Optimization of Systems modeled by PDEs

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

    Multi-Secant Equations, Approximate Invariant Subspaces and Multigrid Optimization

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

    New approximate secant equations are shown to result from the knowledge of (problem dependent) invariant subspace information, which in turn suggests improvements in quasi-Newton methods for unconstrained minimization. It is also shown that this type of information may often be extracted from the multigrid structure of discretized infinite dimensional problems. A new limited-memory BFGS using … Read more

    A multilevel algorithm for solving the trust-region subproblem

  • Philippe L. Toint
  • Melissa Weber-Mendonca
  • Dimitri Tomanos
    • Categories Infinite Dimensional Optimization, Optimization of Systems modeled by PDEs, Unconstrained Optimization Tags , , ,

    We present a multilevel numerical algorithm for the exact solution of the Euclidean trust-region subproblem. This particular subproblem typically arises when optimizing a nonlinear (possibly non-convex) objective function whose variables are discretized continuous functions, in which case the different levels of discretization provide a natural multilevel context. The trust-region problem is considered at the highest … Read more

    Processor Speed Control with Thermal Constraints

  • David Atienza
  • Stephen Boyd
  • Giovanni De Micheli
  • Rajesh Gupta
  • Srinivasan Murali
  • Almir Mutapcic
    • Categories Control Applications, Optimization of Systems modeled by PDEs Tags , , , , ,

    We consider the problem of adjusting speeds of multiple computer processors sharing the same thermal environment, such as a chip or multi-chip package. We assume that the speed of processor (and associated variables, such as power supply voltage) can be controlled, and we model the dissipated power of a processor as a positive and strictly … Read more

    Robust Nonconvex Optimization for Simulation-based Problems

  • Dimitris Bertsimas
  • Kwong Meng Teo
  • Omid Nohadani
    • Categories Optimization of Simulated Systems, Optimization of Systems modeled by PDEs, Robust Optimization

    In engineering design, an optimized solution often turns out to be suboptimal, when implementation errors are encountered. While the theory of robust convex optimization has taken significant strides over the past decade, all approaches fail if the underlying cost function is not explicitly given; it is even worse if the cost function is nonconvex. In … Read more

    Data Assimilation in Weather Forecasting: A Case Study in PDE-Constrained Optimization

  • M. Fisher
  • Jorge Nocedal
  • Stephen Wright
  • Y. Tremolet
    • Categories Nonlinear Optimization, Optimization of Systems modeled by PDEs Tags ,

    Variational data assimilation is used at major weather prediction centers to produce the initial conditions for 7- to 10-day weather forecasts. This technique requires the solution of a very large data-fitting problem in which the major element is a set of partial differential equations that models the evolution of the atmosphere over a time window … Read more

    Second-order convergence properties of trust-region methods using incomplete curvature information, with an application to multigrid optimization

  • Serge Gratton
  • Annick Sartenaer
  • Philippe L. Toint
    • Categories Optimization of Systems modeled by PDEs, Systems governed by Differential Equations Optimization, Unconstrained Optimization Tags , ,

    Convergence properties of trust-region methods for unconstrained nonconvex optimization is considered in the case where information on the objective function’s local curvature is incomplete, in the sense that it may be restricted to a fixed set of “test directions” and may not be available at every iteration. It is shown that convergence to local “weak” … Read more

    Pricing a class of exotic options via moments and SDP relaxations

  • Jean-Bernard Lasserre
  • Thomas Prieto-Rumeau
  • Mihail Zervos
    • Categories Finance and Economics, Optimization of Systems modeled by PDEs

    We present a new methodology for the numerical pricing of a class of exotic derivatives such as Asian or barrier options when the underlying asset price dynamics are modelled by a geometric Brownian motion or a number of mean-reverting processes of interest. This methodology identifies derivative prices with infinite-dimensional linear programming problems involving the moments … Read more