Jan Riehme

A differentiation-enabled Fortran 95 compiler

The availability of first derivatives of vector functions is crucial for the robustness and efficiency of a large number of numerical algorithms. An upcoming new version of the differentiation-enabled NAGWare Fortran 95 compiler is described that uses programming language extensions and a semantic code transformation known as automatic differentiation to provide Jacobians of numerical programs with machine accuracy. We describe a new user interface as well as the relevant algorithmic details. In particular, we focus on the source transformation approach that generates locally optimal gradient code for single assignments by vertex elimination in the linearized computational graph. Extensive tests show the superiority of this method over the current overloading-based approach. The robustness and convenience of the new compiler-feature is illustrated by various case studies.

On the use of discrete adjoints in goal error estimation for shallow water equations

Goal oriented dual weight error estimation has been used in context of computational fluid dynamics for several years. The adaptation of this method to geophysical models is the subject of this paper. A differentiation-enabled prototype of the NAG Fortran compiler is used to generate a discrete adjoint version of such a geophysical model and allows to compute the required goal sensitivities. Numerical results are presented for a shallow water configuration of the Icosahedral Non-hydrostatic General Circulation Model (ICON). A special treatment of the underlying linear solver is discussed yielding improved scalability of this approach and a significant reduction in memory consumption and runtime.

A Framework for Proving Correctness of Adjoint Message-Passing Programs

We propose a technique for proving correctness of adjoint message passing programs that relies on data dependences in partitioned global address space. As an example we discuss asynchronous unbuffered send/receive using MPI.

Extension of fixed point PDE solvers for optimal design by one-shot method - With first applications to aerodynamic shape optimization

This paper concerns mathematical methods, algorithmic techniques and software tools for the transition from simulation to optimization. We focus in particular on applications in aerodynamics. The methodology is applicable to all areas of scientific computing, where large scale governing equations involving discretized PDEs are treated by custom made fixed point solvers. To exploit the domain specific experience and expertise invested in these simulation tools we propose to extend them in a semi-automated fashion. First they are augmented with an adjoint solvers to obtain (reduced) derivatives and then this sensitivity information is immediately used to determine optimization corrections.

Towards differentiation-enabled Fortran 95 compiler technology

We present a novel approach to generating derivative code for mathematical models implemented as Fortran 95 programs using Automatic Differentiation inside a compiler. This technique allows us to combine the advantages of both operator overloading and source transformation based tools for Automatic Differentiation. Furthermore, the compilers infrastructure for syntactic, semantic, and static data flow analysis can be built on.

Adjoints for time-dependent optimal control

The use of discrete adjoints in the context of a hard time-dependent optimal control problem is considered. Gradients required for the steepest descent method are computed by code that is generated automatically by the differentiation-enabled NAGWare Fortran compiler. Single time steps are taped using an overloading approach. The entire evolution is reversed based on an efficient checkpointing schedule that is computed by revolve. The feasibility of nonlinear optimization based on discrete adjoints is supported by our numerical results.

First-order uncertainty analysis using Algorithmic Differentiation of morphodynamic models

We present here an efficient first-order second moment method using Algorithmic Differentiation (FOSM/AD) which can be applied to quantify uncertainty/sensitivities in morphodynamic models. Changes with respect to variable flow and sediment input parameters are estimated with machine accuracy using the technique of Algorithmic Differentiation (AD). This method is particularly attractive for process-based morphodynamic models like the Telemac-2D/Sisyphe model considering the large number of input parameters and CPU time associated to each simulation.The FOSM/AD method is applied to identify the relevant processes in a trench migration experiment (van Rijn, 1987). A Tangent Linear Model (TLM) of the Telemac-2D/Sisyphe morphodynamic model (release 6.2) was generated using the AD-enabled NAG Fortran compiler. One single run of the TLM is required per variable input parameter and results are then combined to calculate the total uncertainty.The limits of the FOSM/AD method have been assessed by comparison with Monte Carlo (MC) simulations. Similar results were obtained assuming small standard deviation of the variable input parameters. Both settling velocity and grain size have been identified as the most sensitive input parameters and the uncertainty as measured by the standard deviation of the calculated bed evolution increases with time. A first-order second moment method (FOSM) is applied to quantify uncertainty.This method uses Algorithmic Differentiation (AD) and a Tangent Linear Model (TLM).The method is compared with Monte Carlo analysis in a trench migration test case.A TLM of the Telemac-2d/Sisyphe morphodynamic model has been applied.The FOSM/AD method is an efficient alternative to Monte Carlo simulations.

Towards automatic significance analysis for approximate computing

Several applications may trade-off output quality for energy efficiency by computing only an approximation of their output. Current approaches to software-based approximate computing often require the programmer to specify parts of the code or data structures that can be approximated. A largely unaddressed challenge is how to automate the analysis of the significance of code for the output quality. To this end, we propose a methodology and toolset for automatic significance analysis. We use interval arithmetic and algorithmic differentiation in our profile-driven yet mathematical approach to evaluate the significance of input and intermediate variables for the output of a computation. Our methodology effectively matches decisions of a domain expert in significance characterization for a set of benchmarks, and in some cases offers new insights. Evaluation of the software infrastructure on a multicore x86 platform shows energy reduction (from 31% up to 91% with a mean of 56%) compared to fully accurate execution, with graceful quality degradation.

Structure-Exploiting Automatic Differentiation of Finite Element Discretizations

A common way to solve PDE constrained optimal control problems by automatic differentiation (AD) is the full black box approach. This technique may fail because of the large memory requirement. In this paper we present two alternative approaches. First, we exploit the structure in time yielding a reduced memory requirement. Second, we additionally exploit the structure in space by providing derivatives on a reference finite element. This approach reduces the memory requirement once again compared to the exploitation in time. We present numerical results for both approaches, where the derivatives are determined by the AD-enabled NAGWare Fortran compiler.

Estimation of data assimilation error: A shallow-water model study

AbstractFour-dimensional variational data assimilation (4D-Var) produces unavoidable inaccuracies in the models initial state vector. In this paper the authors investigate a novel variational error estimation method to calculate these inaccuracies. The impacts of model, background, and observational errors on the state estimate produced by 4D-Var are analyzed by applying the variational error estimation method. The structure of the method is similar to the conventional 4D-Var, with the differences in that (i) instead of observations it assimilates observational errors, and (ii) the original model equations (used in 4D-Var as constraints) are first linearized with respect to a small perturbation in the initial state vector and then used as the constraints. The authors then carry out a proof-of-concept study and validate the reliability of this method through multiple twin experiments on the basis of a 2D shallow-water model. All required differentiated models were generated by means of algorithmic differen...