Programme of the Tenth Euro AD Workshop

We have used the ROSE Open Source Compiler Infrastructure developed at Lawrence Livermore National Laboratory to develop two source-transformation tools. PBound estimates upper performance bounds of C/C++ applications. It combines application signatures with architectural information to generate parametrized expressions for different types of memory accesses and integer and floating-point computations. ADIC2 is a tool for the automatic differentiation of C and C++ code through source-to-source transformation. I will discuss the program representation provided by ROSE and how the tools interact with ROSE.

The microscopic building blocks of matter, i.e., quarks and gluons are subject of a fundamental force, the strong interaction. Similar to water that appears in different aggregate states under different conditions strongly interacting matter is expected to occur in different phases at different temperatures and densities. Experimentally these phases are probed in heavy ion collisions e.g., at the LHC. In theoretical physics the strong interaction is described by Quantum Chromodynamics. However, a theoretical description of the phase structure is not straightforward as standard approaches fail. Lattice simulations using Monte-Carlo methods, carried out on supercomputers, are one possible ab-inito approach. These are however limited to vanishing densities. To circumvent this restriction Taylor expansion methods have been proposed to extrapolate to finite density. In this talk I present results of the calculation of higher order coefficients, which became accessible by means of the inverse Taylor expansion method available in ADOL-C.

We will present some recent work on univariate Taylor arithmetic applied to matrix factorizations. Particular focus will be the symmetric eigenvalue decomposition with potentially multiple eigenvalues, the cholesky decomposition and the QR decomposition. The algorithms for the forward and reverse mode have been implemented and tested in the Python AD tool ALGOPY. We will present some numerical tests that indicate how accurate the matrix factorizations are. If time permits, some live examples will be shown to show how easy it is to differentiate matrix valued functions.

The efficient provision of derivatives is indispensable for numerous applications in flow control. Thus, we use adjoint information in order to reduce the numerical effort. We present our approach for flow control using Tollmien-Schlichting waves as an example. Amongst others effects the boundary layer increases the resistance of a body. In our example the boundary layer flows smoothly over the streamlined shape of the body. Therefore, the resistance is minimal. Small disturbance waves (Tollmien-Schlichting waves) extend in the boundary layer during time. Hence, Tollmien-Schlichting waves cause the transition from a laminar to a turbulent boundary layer and therewith an increasing resistance. We want to use electromagnetic forces to achieve a flow profile which is more stable. More precisely, we want to achieve a larger Reynolds number for which the flow is still stable. Numerical results for the flow control problem illustrate the resulting numerical effort for the computation of the required adjoint information and the optimization.

In many CFD applications it is desired to generate adjoint CFD solvers for the optimization of given design problems. Usually CFD tools consist of models for complex physical phenomena such that hand writing an adjoint code in order to compute sensitivities is a rather hard task. Apart from hand coded discrete adjoints and continuous adjoint approaches, reverse mode of AD offers a possibility in order to generate an adjoint solver in a (semi-)automatic fashion. However, the enormous memory demand of reverse AD makes it quite infeasible for practical problems. Besides, usually AD tools are uncapable of treating parallel codes which are almost a must in CFD applications. To overcome this kind of problems, in this talk we focus on the reverse accumulation approach to reduce the memory demand and treatment of MPI directives for the reverse mode of AD with an application to a Navier-Stokes solver. We present also some results based on a benchmark flow control problem.

In this talk are presented the concepts to employ Algorithmic Differentiation (AD) in optimal control problems. Especially two problems are addressed. The first one is checkpointing. The computation of the adjoint equation requires the complete forward solution to be stored. This may be impossible due to lack of storage capacities. We present a multistage binomial checkpointing approach (storing parts of the forward solution in the main memory and on disc for an a priori known number of time steps) and an online checkpointing approach (storing parts of the forward solution in the main memory for an a priori unknown number of time steps) in order to achieve an optimal runtime. The second problem to be addressed is the determination of AD adjoints in optimal control problems, especially the possibilities of structure exploitation. We provide an error analysis for the AD adjoints w.r.t. continuous adjoints and show that the AD adjoints yield a version of the adjoint of the discretize-then-optimize approach. Additionally, we show that exploiting spatial and time structure for AD adjoints yield a vast memory reduction and gains in runtime. Numerical results underline the theoretical results.

We present and compare different approaches to the differentiation of a finite element code for stationary incompressible Navier-Stokes. The solver utilises the problem structure to a high degree (preconditioning, multigrid) to gain efficiency. AD using ADOL-C is applied at different abstraction levels of code and the resulting behaviour is compared to hand differentiated code. Problems with size of up to 37 million degrees of freedom are used in the comparison, thus structure exploitation is mandatory for efficiency.

Structural optimisation currently relies heavily on methods based on discretisation. In simpler cases like the simulation of frames and trusses, where discretisation is not necessary, only the elongation or compression is considered and the joints are free, like ball and socket joints, in order to avoid bending the trusses. In my dissertation a discretisation free method for the modelling and optimisation of frames is developed which considers bending of the beams along with compression or elongation with joints between the beams being rigid. The main difficulty in derivative computation for this model is the necessity for Nested differentiation which normally requires special handling. I shall present two aproaches for computing these derivatives.