Simultaneous Pseudo-Timestepping for PDE-Model Based Optimization Problems

In this paper we present a new method for the solution of optimization problems with PDE constraints. It is based on simultaneous pseudo-time stepping for evolution equations. The new method can be viewed as a continuous reduced SQP method in the sense that it uses a preconditioner derived from that method. The reduced Hessian in the preconditioner is approximated by a pseudo-differential operator, whose symbol can be investigated analytically. We apply our method to a boundary control model problem. The new optimization method needs 3.2-times the overall computational effort of the solution of simulation problem alone.     

Preconditioning Indefinite Systems in Interior Point Methods for Optimization

Every Newton step in an interior-point method for optimization requires a solution of a symmetric indefinite system of linear equations. Most of today's codes apply direct solution methods to perform this task. The use of logarithmic barriers in interior point methods causes unavoidable ill-conditioning of linear systems and, hence, iterative methods fail to provide sufficient accuracy unless appropriately preconditioned. Two types of preconditioners which use some form of incomplete Cholesky factorization for indefinite systems are proposed in this paper. Although they involve significantly sparser factorizations than those used in direct approaches they still capture most of the numerical properties of the preconditioned system. The spectral analysis of the preconditioned matrix is performed: for convex optimization problems all the eigenvalues of this matrix are strictly positive. Numerical results are given for a set of public domain large linearly constrained convex quadratic programming problems with sizes reaching tens of thousands of variables. The analysis of these results reveals that the solution times for such problems on a modern PC are measured in minutes when direct methods are used and drop to seconds when iterative methods with appropriate preconditioners are used.     

Reconstruction of Geophysical Layers with Shape Optimization and Level Set Methods

A shape optimization method is used to reconstruct the unknown shape of geophysical layers from boundary heat flux measurements by the use of adjoint fields and level sets. The identification of the shape of the geophysical layers by boundary heat flux measurements is necessary for the efficient use of geothermal energy. The method of speed is used to calculate the shape sensitivities, and the continuous adjoint approach is followed for the computation of the shape derivatives. The unknown shape is described with the help of the level set function; the advantage of the shape function is that no mesh movement or remeshing is necessary, but an additional Hamilton-Jacobi equation has to be solved. This equation is solved in an artificial time, where the velocity represents the movement in the direction of the normal vector of the interface. For large optimization steps, re-initialization of the level set function is also necessary, in order to keep the magnitude of the level set function near unity and also to smooth the level set function. Numerical results are given for measured heat fluxes on the boundary of the domain for different time steps and conductivity ratios. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Comparing Optimization Algorithms for Shape Optimization of Extrusion Dies

The classical approach to extrusion die design relies heavily on the experience of the die designer; Especially the designer's ability to create an initial die design from a product design, the designer's constructional knowledge and performance during the running-in trials. Furthermore, the relative unpredictability of the running-in trials combined with the additional resource usage introduce uncertainties and delays in the time-to-market of a given product. To lower these delays and resource usage, extrusion die design can benefit greatly from numerical shape optimization. In this application, however, plastics melts pose a difficult obstacle, due to their rather unintuitive and nonlinear behavior. These properties complicate the numerical optimization process, which mimics running-in trials and relies on a minimal number of optimization iterations. As part of the Cluster of Excellence Integrative Production Technologies for High-Wage Countries at the RWTH Aachen University, an effort is made to shorten the manual running-in process by the means of numerical shape optimization. Using an in-house numerical shape optimization framework, a set of optimization algorithms, consisting of global, derivative-free and gradient-based optimizers, are evaluated with respect to the best die quality and a minimal number of optimization iterations. This evaluation is an important step on the way to include more computationally intensive material models into the optimization framework and identify the best possible optimization strategy for the numerical design of extrusion dies. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

形状优化的全解析敏度分析

在形状优化设计中,建立了边界元的全解析敏度分析技术,并将该技术与通用的形状优化设计算法相结合,对二维平面应力下的弹性体进行形状优化。在优化该文的例题时,用加权求和法处理该例题的多目标问题,最后获得满意的结果。     

Solution Remapping Technique to Accelerate Flow Convergence for Finite Volume Methods Applied to Shape Optimization Design

A solution remapping technique is applied to transonic airfoil optimization design to provide a fast flow steady state convergence of intermediate shapes for the finite volume schemes in solving the compressible Euler equations. Specifically, once the flow solution for the current shape is obtained, the flow state for the next shape is initialized by remapping the current solution with consideration of mesh deformation. Based on this strategy, the formula of deploying the initial value for the next shape is theoretically derived under the assumption of small mesh deformation. Numerical experiments show that the present technique of initial value deployment can attractively accelerate flow convergence of intermediate shapes and reduce computational time up to 70% in the optimization process.     

Preconditioning the Pressure Tracking in Fluid Dynamics by Shape Hessian Information

Potential flow pressure matching is a classical inverse design aerodynamic problem. The resulting loss of regularity during the optimization poses challenges for shape optimization with normal perturbation of the surface mesh nodes. Smoothness is not enforced by the parameterization but by a proper choice of the scalar product based on the shape Hessian, which is derived in local coordinates for starshaped domains. Significant parts of the Hessian are identified and combined with an aerodynamic panel solver. The resulting shape Hessian preconditioner is shown to lead to superior convergence properties of the resulting optimization method. Additionally, preconditioning gives the potential for level independent convergence.     

A New Iteration and Preconditioning Method for Elliptic PDE-Constrained Optimization Problems

Optimal control problems constrained by a partial differential equation (PDE) arise in various important applications, such as in engineering and natural sciences. Normally the problems are of very large scale, so iterative solution methods must be used. Thereby the choice of an iteration method in conjunction with an efficient preconditioner is essential. In this paper, we consider a new iteration method and a new preconditioning technique for an elliptic PDE-constrained optimal control problem with a distributed control function. Some earlier used iteration methods and preconditioners in the literature are compared, both analytically and numerically with the new iteration method and the preconditioner.     

