A time-parallel method for the solution of PDE-constrained optimization problems

Technical Report · Tue Dec 01 00:00:00 EST 2015

DOI:https://doi.org/10.2172/1430466· OSTI ID:1430466

Hajghassem, Mona ^[1]; Cyr, Eric C. ^[2]; Ridzal, Denis ^[2]

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1. Univ. of Maryland Baltimore County (UMBC), Baltimore, MD (United States)
2. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

We study a time-parallel approach to solving quadratic optimization problems with linear time-dependent partial differential equation (PDE) constraints. These problems arise in formulations of optimal control, optimal design and inverse problems that are governed by parabolic PDE models. They may also arise as subproblems in algorithms for the solution of optimization problems with nonlinear time-dependent PDE constraints, e.g., in sequential quadratic programming methods. We apply a piecewise linear finite element discretization in space to the PDE constraint, followed by the Crank-Nicolson discretization in time. The objective function is discretized using finite elements in space and the trapezoidal rule in time. At this point in the discretization, auxiliary state variables are introduced at each discrete time interval, with the goal to enable: (i) a decoupling in time; and (ii) a fixed-point iteration to recover the solution of the discrete optimality system. The fixed-point iterative schemes can be used either as preconditioners for Krylov subspace methods or as smoothers for multigrid (in time) schemes. We present promising numerical results for both use cases.

• Research Organization: Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
• Sponsoring Organization: USDOE National Nuclear Security Administration (NNSA)
• DOE Contract Number: AC04-94AL85000
• OSTI ID: 1430466
• Report Number(s): SAND-2015-11034R; 618843
• Country of Publication: United States
• Language: English