations of elastic damage mechanics for inhomogeneous body

Copyright @ 2006 Tech Science PressA quasi-static mixed boundary value problem of elastic damage mechanics for a continuously inhomogeneous body is considered. Using the two-operator Green-Betti formula and the fundamental solution of an auxiliary homogeneous linear elasticity with frozen initial, secant or tangent elastic coe±cients, a boundary-domain integro-differential formulation of the elasto-plastic problem with respect to the displacement rates and their gradients is derived. Using a cut-off function approach, the corresponding localized parametrix of the auxiliary problem is constructed to reduce the problem to a nonlinear localized boundary-domain integro-differential equation. Algorithms of mesh-based and mesh-less discretizations are presented resulting in sparsely populated systems of nonlinear algebraic equations for the displacement increments