Programme of the Sixth Euro AD Workshop

Second-order derivatives of PDE-constrained functionals can be obtained through an adjoint formulation and using two differents AD approaches: Tangent-on-Tangent and Tangent-on-Reverse. The two strategies return the same derivatives but differ about their computational cost. I will present the two algorithms with some estimates about their computational costs and some examples of applications using a 3D Euler solver.

AD tools using the operator overloading technique are easy to use in principle, since they only require a change of the data-type of the code (activation). An effective method for the concurrent development and maintenance of the regular and activated version of a source code is the use of generic programming (templates). We take a look at the feasibility of this approach using various AD tools.

Some computations performed by automatic differentiation can be performed more effectively if we can be assured that the control flow of the program does not change from one call of the program to the next. This is the case with: uncertainty estimation via Taylor series, Jacobian sparsity estimation, or reverse mode AD implemented via taping. We present an overloaded branch detection algorithm for the MAD package for AD in Matlab that enables one to automatically check whether any relational operators are evaluated and/or whether their evaluation changes from one call of the program to the next. In the absence of the control-flow analysis facilitated by source-transformation, detection of relational operators is taken to indicate the presence of a branch in the program and hence potential changes in control flow. We further present a sparsity detection class for MAD which gives tight overestimates of Jacobian sparsity patterns even in the presence of program computations involving sparse matrices. The sparsity patterns may then be used for Jacobian compression.

Data assimilation is a method of coupling models and observations. It aims to produce an optimal state of a physical system. The method is widely used in geoscience, namely in physics and chemistry of the atmosphere and ocean. In fact, both a chaotic character of these phenomena and an inherent model error require to periodically constrain a model with observations.

In this presentation we shall focus on Kalman Filtering approach to data assimilation. Firstly, the bases of the standard Kalman Filter algorithm shall be presented. Next, in view of practical applications, the necessity of order reduction shall be raised. The foundations of the Reduced Rank Square Root(RRSQRT) KalmanFilter and Singular Evolutive Extended/Interpolated Kalman(SEEK/SEIK) Filter shall be given. The presence of the tangent linear and adjoint models in the algorithms shall be stressed and the currently used ways of circumventing them discussed. Furthermore, Kalman Filter-based methods (Ensemble Kalman Filter and Particle Filter) which enable tackling nonlinearities in the models shall be discussed. Finally, the differences between Kalman Filtering and variational method shall be emphasised. The equivalence of the latter one and a fixed interval smoother issued from Kalman Filter shall also be mentioned.

Transformation of Algorithms in Fortran (TAF) and Transformation of Algorithms in C++ (TAC++) are FastOpt's Automatic Differentiation tools for code written in Fortran and C, respectively. We will give a brief overview on the tools and their applications. Successful large-scale applications cover all components of state of the art earth system models,such as atmospheric and ocean general circulation, terrestrial and marine biogeochemistry, atmospheric transport and chemistry, sea-ice and the radiative transfer.

Satellite images represent a large amount of data which are currently underused in numerical forecast systems. Extending assimilation techniques to image data become of primary importance to improve the quality of the analysis. Assimilating dynamical informations included in image sequences requires, among other things, the construction of observation operators which map the model state variables space onto a given image space. This mapping will probably induce some difficulties in automatic differentiation.

This work is supported by the ANR in the framework of the ADDISA project ( http://addisa.gforge.inria.fr).

Ned Nedialkov and John Pryce are the authors of DAETS, a C++ code for solving differential-algebraic equations (DAEs). It uses Pryce's structural analysis theory, and expands the solution in Taylor series using AD. Version 1 is now available. DAETS is especially good when high accuracy is required, and at solving problems of high index (we have solved artificial DAEs of index up to 47). It is versatile: higher-order systems do not have to be cast in first-order form; it can solve explicit and implicit ODEs; it can solve purely algebraic (continuation) problems, by simple or by arc-length continuation. The talk will outline the design of the algorithm and the code, and give examples of the code's performance. This includes some puzzling phenomena, e.g. why, on one stiff ODE from a standard test set, does tightening the tolerance give a *decrease* in CPU time and number of steps, uniformly over a range of tolerances from 1e-4 to 1e-12?

