scholarly journals Computation of Lie Derivatives of Tensor Fields Required for Nonlinear Controller and Observer Design Employing Automatic Differentiation

PAMM ◽  
2005 ◽  
Vol 5 (1) ◽  
pp. 181-184 ◽  
Author(s):  
Klaus Röbenack
Keyword(s):  
Automatic Differentiation ◽  
Observer Design ◽  
Nonlinear Controller ◽  
Tensor Fields ◽  
Lie Derivatives ◽  
Derivatives Of

scholarly journals On the classification of isotropic tensors

1987 ◽  
Vol 29 (2) ◽  
pp. 185-196 ◽  
Author(s):  
P. G. Appleby ◽  
B. R. Duffy ◽  
R. W. Ogden
Keyword(s):  
Linear Group ◽  
General Linear Group ◽  
Final Section ◽  
Complete Classification ◽  
Coordinate Transformations ◽  
Tensor Fields ◽  
Isotropic Tensor ◽  
Unimodular Group ◽  
Derivatives Of

A tensor is said to be isotropic relative to a group of transformations if its components are invariant under the associated group of coordinate transformations. In this paper we review the classification of tensors which are isotropic under the general linear group, the special linear (unimodular) group and the rotational group. These correspond respectively to isotropic absolute tensors [4, 8] isotropic relative tensors [4] and isotropic Cartesian tensors [3]. New proofs are given for the representation of isotropic tensors in terms of Kronecker deltas and alternating tensors. And, for isotropic Cartesian tensors, we provide a complete classification, clarifying results described in [3].In the final section of the paper certain derivatives of isotropic tensor fields are examined.

Implications of Using Dual Number Derivatives With a Numerical Solution

2013 ◽  
Author(s):  
Suryanarayana R. Pakalapati ◽  
Hayri Sezer ◽  
Ismail B. Celik
Keyword(s):  
Automatic Differentiation ◽  
Numerical Solutions ◽  
Computer Code ◽  
Model Parameters ◽  
Model Parameter ◽  
Systematic Assessment ◽  
Dual Number ◽  
Advection Diffusion Equation ◽  
Model Equations ◽  
Derivatives Of

Dual number arithmetic is a well-known strategy for automatic differentiation of computer codes which gives exact derivatives, to the machine accuracy, of the computed quantities with respect to any of the involved variables. A common application of this concept in Computational Fluid Dynamics, or numerical modeling in general, is to assess the sensitivity of mathematical models to the model parameters. However, dual number arithmetic, in theory, finds the derivatives of the actual mathematical expressions evaluated by the computer code. Thus the sensitivity to a model parameter found by dual number automatic differentiation is essentially that of the combination of the actual mathematical equations, the numerical scheme and the grid used to solve the equations not just that of the model equations alone as implied by some studies. This aspect of the sensitivity analysis of numerical simulations using dual number auto derivation is explored in the current study. A simple one-dimensional advection diffusion equation is discretized using different schemes of finite volume method and the resulting systems of equations are solved numerically. Derivatives of the numerical solutions with respect to parameters are evaluated automatically using dual number automatic differentiation. In addition the derivatives are also estimated using finite differencing for comparison. The analytical solution was also found for the original PDE and derivatives of this solution are also computed analytically. It is shown that a mathematical model could potentially show different sensitivity to a model parameter depending on the numerical method employed to solve the equations and the grid resolution used. This distinction is important since such inter-dependence needs to be carefully addressed to avoid confusion when reporting the sensitivity of predictions to a model parameter using a computer code. A systematic assessment of numerical uncertainty in the sensitivities computed using automatic differentiation is presented.

Infinitesimal Deformations of ECD Fields

2020 ◽  
pp. 147-154
Author(s):  
Daniel Canarutto
Keyword(s):  
Lie Derivative ◽  
Dirac Fields ◽  
Lie Derivatives ◽  
Infinitesimal Deformations ◽  
Standard Notion ◽  
Derivatives Of

The standard notion of Lie derivative is extended in order to include Lie derivatives of spinors, soldering forms, spinor connections and spacetime connections. These extensions are all linked together, and provide a natural framework for discussing infinitesimal deformations of Einstein-Cartan-Dirac fields in the tetrad-affine setting.

scholarly journals A Simple and Efficient Tensor Calculus

2020 ◽  
Vol 34 (04) ◽  
pp. 4527-4534
Author(s):  
Sören Laue ◽  
Matthias Mitterreiter ◽  
Joachim Giesen
Keyword(s):  
Machine Learning ◽  
Efficient Method ◽  
Automatic Differentiation ◽  
State Of The Art ◽  
Higher Order ◽  
Tensor Representation ◽  
Tensor Calculus ◽  
Correct Algorithm ◽  
Previous State ◽  
Derivatives Of

