AD-based perturbation methods for uncertainties and errors

International audienceThe progress of Automatic Differentiation ({\bf AD}) and its impact on perturbation methods is the object of this paper. AD studies show an important activity for developing methods addressing the management of modern CFD kernels, taking into account the language evolution, and intensive parallel computing. The evaluation of a posteriori error analysis and of resulting correctors will be addressed. Recents works in the AD-based contruction of second-derivatives for building reduced-order models based on a Taylor formula will be presented on the test case of a steady compressible flow around an aircraft

The Tapenade Automatic Differentiation tool: principles, model, and specification

International audienceTapenade is an Automatic Differentiation tool which, given a Fortran or C code that computes a function, creates a new code that computes its tangent or adjoint derivatives. Tapenade puts particular emphasis on adjoint differentiation, which computes gradients at a remarkably low cost. This paper describes the principles of Tapenade, a subset of the general principles of AD. We motivate and illustrate on examples the AD model of Tapenade, i.e. the structure of differentiated codes and the strategies used to make them more efficient. Along with this informal description, we formally specify this model by means of Data-Flow equations and rules of Operational Semantics, making this the reference specification of the tangent and adjoint modes of Tapenade. One benefit we expect from this formal specification is the capacity to study formally the AD model itself, especially for the adjoint mode and its sophisticated strategies. This paper also describes the architectural choices of the implementation of Tapenade. We describe the current performances of Tapenade on a set of codes that include industrial-size applications. We present the extensions of the tool that are planned in a foreseeable future, deriving from our ongoing research on AD

Data Assimilation and Sensitivity of the Black Sea Model to Parameters

An adjoint based technique is applied to a Shallow Water Model in order to estimate influence of the model's parameters on the solution. Among parameters the bottom topography, initial conditions, boundary conditions on rigid boundaries, viscosity coefficients and the amplitude of the wind stress tension are considered. Their influence is analyzed from different points of view. Two configurations have been analyzed: an academic case of the model in a square box and a more realistic case simulating Black Sea currents. It is shown in both experiments that the boundary conditions near a rigid boundary influence the most the solution. This fact points out the necessity to identify optimal boundary approximation during a model development.Comment: Hydrodynamical Modelling of the Black Sea (2011

Automated derivation of the adjoint of high-level transient finite element programs

In this paper we demonstrate a new technique for deriving discrete adjoint and tangent linear models of finite element models. The technique is significantly more efficient and automatic than standard algorithmic differentiation techniques. The approach relies on a high-level symbolic representation of the forward problem. In contrast to developing a model directly in Fortran or C++, high-level systems allow the developer to express the variational problems to be solved in near-mathematical notation. As such, these systems have a key advantage: since the mathematical structure of the problem is preserved, they are more amenable to automated analysis and manipulation. The framework introduced here is implemented in a freely available software package named dolfin-adjoint, based on the FEniCS Project. Our approach to automated adjoint derivation relies on run-time annotation of the temporal structure of the model, and employs the FEniCS finite element form compiler to automatically generate the low-level code for the derived models. The approach requires only trivial changes to a large class of forward models, including complicated time-dependent nonlinear models. The adjoint model automatically employs optimal checkpointing schemes to mitigate storage requirements for nonlinear models, without any user management or intervention. Furthermore, both the tangent linear and adjoint models naturally work in parallel, without any need to differentiate through calls to MPI or to parse OpenMP directives. The generality, applicability and efficiency of the approach are demonstrated with examples from a wide range of scientific applications

Sensitivity Evaluation in Aerodynamic Optimal Design

The possibility to compute first- and second-derivatives of functionals subject to equality constraints given by state equations (and in particular non-linear systems of Partial Derivative Equations) allows us to use efficient techniques to solve several industrial-strength problems. Among possible applications that require knowledge of the derivatives, let us mention: aerodynamic shape optimization with gradient-based descent algorithms, propagation of uncertainties using perturbation techniques, robust optimization, and improvement of the accuracy of a functionnal using the adjoint state. In this work, we develop and analyze several strategies to evaluate the first- and second-derivatives of constrained functionals, using techniques based on Automatic Differentiation. Furthermore, we propose a descent algorithm for aerodynamic shape optimization, that is based on techniques of multi-level gradient, and which can be applied to different kinds of parameterization

The automation of PDE-constrained optimisation and its applications

This thesis is concerned with the automation of solving optimisation problems constrained by partial differential equations (PDEs). Gradient-based optimisation algorithms are the key to solve optimisation problems of practical interest. The required derivatives can be efficiently computed with the adjoint approach. However, current methods for the development of adjoint models often require a significant amount of effort and expertise, in particular for non-linear time-dependent problems. This work presents a new high-level reinterpretation of algorithmic differentiation to develop adjoint models. This reinterpretation considers the discrete system as a sequence of equation solves. Applying this approach to a general finite-element framework results in an automatic and robust way of deriving and solving adjoint models. This drastically reduces the development effort compared to traditional methods. Based on this result, a new framework for rapidly defining and solving optimisation problems constrained by PDEs is developed. The user specifies the discrete optimisation problem in a compact high-level language that resembles the mathematical structure of the underlying system. All remaining steps, including parameter updates, PDE solves and derivative computations, are performed without user intervention. The framework can be applied to a wide range of governing PDEs, and interfaces to various gradient-free and gradient-based optimisation algorithms. The capabilities of this framework are demonstrated through the application to two PDE-constrained optimisation problems. The first is concerned with the optimal layout of turbines in tidal stream farms; this optimisation problem is one of the main challenges facing the marine renewable energy industry. The second application applies data assimilation to reconstruct the profile of tsunami waves based on inundation observations. This provides the first step towards the general reconstruction of tsunami signals from satellite information

Modélisation, simulation numérique et problèmes inverses. Contributions en physique des plasmas de Tokamak, en écologie marine et autres travaux

Dans ce document je décris mon activité scientifique depuis la fin de ma thèse soutenue en octobre 2002. Il est constitué de deux parties. La première est une synthèse de mon travail et la seconde un recueil de mes articles les plus représentatifs en rapport avec les travaux décrits en première partie. La partie synthèse est constituée de deux chapitres. Le premier regroupe mes travaux les plus récents, depuis fin 2007, sur la thématique de la simulation numérique et des problèmes inverses pour les plasmas de Tokamak. Le second regroupe des travaux plus anciens concernant la modélisation et l'assimilation variationnelle de données en écologie marine ainsi que deux autres travaux isolés

Building the Tangent and Adjoint codes of the Ocean General Circulation Model OPA with the Automatic Differentiation tool TAPENADE

The ocean general circulation model OPA is developed by the LODYC team at Paris VI university. OPA has recently undergone a major rewriting, migrating to FORTRAN95, and its adjoint code needs to be rebuilt. For earlier versions, the adjoint of OPA was written by hand at a high development cost. We use the Automatic Differentiation tool TAPENADE to build mechanicaly the tangent and adjoint codes of OPA. We validate the differentiated codes by comparison with divided differences, and also with an identical twin experiment. We apply state-of-the-art methods to improve the performance of the adjoint code. In particular we implement the Griewank and Walther's binomial checkpointing algorithm which gives us an optimal trade-off between time and memory consumption. We apply a specific strategy to differentiate the iterative linear solver that comes from the implicit time stepping schem