Solution Of PDE-constrained Optimization Problems Research Articles

Parallel generalized Lagrange–Newton method for fully coupled solution of PDE-constrained optimization problems with bound-constraints

In large-scale simulations of optimization problems constrained by partial differential equations (PDEs), the class of fully coupled methods is attracting great attention due to their impressive robustness and scalability. In this study, we introduce and investigate a parallel generalized Lagrange–Newton solver, which is based on the Lagrange active–set reduced–space (LASRS) method and the domain decomposition technique, for solving a family of PDE-constrained optimization problems with inequality constraints, i.e., the distributed control problem with bound-constraints. In the approach, a Lagrangian functional is constructed and the corresponding first-order optimality conditions are derived. After that, the resultant nonlinear algebraic system is solved by the active–set reduced–space method to guarantee the nonlinear consistency of the fully coupled system in a monolithic way. For the linear system arising at each generalized Newton iteration, the overlapping additive Schwarz preconditioners are employed to enhance the convergence of the linear iterations and the scalability of the large-scale simulations. Numerical results on the Tianhe supercomputer show that the fully coupled methods are robust and effective for some two- and three-dimensional distributed control problems with bound-constraints.

Preconditioning for PDE-constrained optimization with total variation regularization

The efficient solution of PDE-constrained optimization problems is of great significance in many scientific and engineering applications. The key point of achieving this goal is to solve the linear systems arising from such problems more efficiently by preconditioning. In this paper, we consider preconditioning the linear systems arising from PDE-constrained optimization problems with total variation (TV) regularization. Based on the block structure of the coefficient matrix or its row and column transformed forms, we propose three kinds of new preconditioners by approximating the Schur complements in different ways. We analyze the eigenvalue properties of the preconditioned coefficient matrices and provide the eigenvalue bounds independent of mesh size. We also show the relations of eigenvalue bounds corresponding to different preconditioners. Numerical results illustrate the efficiency of the proposed preconditioners by comparison with other existing methods.

Three-dimensional trajectory optimization for multi-stage launch vehicle mission using a full-space quasi-Lagrange–Newton method

Many aerospace industrial applications require robust and efficient numerical solutions of large sparse nonlinear constrained parameter optimization problems arising from optimal trajectory problems. A three-dimensional multistage launcher problem is taken as a numerical example for studying the performance and applicability of the full-space Lagrange–Newton–Krylov method. The typical optimal trajectory, control history and other important physical qualities are presented, and the efficiency of the algorithm is also investigated.
 
 References J. T. Betts. Practical methods for optimal control and estimation using nonlinear programming. Advances in Design and Control. SIAM, 2nd edition, 2010. doi:10.1137/1.9780898718577. R. T. Marler and J. S. Arora. Survey of multi-objective optimization methods for engineering. Struct. Multidiscip. Opt., 26(6):369395, 2004. doi:10.1007/s00158-003-0368-6. W. Roh and Y. Kim. Trajectory optimization for a multi-stage launch vehicle using time finite element and direct collocation methods. Eng. Opt., 34:1532, 2002. doi:10.1080/03052150210912. G. D. Silveira and V. Carrara. A six degrees-of-freedom flight dynamics simulation tool of launch vehicles. J. Aero. Tech. Manag., 7:231239, 2015. doi:10.5028/jatm.v7i2.433. H.-H. Wang, Y.-S. Lo, F.-T. Hwang, and F.-N. Hwang. A full-space quasi-LagrangeNewtonKrylov algorithm for trajectory optimization problems. Electron. T. Numer. Anal., 49:103125, 2018. doi:10.1553/etna_vol49s103. H. Yang, F.-N. Hwang, and X.-C. Cai. Nonlinear preconditioning techniques for full-space Lagrange-Newton solution of PDE-constrained optimization problems. SIAM J. Sci. Comput., 38:A2756A2778, 2016. doi:10.1137/15M104075X.