Computing derivatives of tensor expressions, also known as tensor calculus, is a fundamental task in machine learning. A key concern is the efficiency of evaluating the expressions and their derivatives that hinges on the representation of these expressions. Recently, an algorithm for computing higher order derivatives of tensor expressions like Jacobians or Hessians has been introduced that is a few orders of magnitude faster than previous state-of-the-art approaches. Unfortunately, the approach is based on Ricci notation and hence cannot be incorporated into automatic differentiation frameworks like TensorFlow, PyTorch, autograd, or JAX that use the simpler Einstein notation. This leaves two options, to either change the underlying tensor representation in these frameworks or to develop a new, provably correct algorithm based on Einstein notation. Obviously, the first option is impractical. Hence, we pursue the second option. Here, we show that using Ricci notation is not necessary for an efficient tensor calculus and develop an equally efficient method for the simpler Einstein notation. It turns out that turning to Einstein notation enables further improvements that lead to even better efficiency.

HYPERCHAOS GENERATED FROM QI SYSTEM AND ITS OBSERVER

2009 ◽  
Vol 23 (07) ◽  
pp. 963-974 ◽  
Author(s):  
XINGYUAN WANG ◽  
YAOXIAN ZHANG ◽  
YONGFENG GAO
Keyword(s):  
Dynamical System ◽  
Time Domain ◽  
Global Exponential Stability ◽  
Observer Design ◽  
The Other ◽  
The Novel ◽  
Nonlinear Controller ◽  
Error System ◽  
The Time Domain ◽  
Lyapunov Exponent Spectrum

This paper reports a novel four-dimensional hyperchaos generated from Qi system, obtained by adding nonlinear controller to Qi chaos system. The novel hyperchaos is studied by bifurcation diagram, Lyapunov exponent spectrum and phase diagram. Numerical simulations show that the new system's behavior can be periodic, chaotic and hyperchaotic as the parameter varies. Based on the time-domain approach, a simple observer for the hyperchaotic is proposed to guarantee the global exponential stability of the resulting error system. The scheme is easy to implement and different from the other observer design since it does not need to transmit all signals of the dynamical system.

scholarly journals Computational Algorithm for Covariant Series Expansions in General Relativity

2018 ◽  
Vol 173 ◽  
pp. 03021 ◽  
Author(s):  
Ivan Potashov ◽  
Alexander Tsirulev
Keyword(s):  
Tensor Field ◽  
Computational Algorithm ◽  
Series Expansions ◽  
Tensor Fields ◽  
Normal Coordinates ◽  
Covariant Derivatives ◽  
Taylor Polynomials ◽  
Real Analytic ◽  
Derivatives Of ◽  
Torsion Free

We present a new algorithm for computing covariant power expansions of tensor fields in generalized Riemannian normal coordinates, introduced in some neighborhood of a parallelized k-dimensional submanifold (k = 0, 1, . . .< n; the case k = 0 corresponds to a point), by transforming the expansions to the corresponding Taylor series. For an arbitrary real analytic tensor field, the coefficients of such series are expressed in terms of its covariant derivatives and covariant derivatives of the curvature and the torsion. The algorithm computes the corresponding Taylor polynomials of arbitrary orders for the field components and is applicable to connections that are, in general, nonmetric and not torsion-free. We show that this computational problem belongs to the complexity class LEXP.

scholarly journals Higher Order Automatic Differentiation with Dual Numbers

2020 ◽  
Author(s):  
László Szirmay-Kalos
Keyword(s):  
Gaussian Curvature ◽  
Automatic Differentiation ◽  
Arbitrary Order ◽  
Higher Order ◽  
Dual Numbers ◽  
Computer Implementation ◽  
Particular Solution ◽  
Derivatives Of ◽  
Curve Fairing ◽  
Curvature Computation

In engineering applications, we often need the derivatives of functions defined by a program. The approach chosen for derivative computation must be algebraic to allow computer implementation. A particular solution to obtain first derivatives is the application of dual numbers. This paper proposes simple and compact generalizations of this idea to obtain derivatives of arbitrary order for single or multi-variate functions and the automatic handling of 0/0 ambiguities in the calculations. We also provide the C++ code that takes advantage of operator overloading and recursion. The method is demonstrated by path animation, Gaussian curvature computation, and curve fairing.

scholarly journals Lie derivatives of sectorform fields

1988 ◽  
Vol 55 (1) ◽  
pp. 71-78 ◽  
Author(s):  
Ivan Kolář
Keyword(s):  
Lie Derivatives ◽  
Derivatives Of
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