We investigate two techniques for estimating the sparsity pattern of Jacobians of user-supplied Matlab functions. The first approach uses Matlab's sparse logical matrices to hold and combine sparsity patterns. The second encodes and propagates the sparsity pattern using Matlab's bit manipulation operations. In both approaches care must be taken when the function's intermediate variables are themselves sparse matrices. Some initial results are presented.
This is joint work with Anand Chaturvedi (IIT Roorkee).
Chebfun is a software system in Matlab which aims at enabling computations with functions and operators rather then vectors and matrices. Using AD, we are able to find derivative of any function with respect to any other function. The derivatives involved are Fréchet derivatives, linear operators which are the continuous analogue of Jacobian matrices. This allows automatic Newton iteration for solving nonlinear BVPs. We present our AD implementation, details of the Newton iteration and show various examples of solving nonlinear BVPs with Chebfun.
An investigation on how to use the information provided by gradient direction, computed using MAD tool, with a swarm based optimization method is presented. That technique has been labelled GRASP. The results of applying both GRASP and PSO (Particle Swarm Optimization) on the CEC'05 benchmark are discussed and compared.
In general differentiation of a function that can only be evaluated pointwise is ill conditioned. For functions that are defined by an evalutation procedure as a composition of arithmetic operators and elementary functions, it can be shown that algorithmic differentiation is backward stable in the sense of Wilkinson. More specifically, the derivative values obtained are exact at the level of machine precision. In order to show this one needs that all the elementals and their derivatives exhibit ultimate precision with argument perturbation. The IEEE standard imposes ultimate precision for arithmetic operations but the extra work required to fulfill such a requirement for elementals like sin and exp would be considerable without much pay off in terms of algorithmic stability. In this talk I shall present the results of numerical experiments with an arbitrary
precision library that show that argument perturbation is indeed necessary for certain elementals. This is joint work with A. Griewank
and A. Walther.
EADS Innovation Works, as research center of EADS, has been involved recently in the development of inverse methods for trajectory tools.
Two aerospace applications will be presented with a focus on inverse problem and first automatic differentiation (AD) experiments using the Tapenade tool, developed at INRIA Sophia-Antipolis TROPICS team.
- Identification of the aerodynamics coefficients of a probe for different flight conditions (Mach number, angle of attack), given trajectories measurements : the final objective is to obtain a reliable prediction of dynamic stability of this probe, using an adapted numerical trajectory scheme (Runge Kutta) and large scale coefficients parameterization.
- Identification of heat flux coefficient from time domain temperature measurements. The industrial objective is the thermal protection system dimensioning problem for atmospheric re-entry missions of aerospace vehicles. The problem is applied on a transient one-dimensional thermal model with one moving boundary (ablative surface), with degradable thermal materials.
For these two applications, the inverse problem is also formulated as a minimization problem with optimal control formulation (lagrangian, adjoint and gradients computations, with optimization loop). To compute numerically the adjoint and gradient quantities, we have used both analytical differentiation and Automatic Differentiation. First encouraging results of automatic differentiation procedure on these industrial applications will be presented, with validations in agreement with measurements.
The construction of a surrogate model for the purposes of design optimisation often involves some form of sub-optimisation of the surrogate's controlling parameters. The construction of a kriging model, for example, can require a series of O(n^3) factorisations of the correlation matrix when performing the likelihood maximisation. Due to the smooth nature of the likelihood function, gradient information can be used to accelerate the likelihood optimisation when employed within a gradient enhanced global optimisation strategy. To this end a series of adjoints of the likelihood function of a variety of kriging based surrogate models are presented.
An adjoint of the likelihood function derived via algorithmic differentiation is presented for traditional kriging. Recent extensions of this formulation to the likelihood functions for co-kriging and gradient enhanced kriging are also presented. Gradient enhanced kriging may be of particular interest to those wishing to employ derivative information from computational simulations, which itself may be a result of an algorithmic differentiation, within a design optimisation.
A methodology for constructing the sensitivity of the incompressible Navier-Stokes equations is presented as the context for differentiating high-level Fortran source code using TAPENADE from INRIA and CompAD from NAG/RWTH Aachen. The methodology aims to be scalable and to retain all the compile-time and run-time safety the language offers. The incompressible solver is presented as it is the standard kernel in industrial CFD software. To complement this work, the software used has been made available on http://www.gpde.net.
1600 –1630 Coffee and Close
1830 Informal Dinner and Drinks- Bell Hotel Faringdon