I will present an algorithm for optimal chain rule-based derivative accumulation on a subclass of directed acyclic graphs. The algorithm is based on the series-parallel decomposition tree, which can be constructed in linear time.

In the past, various approaches have been taken to make AD tools aware of MPI calls and generate the proper adjoint code. Considering the correctness of the adjoint and its performance there is so far no generic recipe for source transformation tools. We will look at some of the problems and discuss first ideas how to arrive at an MPI code that can be adjoined, is guaranteed to be correct and does not loose too much performance.

In several (potential) applications of algorithmic differentiation one needs to evaluate or bound cross-derivatives, i.e. mixed partials that are obtained by differentiating with respect to each one of <i>n</i> variables at most once. We observe that the 2ⁿ cross-derivatives of a scalar function <i>f</i> can be evaluated in the forward mode with an operation count of no more than 3ⁿ times that for evaluating <i>f</i> by itself. For the error estimation and design of certain quasi-multi Carlo quadratures it is sufficient to bound the cross-derivatives by products of 2,n order and variable weights. We demonstrate how they can be calculated with an effort that grows at worst quadratically with respect to the dimension <i>n</i>, again relative to the cost of evaluating <i>f</i> by itself.

For identifying the blood model which is appropriate for the shape optimization of a blood pump we want to get the sensitivities of the optimal geometry with respect to the parameters of different blood models. For getting this information we want to differentiate through an optimization algorithm. In this talk the first results of differentiating the optimizer DonLP2 will be shown.

The study of vegetation at landspace scale often implies the use of simulation models, observed data and the resolution of inverse problems. We will briefly present several applications based on adjoint methods and using the AD tool TAPENADE. We will particularly focus on the interest of using spatial and /or temporal constraints in the resolution of inverse problems and on the experience of using TAPENADE on different models.

Rigorous use of automatic differentiation with the AD tool TAF has enabled numerous applications of the MITgcm adjoint modeling framework. They include state estimation and sensitivity studies in an ocean-only, coupled ocean-biogeochemistry and coupled ocean-seaice context, at coarse and high (eddy-ermitting) resolution for various domains and grid topologies of the world ocean. Applications ran on various HPC platforms, some of which scaled up to 600 processors. Successful MITgcm adjoint code generation with the new open-source tool OpenAD has enabled us to compare adjoint code reliability among different AD tools in the complex framwork of a fully-fledged GCM.

We present the construction by Tapenade of the adjoint code of the oceanography code OPA. OPA 9.0 is a major rewrite of OPA, now written in FORTRAN 95. We discuss some techniques used to improve the adjoint code, such as binomial checkpointing and a specific differentiation strategy for the iteative solver. We present validation results and some preliminary applications.

Accurate yet efficient computation of derivatives is essential for solution of inverse problems in atmospheric remote sensing. A line-by-line radiative transfer code has been developed for high resolution infrared and microwave atmospheric sounding. Derivatives of spectra with respect to atmospheric state parameters (e.g. molecular concentration or temperature) are obtained by means of automatic differentiation using ADIFOR. Currently the code is converted to modern Fortran. Experience with ADIFOR and TAPENADE is discussed.

In electrical engineering (electromechanical devices, power electronics converters, electrical drives, micro-systems, etc.), analytical models are usually used for fast computation, mainly in sizing processes. Such a process is often a constrained optimization problem with numerous constraints (some tens to some hundreds). So, gradient optimization algorithms are useful. However, such a model may contain algorithms, complex equations (not real), specific routines such as non-linear implicit equations, differential and integral equations, etc. which require numerical solving methods. In the presentation, the authors will present an approach with the use of ADOLC to derive symbolic sizing model described in C, and specifically equations and simple algorithms. The authors will focus on the mathematical aspects of their work. The computation results are compared with results obtained by derivatives provided by a symbolic approach (when it is possible). The approach with ADOLC is integrated in CADES framework (developed in our laboratory for device sizing), to be used by researchers in electrical engineering.