Generalized adjoint for physical processes with parameterized discontinuities. Part VI: Minimization problems in multidimensional space

Generalized adjoint with modified discretization and generalized coarse-gra in adjoint are derived for a vector system of equations that contains param eterized on/off switches. With vector examples, it is shown that the conven tional adjoint minimization may have a convergence problem in multidimensio nal space. The problem can be solved by the generalized adjoint with modifi ed discretization or by the generalized coarse-grain adjoint without modify ing the traditional discretization in the forward model.