Many inverse problems for differential equations
can be formulated as optimal control problems.
It is well known that inverse problems often need to
be regularized to obtain good approximations.
This work presents a systematic method to regularize
and to establish error estimates for approximations to
some control problems in high dimension,
based on symplectic approximation
of the Hamiltonian system for the control problem. In particular
the work derives error estimates
and constructs regularizations for numerical approximations to
optimally controlled ordinary differential equations in
with non smooth control.
Though optimal controls in general become
viscosity solutions to the corresponding Hamilton-Jacobi-Bellman
equation provide good theoretical foundation, but poor computational efficiency
in high dimensions.
The computational method here uses the adjoint variable and works
efficiently also for high dimensional problems with d >> 1.
Controls can be discontinuous due to a lack of regularity
in the Hamiltonian or due to colliding backward paths, i.e. shocks.
error analysis, for both these cases, is based on consistency with the
Hamilton-Jacobi-Bellman equation, in the viscosity solution sense,
and a discrete Pontryagin principle:
the bi-characteristic Hamiltonian ODE system is solved
with a C
2 approximate Hamiltonian.
The error analysis leads to estimates
useful also in high dimensions since the bounds depend on the Lipschitz
norms of the Hamiltonian and the gradient of the value function
but not on d explicitly.
Applications to inverse implied volatility estimation, in mathematical finance,
and to a topology optimization problem are presented.
An advantage with the Pontryagin based method
is that the Newton method can be applied to efficiently
solve the discrete nonlinear Hamiltonian system,
with a sparse Jacobian that can be calculated explicitly.