Showing papers by "Andrea Walther published in 2006"

An optimal memory‐reduced procedure for calculating adjoints of the instationary Navier‐Stokes equations

Abstract: This paper discusses approximation schemes for adjoints in control of the instationary Navier–Stokes system. It tackles the storage problem arising in the numerical calculation of the appearing adjoint equations by proposing a low-storage approach which utilizes optimal checkpointing. For this purpose, a new proof of optimality is given. This new approach gives so far unknown properties of the optimal checkpointing strategies and thus provides new insights. The optimal checkpointing allows a remarkable memory reduction by accepting a slight increase in run-time caused by repeated forward integrations as illustrated by means of the Navier–Stokes equations. In particular, a memory reduction of two orders of magnitude causes only a slow down factor of 2–3 in run-time. Copyright © 2005 John Wiley & Sons, Ltd.

Online checkpointing for parallel adjoint computation in PDEs: application to goal-oriented adaptivity and flow control

Abstract: The computation of derivatives for the optimization of time-dependent flow problems is based on the integration of the adjoint differential equation For this purpose, the knowledge of the complete forward solution is required Similar information is needed for a posteriori error estimation with respect to a given functional In the area of flow control, especially for three dimensional problems, it is usually impossible to store the full forward solution due to the lack of memory capacities Additionally, adaptive time-stepping procedures are needed for efficient integration schemes in time Therefore, standard optimal offline checkpointing strategies are usually not well-suited in that framework We present a new online procedure for determining the checkpoint distribution on the fly Complexity estimates and consequences for storing and retrieving the checkpoints using parallel I/O are discussed The resulting checkpointing approach is integrated in HiFlow, a multipurpose parallel finite-element package with a strong emphasis in computational fluid dynamic, reactive flows and related subjects Using an adjoint-based error control for prototypical three dimensional flow problems, numerical experiments demonstrate the effectiveness of the proposed approach

Optimal checkpointing for time-stepping procedures in ADOL-C

Abstract: Using the basic reverse mode of automatic differentiation, the memory needed for the computation of discrete adjoints is proportional to the number of operations performed. This behavior is frequently not acceptable, especially for large-scale problems that involve a kind of time-stepping procedure. Therefore, we integrate a checkpointing mechanism into ADOL-C, a tool for the automatic differentiation of C and C++ programs. This checkpointing procedure is optimal for a given number of checkpoints in the sense that it yields the minimal number of recomputations. The resulting effects on the run-time behavior are illustrated by means of the derivative computation for an ODE-based optimization problem.

On the efficient generation of taylor expansions for DAE solutions by automatic differentiation

Abstract: Under certain conditions the signature method suggested by Pantiledes and Pryce facilitates the local expansion of DAE solutions by Taylor polynomials of arbitrary order. The successive calculation of Taylor coefficients involves the solution of nonlinear algebraic equations by some variant of the Gauss-Newton method. Hence, one needs to evaluate certain Jacobians and several right hand sides. Without advocating a particular solver we discuss how this information can be efficiently obtained using ADOL-C or similar automatic differentiation packages.

Application of AD-based Quasi-Newton Methods to Stiff ODEs

Abstract: Systems of stiff ordinary differential equations (ODEs) can be integrated properly only by implicit methods. For that purpose, one usually has to solve a system of nonlinear equations at each time step. This system of equations may be solved by variants of Newton’s method. The main computing effort lies in forming and factoring the Jacobian or a suitable approximation to it. We examine a new approach of constructing an appropriate quasi-Newton approximation for solving stiff ODEs. The method makes explicit use of tangent and adjoint information that can be obtained using the forward and the reverse modes of algorithmic differentiation (AD). We elaborate the conditions for invariance with respect to linear transformations of the state space and thus similarity transformations of the Jacobian. We present one new updating variant that yields such an invariant method. Numerical results for Runge-Kutta methods and linear multi-step methods are discussed.