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Adjoint Sensitivity Computation for the Parallel Multigrid Reduction in Time Software Library XBraid Günther, Stefanie


In this paper we present an adjoint solver for the multigrid in time software library XBraid. XBraid provides a non-intrusive approach for simulating unsteady dynamics on multiple processors while parallelizing not only in space but also in the time domain. It applies an iterative multigrid reduction in time algorithm to existing spatially parallel classical time propagators and computes the unsteady solution parallel in time. However, in many engineering applications not only the primal unsteady flow computation is of interest but also the ability to compute sensitivities that determine the influence of design changes to some output quantity. In recent years, adjoint solvers have widely been developed which propagate sensitivity information backwards through the time domain. We develop an adjoint solver for XBraid that enhances the primal iterations by an iteration for computing adjoint sensitivities. In each iteration, the adjoint code runs backwards through the primal XBraid actions and computes the consistent discrete adjoint sensitivities parallel in time. It is highly non-intrusive as existing adjoint time propagators can easily be integrated through the adjoint interface. We validate the adjoint code by applying it to an unsteady partial differential equation that mimics the behavior of separated flows past bluff bodies. In our 1D model, the near wake is mimicked by a nonlinear ODE, namely the Lorenz attractor which exhibits self-excited oscillations. The far wake is modeled by an advection - diffusion equation whose upstream boundary condition is determined by the ODE mimicking the near wake. We demonstrate the integration of a serial time stepping algorithm, that solves the PDE forward in time, into the parallel-in-time XBraid framework as well as the development of the corresponding adjoint interface. The resulting sensitivities are in good agreement with those computed from finite differences. Nevertheless, there is still great potential for optimizing the performance of the adjoint code using advanced algorithmic differentiation techniques such as reverse accumulation and checkpointing. Due to the iterative nature of the primal and the adjoint flow computation, the method is very well suited for simultaneous optimization algorithms like the One-shot method which solve the optimization problem in the full space. They have proven to be very efficient for optimization with steady-state PDE constraints while its application to unsteady PDE is still under development. The non-intrusive adjoint XBraid solver is therefore highly desirable and will extend its application range from pure simulation to optimization.

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