PDE-constrained optimization in medical image analysis

PDE-constrained optimization problems find many applications in medical image analysis, for example, neuroimaging, cardiovascular imaging, and oncological imaging. We review related literature and give examples on the formulation, discretization, and numerical solution of PDE-constrained optimization problems for medical imaging. We discuss three examples. The first one is image registration. The second one is data assimilation for brain tumor patients, and the third one data assimilation in cardiovascular imaging. The image registration problem is a classical task in medical image analysis and seeks to find pointwise correspondences between two or more images. The data assimilation problems use a PDE-constrained formulation to link a biophysical model to patient-specific data obtained from medical images. The associated optimality systems turn out to be sets of nonlinear, multicomponent PDEs that are challenging to solve in an efficient way. The ultimate goal of our work is the design of inversion methods that integrate complementary data, and rigorously follow mathematical and physical principles, in an attempt to support clinical decision making. This requires reliable, high-fidelity algorithms with a short time-to-solution. This task is complicated by model and data uncertainties, and by the fact that PDE-constrained optimization problems are ill-posed in nature, and in general yield high-dimensional, severely ill-conditioned systems after discretization. These features make regularization, effective preconditioners, and iterative solvers that, in many cases, have to be implemented on distributed-memory architectures to be practical, a prerequisite. We showcase state-of-the-art techniques in scientific computing to tackle these challenges.

Energy corrected FEM for optimal Dirichlet boundary control problems

In the presence of re-entrant corners, the solution of PDE-constrained optimization problems, in general, has singular components, even when the given data are smooth. Consequently, the use of standard finite element discretization gives rise to the so-called ’pollution effect’. This means that only a reduced convergence rate, compared to the best approximation error, is obtained. We discuss how optimal convergence rates in weighted norms can be achieved using the idea of energy corrected finite element methods, applied to optimal Dirichlet boundary control problem in the energy space. We present optimal error estimates in weighted norms for the state variable and for the control. Several numerical examples illustrate the obtained theoretical results.

Nonlinear Preconditioning Techniques for Full-Space Lagrange--Newton Solution of PDE-Constrained Optimization Problems

The full-space Lagrange--Newton algorithm is one of the numerical algorithms for solving problems arising from optimization problems constrained by nonlinear partial differential equations. Newton-type methods enjoy fast convergence when the nonlinearity in the system is well-balanced; however, for some problems, such as the control of incompressible flows, even linear convergence is difficult to achieve and a long stagnation period often appears in the iteration history. In this work, we introduce a nonlinearly preconditioned inexact Newton algorithm for the boundary control of incompressible flows. The system has nine field variables, and each field variable plays a different role in the nonlinearity of the system. The nonlinear preconditioner approximately removes some of the field variables, and as a result, the nonlinearity is balanced and inexact Newton converges much faster when compared to the unpreconditioned inexact Newton method or its two-grid version. Some numerical results are presented to d...

Block-Symmetric and Block-Lower-Triangular Preconditioners for PDE-Constrained Optimization Problems

Optimization problems with partial differential equations as constraints arise widely in many areas of science and engineering, in particular in problems of the design. The solution of such class of PDE-constrained optimization problems is usually a major computational task. Because of the complexion for directly seeking the solution of PDE-constrained optimization problem, we transform it into a system of linear equations of the saddle-point form by using the Galerkin finite-element discretization. For the discretized linear system, in this paper we construct a block-symmetric and a block-lower-triangular preconditioner, for solving the PDE-constrained optimization problem. Both preconditioners exploit the structure of the coefficient matrix. The explicit expressions for the eigenvalues and eigenvectors of the corresponding preconditioned matrices are derived. Numerical implementations show that these block preconditioners can lead to satisfactory experimental results for the preconditioned GMRES methods when the regularization parameter is suitably small.

Algorithms for PDE‐constrained optimization

AbstractSome first and second order algorithmic approaches for the solution of PDE‐constrained optimization problems are reviewed. An optimal control problem for the stationary Navier‐Stokes system with pointwise control constraints serves as an illustrative example. Some issues in treating inequality constraints for the state variable and alternative objective functions are also discussed (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)