Browsing by Author "Kroeger, Nathaniel James"

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    ADMM and Diagonalization Based Parallel-in-Time Methods for Optimal Control Problems
    (2023-12-12) Kroeger, Nathaniel James; Heinkenschloss, Matthias
    This thesis investigates alternating direction method of multipliers (ADMM) and diagonalization - based parallel-in-time methods for linear-quadratic partial differential equation (PDE)-constrained optimization problems. The solution of such optimization problems is computing time and memory intensive, and efficient methods are essential to making such problems computationally tractable. Two parallel-in-time approaches are considered. In the first approach, ADMM is applied to a time domain decomposition (TDD) formulation. ADMM tailored to the TDD formulation requires the parallel solution of smaller subdomain problems and reduces the number of variables that need to be kept in memory globally. Thus, ADMM carries out the parallelization-in-time because the ADMM subproblems are able to be broken down by time subdomain. In the second approach, a diagonalization technique is used to parallelize-in-time. This approach is then extended to handle inequality constraints. The inequality constraints extension is handled by a combination of diagonalization and ADMM - the ADMM algorithm is the “main” algorithm, while the diagonalization method handles the computationally expensive substep in ADMM. Here, the diagonalization provides the parallelism in time, while the ADMM algorithm decouples the inequality constraints from the rest of the optimal control problem. Numerical results are provided to show the effectiveness of these methods.
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    ADMM Based Methods for Time-Domain Decomposition Formulations of Optimal Control Problems
    (2020-08-03) Kroeger, Nathaniel James; Heinkenschloss, Matthias
    This thesis investigates alternating direction method of multipliers (ADMM)-based methods for time-domain decomposition (TDD) formulations of linear-quadratic partial differential equation (PDE)-constrained optimization problems. The solution of such optimization problems is computing time and memory intensive. TDD formulations split the time-dependent PDE into coupled subdomain equations and introduce potential for parallelism and global memory reduction. This thesis tailors ADMM to the TDD structure. ADMM requires the parallel solution of smaller subdomain problems and reduces the number of variables that need to be kept in memory globally. Different TDD splittings lead to different ADMM variants. ADMM convergence analyses are derived from a matrix-splitting view and from the equivalence to the Douglas-Rachford algorithm applied to the dual problem, and are applied to these different variants. The effectiveness of ADMM as a preconditioner within GMRES is investigated. Computational results are presented for several variants of ADMM applied to an advection-diffusion problem.