| I will introduce the PETSc software package for the scalable
solution of partial differential equations and discuss it's
design. This will include results from several application codes
that use PETSc. Then I will introduce a matrix-free multigrid
method for the solution of PDE constrained optimization problems
that can provide quadratic convergence but does not require the
formation of the Hessian. This approach runs multigrid over all
of the variables, including the control variables and the
Lagrange multipliers. The software framework for the PDE
optimization algorithm is almost identical to that for just the
PDE solution. |