Optimization of Systems modeled by PDEs

Newton–Picard-Based Preconditioning for Linear-Quadratic Optimization Problems with Time-Periodic Parabolic PDE Constraints

  • Hans Georg Bock
  • Andreas Potschka
  • M.S. Mommer
  • Johannes P. Schlöder
    • Categories Optimization of Systems modeled by PDEs, Quadratic Programming Tags , , , ,

    We develop and investigate two preconditioners for a basic linear iterative splitting method for the numerical solution of linear-quadratic optimization problems with time-periodic parabolic PDE constraints. The resulting real-valued linear system to be solved is symmetric indefinite. We propose all-at-once symmetric indefinite preconditioners based on a Newton–Picard approach which divides the variable space into slow … Read more

    Using approximate secant equations in limited memory methods for multilevel unconstrained optimization

  • Serge Gratton
  • Vincent Malmedy
  • Philippe L. Toint
    • Categories Optimization of Systems modeled by PDEs, Unconstrained Optimization Tags , , , ,

    The properties of multilevel optimization problems defined on a hierarchy of discretization grids can be used to define approximate secant equations, which describe the second-order behaviour of the objective function. Following earlier work by Gratton and Toint (2009), we introduce a quasi-Newton method (with a linesearch) and a nonlinear conjugate gradient method that both take … Read more

    Asymptotic expansion for the solution of a penalized control constrained semilinear elliptic problems

  • Joseph Frédéric Bonnans
  • Francisco J. Silva
    • Categories Optimization of Systems modeled by PDEs Tags , , ,

    In this work we consider the optimal control problem of a semilinear elliptic PDE with a Dirichlet boundary condition, where the control variable is distributed over the domain and is constrained to be nonnegative. The approach is to consider an associated parametrized family of penalized problems, whose solutions define a central path converging to the … Read more

    Band Gap Optimization of Two-Dimensional Photonic Crystals Using Semidefinite Programming and Subspace Methods

  • Robert M. Freund
  • Han Men
  • Ngoc-Cuong Nguyen
  • Pablo A. Parrilo
  • Jaime Peraire
    • Categories Optimization of Systems modeled by PDEs, Semi-definite Programming Tags , ,

    In this paper, we consider the optimal design of photonic crystal band structures for two-dimensional square lattices. The mathematical formulation of the band gap optimization problem leads to an infinite-dimensional Hermitian eigenvalue optimization problem parametrized by the dielectric material and the wave vector. To make the problem tractable, the original eigenvalue problem is discretized using … Read more

    Control problems with mixed constraints and application to an optimal investment problem

  • Joseph Frédéric Bonnans
  • Dan Tiba
    • Categories Optimization of Systems modeled by PDEs Tags , , , ,

    We discuss two optimal control problems of parabolic equations, with mixed state and control constraints, for which the standard qualification condition does not hold. Our first example is a bottleneck problem, and the second one is an optimal investment problem where a utility type function is to be minimized. By an adapted penalization technique, we … Read more

    An Interior-Point Algorithm for Large-Scale Nonlinear Optimization with Inexact Step Computations

  • Frank E. Curtis
  • Olaf Schenk
    • Categories Nonlinear Optimization, Optimization of Systems modeled by PDEs Tags , , , , , ,

    We present a line-search algorithm for large-scale continuous optimization. The algorithm is matrix-free in that it does not require the factorization of derivative matrices. Instead, it uses iterative linear system solvers. Inexact step computations are supported in order to save computational expense during each iteration. The algorithm is an interior-point approach derived from an inexact … Read more

    The Advanced Step NMPC Controller: Optimality, Stability and Robustness

  • Lorenz T. Biegler
  • Victor M. Zavala
    • Categories Control Applications, Nonlinear Optimization, Optimization of Systems modeled by PDEs

    Widespread application of dynamic optimization with fast optimization solvers leads to increased consideration of first-principles models for nonlinear model predictive control (NMPC). However, significant barriers to this optimization-based control strategy are feedback delays and consequent loss of performance and stability due to on-line computation. To overcome these barriers, recently proposed NMPC controllers based on nonlinear … Read more

    Approximating Hessians in multilevel unconstrained optimization

  • Vincent Malmedy
  • Philippe L. Toint
    • Categories Optimization of Systems modeled by PDEs, Unconstrained Optimization Tags , , , ,

    We consider Hessian approximation schemes for large-scale multilevel unconstrained optimization problems, which typically present a sparsity and partial separability structure. This allows iterative quasi-Newton methods to solve them despite of their size. Structured finite-difference methods and updating schemes based on the secant equation are presented and compared numerically inside the multilevel trust-region algorithm proposed by … Read more

    On fast integration to steady state and earlier times

  • Uri Ascher
  • Hui Huang
  • Benar Fux Svaiter
  • Kees van den Doel
    • Categories Basic Sciences Applications, Optimization of Systems modeled by PDEs, Systems governed by Differential Equations Optimization Tags , , , , ,

    The integration to steady state of many initial value ODEs and PDEs using the forward Euler method can alternatively be considered as gradient descent for an associated minimization problem. Greedy algorithms such as steepest descent for determining the step size are as slow to reach steady state as is forward Euler integration with the best … Read more

    Numerical Experience with a Recursive Trust-Region Method for Multilevel Nonlinear Optimization

  • Serge Gratton
  • Annick Sartenaer
  • Philippe L. Toint
  • Mélodie Mouffe
  • Dimitri Tomanos
    • Categories Bound-constrained Optimization, Distributed Control, Optimization of Systems modeled by PDEs Tags , , , , ,

    We consider an implementation of the recursive multilevel trust-region algorithm proposed by Gratton, Mouffe, Toint, Weber (2008) for bound-constrained nonlinear problems, and provide numerical experience on multilevel test problems. A suitable choice of the algorithm’s parameters is identified on these problems, yielding a satisfactory compromise between reliability and efficiency. The resulting default algorithm is then … Read more