Thomas Slawig

Generating efficient derivative code with TAF adjoint and tangent linear Euler flow around an airfoil

FastOpts new automatic differentiation tool TAF is applied to the two-dimensional Navier-Stokes solver NSC2KE. For a configuration that simulates the Euler flow around an NACA airfoil, TAF has generated the tangent linear and adjoint models as well as the second derivative (Hessian) code. Owing to TAFs capability of generating efficient adjoints of iterative solvers, the derivative code has a high performance: running both the solver and its adjoint requires 3.4 times as long as running the solver only. Further examples of highly efficient tangent linear, adjoint, and Hessian codes for large and complex three-dimensional Fortran 77-90 climate models are listed. These examples suggest that the performance of the NSC2KE adjoint may well be generalised to more complex three-dimensional CFD codes. We also sketch how TAF can improve the adjoints performance by exploiting self-adjointness, which is a common feature of CFD codes.

Modeling the influence of changing storm patterns on the ability of a salt marsh to keep pace with sea level rise

[1] Previous predictions on the ability of coastal salt marshes to adapt to future sea level rise (SLR) neglect the influence of changing storm activity that is expected in many regions of the world due to climate change. We present a new modeling approach to quantify this influence on the ability of salt marshes to survive projected SLR, namely, we investigate the separate influence of storm frequency and storm intensity. The model is applied to a salt marsh on the German island of Sylt and is run for a simulation period from 2010 to 2100 for a total of 13 storm scenarios and 48 SLR scenarios. The critical SLR rate for marsh survival, being the maximum rate at which the salt marsh survives until 2100, lies between 19 and 22 mm yr-1. Model results indicate that an increase in storminess can increase the ability of the salt marsh to accrete with sea level rise by up to 3 mm yr-1, if the increase in storminess is triggered by an increase in the number of storm events (storm frequency). Meanwhile, increasing storminess, triggered by an increase in the mean storm strength (storm intensity), is shown to increase the critical SLR rate for which the marsh survives until 2100 by up to 1 mm yr-1 only. On the basis of our results, we suggest that the relative importance of storm intensity and storm frequency for marsh survival strongly depends on the availability of erodible fine-grained material in the tidal area adjacent to the salt marsh.

Strategies for time-dependent PDE control with inequality constraints using an integrated modeling and simulation environment

In Neitzel et al. (Strategies for time-dependent PDE control using an integrated modeling and simulation environment. Part one: problems without inequality constraints. Technical Report 408, Matheon, Berlin, 2007) we have shown how time-dependent optimal control for partial differential equations can be realized in a modern high-level modeling and simulation package. In this article we extend our approach to (state) constrained problems. “Pure” state constraints in a function space setting lead to non-regular Lagrange multipliers (if they exist), i.e. the Lagrange multipliers are in general Borel measures. This will be overcome by different regularization techniques. To implement inequality constraints, active set methods and barrier methods are widely in use. We show how these techniques can be realized in a modeling and simulation package. We implement a projection method based on active sets as well as a barrier method and a Moreau Yosida regularization, and compare these methods by a program that optimizes the discrete version of the given problem.

Automated Extension of Fixed Point PDE Solvers for Optimal Design with Bounded Retardation

We study PDE-constrained optimization problems where the state equation is solved by a pseudo-time stepping or fixed point iteration. We present a technique that improves primal, dual feasibility and optimality simultaneously in each iteration step, thus coupling state and adjoint iteration and control/design update. Our goal is to obtain bounded retardation of this coupled iteration compared to the original one for the state, since the latter in many cases has only a Q-factor close to one. For this purpose and based on a doubly augmented Lagrangian, which can be shown to be an exact penalty function, we discuss in detail the choice of an appropriate control or design space preconditioner, discuss implementation issues and present a convergence analysis. We show numerical examples, among them applications from shape design in fluid mechanics and parameter optimization in a climate model.