Regularized Quadratic Penalty Methods for Shape from Shading

Shape from shading (SFS) denotes the problem of reconstructing a 3D surface, starting from a single shaded image which represents the surface itself. Minimization techniques are commonly used for solving the SFS problem, where the objective function is a weighted combination of the brightness error, plus one or more terms aiming to obtain a valid solution. We present a regularized quadratic penalty method where quadratic penalization is used to adaptively adjust the smoothing weights, and regularization improves the robustness and reliability of the procedure. A nonmonotone Barzilai–Borwein method is employed to efficiently solve the arising subproblems. Numerical results are provided showing the reliability of the proposed approach.     

Shape Optimization of Elasto-Plastic Bodies

Existence of an optimal shape of a deformable body made from a physically nonlinear material obeying a specific nonlinear generalized Hooke's law (in fact, the so called deformation theory of plasticity is invoked in this case) is proved. Approximation of the problem by finite elements is also discussed.     

On Shape Optimization for an Evolution Coupled System

A shape optimization problem in three spatial dimensions for an elasto-dynamic piezoelectric body coupled to an acoustic chamber is introduced. Well-posedness of the problem is established and first order necessary optimality conditions are derived in the framework of the boundary variation technique. In particular, the existence of the shape gradient for an integral shape functional is obtained, as well as its regularity, sufficient for applications e.g. in modern loudspeaker technologies. The shape gradients are given by functions supported on the moving boundaries. The paper extends results obtained by the authors in (Math. Methods Appl. Sci. 33(17):2118–2131, 2010) where a similar problem was treated without acoustic coupling.     

Shape Optimization for Stokes Flows using Conformal Metrics

We derive a shape optimization approach for a two-dimensional Stokes flow problem. Our goal is to compute a geometry whose wall shear stress is L₂-close to a desired target stress. Computations are carried out on a fixed domain, while shape variations are described through conformal metric changes. Tools from differential geometry are used to handle metric dependent operators. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Shape Optimization and Electron Bubbles

We present an analytical treatment of the shape optimization problem that arises in the study of electron bubbles. The problem is to minimize a weighted sum of a Laplace eigenvalue, volume, and surface area with respect to the shape of the domain. The analysis employs the calculus of moving surfaces and yields surprising conclusions regarding the stability of equilibrium spherical configurations. Namely, all but the lowest eigenvalue result in unstable configurations and certain combinations of parameters, near-spherical equilibrium stable configurations exist. Two-dimensional and three-dimensional problems are considered and numerical results are presented for the two-dimensional case.     

Implementation of Some Methods of Shape Design for Variational Inequalities

In this paper we present some results concerning the optimal shape design problem governed by the fourth-order variational inequalities. The problem can be considered as a model example for the design of the shapes for elastic-plastic problem. The computations are done by finite element method, and the performance criterion is minimized by the material derivative method. We also discuss the error estimates in the appropriate norm and present some numerical results. An example is used to clearly illustrate the essential elements of shape design problems.     

Quasi-Newton Methods for Unconstrained Optimization

A revised algorithm is given for unconstrained optimizationusing quasi-Newton methods. The method is based on recurringthe factorization of an approximation to the Hessian matrix.Knowledge of this factorization allows greater flexibility whenchoosing the direction of search while minimizing the adverseeffects of rounding error. The control of rounding error isparticularly important when analytical derivatives are unavailable,and a modification of the algorithm to accept finite-differenceapproximations to the derivatives is given.     

Numerical Methods for Non-Linear Optimization

Journal of the Operational Research Society -     

Shape Optimization Under Width Constraint

We introduce new numerical methods to solve optimization problems among convex bodies which satisfy standard width geometrical constraints. We describe two different numerical approaches to handle width equality and width inequality constraints. To illustrate the efficiency of our method, our algorithms are used to approximate optimal solutions of Meissner’s problem and to confirm two conjectures of Heil.     

Towards Matrix-Free AD-Based Preconditioning of KKT Systems in PDE-Constrained Optimization

The presented approach aims at solving an equality constrained, finite-dimensional optimization problem, where the constraints arise from the discretization of some partial differential equation (PDE) on a given space grid. For this purpose, a stationary point of the Lagrangian is computed using Newton's method, which requires the repeated solution of KKT systems. The proposed algorithm focuses on two topics: Firstly, Algorithmic Differentiation (AD) will be used to evaluate the necessary computations of gradients, Jacobian-vector products, and Hessian-vector products, so that only the objective f (y , u ) and the PDE constraint e (y , u ) = 0 have to be specified by the user. Secondly, we solve the KKT system iteratively using the QMR algorithm, with preconditioning provided by a multigrid approach. We wish to explore whether the Jacobian-vector products provided by AD are sufficient to construct suitable multigrid preconditioners. Our approach is then embedded into a globalized optimization routine. Numerical results for optimization problems involving a nonlinear reaction-diffusion model will be given. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Shape Methods for the Transmission Problem with a Single Measurement

In the current work, we consider the inverse conductivity problem of recovering inclusion with one measurement. First, we use conformal mapping techniques for determining the location of the anomaly and estimating its size. We then get a good initial guess for quasi-Newton type method. The inverse problem is treated from the shape optimization point of view. We give a rigorous proof for the existence of the derivative of the state function and of shape functionals. We consider both least squares fitting and Kohn and Vogelius functionals. For the numerical implementation, we use a parameterization of shapes coupled with a boundary element method. Several numerical examples indicate the superiority of the Kohn and Vogelius functional over least squares fitting.