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Abstract:
Many differential equations of interest in the physical sciences and engineering exhibit geometric properties that are preserved by the dynamics. Recently, there has been a trend towards the construction of numerical schemes that preserve as many of these geometric invariants as possible.
Such methods are of particular interest when simulating mechanical systems that arise from Lagrangian or Hamiltonian mechanics, wherein the preservation of physical invariants such as the energy, momentum, and symplectic form can be important when simulating long-time dynamics of such systems.
In applications arising from astrodynamics and robotics, the dynamics evolve on nonlinear manifolds such as Lie groups, and in particular the rotation group, and the special Euclidean group. Numerical schemes that respect the underlying nonlinear manifold structure will also be discussed.
This course will begin with an overview of classical numerical integration schemes, and their analysis, followed by a more in depth discussion of the various geometric properties that are of importance in many practical applications, followed by a survey of the various geometric integration schemes that have been developed in recent years. Issues pertaining to the analysis and implementation of such schemes will also be addressed.
Abstract:
This graduate course in numerical analysis will focus on the mathematical analysis and derivation of numerical methods for the solution of ordinary differential equations. Issues include order of accuracy, convergence, stability.
Abstract:
This undergraduate course in numerical analysis will focus on the mathematical analysis and derivation of numerical methods for numerical differentiation and integration, and the solution of ordinary differential equations and boundary-value problems. Issues include order of accuracy, convergence, stability.
Abstract:
This course surveys numerical methods for physical modeling, and discusses floating point arithmetic, direct and iterative solution of linear equations, iterative solution of nonlinear equations, optimization, approximation theory, interpolation, quadrature, numerical methods for initial and boundary value problems in ordinary differential equations.
Abstract:
Smooth optimization on manifolds naturally generalizes smooth optimization in Euclidean spaces in a manner that is of interest in a variety of applications, including modal analysis, blind source separation (via independent component analysis), pose estimation in computer vision, and model reduction in dynamical systems. Manifolds of interest include the Stiefel manifolds and Grassmann manifolds.
After presenting a number of motivating applications, we will introduce the basics of differential manifolds and Riemannian geometry, and describe how methods in optimization, like line-search, Newton's method, and trust region methods can be generalized to the case of manifolds. The course will assume only a basic knowledge of matrix algebra and real analysis.
Abstract:
Information geometry involves the use of differential geometric tools to describe the manifold of probability density functions, and allows one to investigate the intrinsic properties of statistical models as opposed to their parametric representations. In particular, we will discuss how divergence functions, and their induced geometric structures, like the Riemannian metric, dually flat affine connections, and curvature relate to statistical issues like asymptotic efficiency of maximum likelihood estimators as well as optimization algorithms on the manifold of probability distributions.
feed/start.txt ยท Last modified: 2023/11/12 22:57 by Horea Caramizaru