A Smooth Regularization of the Projection Formula for Constrained Parabolic Optimal Control Problems

We present a smooth, that is, differentiable regularization of the projection formula that occurs in constrained parabolic optimal control problems. We summarize the optimality conditions in function spaces for unconstrained and control-constrained problems subject to a class of parabolic partial differential equations. The optimality conditions are then given by coupled systems of parabolic PDEs. For constrained problems, a non-smooth projection operator occurs in the optimality conditions. For this projection operator, we present in detail a regularization method based on smoothed sign, minimum and maximum functions. For all three cases, that is, (1) the unconstrained problem, (2) the constrained problem including the projection, and (3) the regularized projection, we verify that the optimality conditions can be equivalently expressed by an elliptic boundary value problem in the space-time domain. For this problem and all three cases we discuss existence and uniqueness issues. Motivated by this elliptic problem, we use a simultaneous space-time discretization for numerical tests. Here, we show how a standard finite element software environment allows to solve the problem and, thus, to verify the applicability of this approach without much implementation effort. We present numerical results for an example problem.

PDE-constrained control using Femlab – Control of the Navier–Stokes equations

We show how the software Femlab can be used to solve PDE-constrained optimal control problems. We give a general formulation for such kind of problems and derive the adjoint equation and optimality system. Then these preliminaries are specified for the stationary Navier–Stokes equations with distributed and boundary control. The main steps to define and solve a PDE with Femlab are described. We describe how the adjoint system can be implemented, and how the optimality system can be used by Femlab’s built-in functions. Special crucial topics concerning efficiency are discussed. Examples with distributed and boundary control for different type of cost functionals in 2 and 3 space dimensions are presented.

A Formula for the Derivative with Respect to Domain Variations in Navier-Stokes Flow Based on an Embedding Domain Method

Frechet differentiability and a formula for the derivative with respect to domain variation of a general class of cost functionals under the constraint of the two-dimensional stationary incompressible Navier--Stokes equations are shown. An embedding domain technique provides an equivalent formulation of the problem on a fixed domain and leads to a simple and computationally cheap line integral formula for the derivative of the cost functional with respect to domain variation. Existence of a solution to the corresponding domain optimization problems is proved. A numerical example shows the effectivity of the derivative formula.

Adjoint gradients compared to gradients from algorithmic differentiation in instantaneous control of the Navier-Stokes equations

Gradients computed via an adjoint equation and obtained algorithmically by automatic differentiation (AD) tools are compared with respect to accuracy and performance. As model application, the method of instantaneous control for the time-dependent incompressible Navier-Stokes equations with distributed control is used. The method of instantaneous control and both approaches to obtain the gradient are described in detail. Implementation issues of the AD process are given. Numerical results for gradient evaluations and complete control runs on different grids are presented.

Marine ecosystem model calibration with real data using enhanced surrogate-based optimization

We have already shown in a previous methodological work that the surrogate-based optimization (SBO) approach can be successful and computationally very efficient when reconstructing parameters in a typical nonlinear, time-dependent marine ecosystem model, where a one-dimensional application has been considered to test the methods functionality in a first step. The application on real (measurement) data is covered in this paper. Essential here are a special model data treatment and further methodological enhancements which allow us to improve the robustness of the algorithm and the accuracy of the solution. By numerical experiments, we demonstrate that SBO is able to yield a solution close to the original models optimum while time savings are again up to 85% when compared to a conventional direct optimization of the original model.

Surrogate-based optimization of climate model parameters using response correction

Abstract We present a computationally efficient methodology for the optimization of climate model parameters applied to a (one-dimensional) representative of a class of marine ecosystem models. We use a response correction technique to create a surrogate from a temporarily coarser discretized physics-based low-fidelity model. We demonstrate that replacing the direct parameter optimization of the high-fidelity ecosystem model by iteratively updating and re-optimizing the surrogate leads to a very satisfactory solution while yielding significant cost saving – about 84% when compared to the direct high-fidelity model optimization.