An Introduction to the Design of the PETSc Package and 

a New Approach to PDE Constrained Optimization

Abstract

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.