Multigrid Methods for the Constraints

Steepest Descent and BFGS methods. A natural way to use multigrid to accelerate optimization procedures is to accelerate the solver parts for both the state and the adjoint equations. In such a case the number of optimization iteration required to solve the problem will remain unchanged, but each optimization step will require significantly less computer time due to the fast solver. This approach is straightforward and does not requires additional analysis into the numerical processes. This approach may be viewed as a loosely coupled optimization procedure, in which the solvers are separated from the optimizer. It works with any gradient based method, either a steepest descent or a quasi-Newton method such as BFGS.

ALGORITHM


(1) Solve the State equation using a Multigrid solver
(2) Solve the adjoint equation using a Multigrid solver
(3) Calculate Gradients
(4) Update Design Variables

Multigrid for infinite dimensional preconditioners. Infinite dimensional preconditioners can be extremely efficient as was explained in a previous lecture. In that approach one uses a preconditioner which is a partial differential operator in 3D, or an ODE in 2D design problems. Sometime a pseudo-differential preconditioner may also result, such as in transonic calculations with either the full potential or the Euler equations. In any case, in order to apply the desired preconditioner one has to solve a large scale sparse system. This system is defined on the boundary of the original domain for boundary control and shape design problems. In addition to this there is also the state and costate equations that needs to be solved per each optimization iteration.

Multigrid can be used in these type of methods as a fast solver for the state equations, adjoint equations and the equation for the preconditioner step. These multigrid solvers for the different parts can be completely independent and actually one may be replaced by some other fast solver. A typical algorithm may looks as follows,

ALGORITHM


(1) Solve the State equation using a Multigrid solver
(2) Solve the adjoint equation using a Multigrid solver
(3) Calculate Gradients
(4) Apply Infinite dimensional Preconditioner Using a Multigrid Solver
(5) Update Design Variables