Accelerating direct-adjoint studies using a parallel-in-time approach

Adjoint methods are widely used in fluid mechanics, for example in studies of parametric sensitivity and optimization. They require the iterative solution of direct and adjoint equations from which gradient information is extracted via an optimality condition. This information is used by itself for sensitivity studies, or processed in an optimization algorithm to improve a prescribed cost functional. The direct-adjoint looping involves solving the governing and adjoint equations multiple times and the computational cost can increase rapidly. In addition, checkpointing is required for nonlinear governing equations or specific cost functionals, further increasing the demands on computational resources.

In this talk we explore a way to accelerate the direct-adjoint looping method using a parallel-in-time approach. Parallel-in-time integration methods have been previously employed for the solution of forward-in-time problems and have shown remarkable gains in efficiency. By fitting the adjoint component into this approach we are able to significantly cut down on the cost of a direct-adjoint loop, thus accelerating studies which use this approach. This method is discussed and illustrated using a simple ODE problem for a linear and nonlinear direct equation.