Convergence rates of symplectic Pontryagin approximations in optimal control theory

Abstract

Abstract. 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 R d , with non smooth control. Though optimal controls in general become non smooth, 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. The 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.Mathematics Subject Classification. 49M29, 49L25.

Symplectic Pontryagin approximations for optimal design

“…see [33]. The value ofū δ for a case with constant solutionsφ * andλ * to (2.22) is approximately T · L(φ * , 0) when T is large (so that we can neglectḡ(φ T )).…”

“…Our method [33] reduces to solve a Hamiltonian system, where the Hamiltonian is a C 2 -regularized version of the original Hamiltonian. This Hamiltonian system is a nonlinear partial differential equation, where the Newton method with a sparse Jacobian becomes efficient and simple to use, e.g.…”

“…In this sense, our approximation in space and regularization shares this symplectic property. An equivalent definition of symplectic time discretization methods for Hamiltonian systems derived from optimal control problems is that the first variation of the discrete value function agrees with the discretization of the Lagrange multiplier, see [33] where symplectic time discretizations are analyzed for optimal control problems with the the similar use of viscosity solution theory as here. This property that the first variation of the discrete value function agrees with the discretization of the Lagrange multiplier only makes sense for time dependent problems, which is one reason our analysis starts by extending the original time independent optimal control problem to an artificial time dependent dynamic programming formulation.…”

Numerical Methods For Optimal Control Based Schemes For Volatility Calibration And Algorithmic Trading

“…There are different regularization schemes and the choice depends on the problem and we contend herewith Tikhonov regularization. See (Bouchouev and Isakov1997;Sandberg and Szepessy 2006) for different regularization schemes related to different problems. We construct a regularization of the Hamiltonian with appropriate function s and its regularized version sδ which are described below.…”

“…for which the Lax Friedrich scheme (Sandberg and Szepessy 2006;Bouchaouv and Isakov 1997 ) is So for our problem which is a four dimensional case that scheme will finally reduce to finite difference schemes (Almgren 2009;Bouchard et al 2010).…”