An efficient algorithm for Hamilton–Jacobi equations in high dimension

Optimal control of molecular dynamics using Markov state models

“…Available methods for the numerical solution of high dimensional HJB equations include Markov chain approximations [2], monotone schemes [3,4], or methods designed for relatively specific problems [5,6]. Nonetheless we are not aware of a single article on optimal control of molecular dynamics (MD) using dynamic programming principles, although the interest in controlling molecular dynamics simulations has already started more than a decade ago with the development of Targeted MD or Steered MD [7,8], and laser control of (open) quantum systems [9]; see [10] for a recent survey of approaches from molecular physics, chemistry, and optical control of quantum molecular dynamics.…”

An Adaptive Sparse Grid Semi-Lagrangian Scheme for First Order Hamilton-Jacobi Bellman Equations

“…In case this number is too large, or if the set B is infinite, then a minimisation procedure which uses only evaluations of the objective function, and not its derivatives, could be performed without changing the main steps of the SL-SG scheme. We refer for instance to [4], where this approach was successfully employed for a semi-Lagrangian scheme using Brent's minimization algorithm.…”

“…To obtain an exact solution to compare with, we consider the case ϕ(x) := q( x ) where q : R + → R is a non-decreasing function. Then, we can show that 4 (30) v(t, x) = q(( x − t) + ).…”

“…[5,7,10]. Finally, adaptive schemes for solving the HJ(B) equation have been developed, see for instance [4,16]. But altogether, the numerical treatment of HJ(B) equations remains a challenging problem, in particular in higher dimensions.…”

Differential Games Controllers That Confine a System to a Safe Region in the State Space, With Applications to Surge Tank Control

“…Attention has been given, in particular, to computation saving schemes for control systems with higher dimensional state spaces [7], [11], [1], [15], [22], of which variants of the surge tank controller design problem involving colored noise disturbances are examples. We point out that the explicit solution provided in this paper to an exit problem arising in process systems, besides providing a new paradigm for surge tank control, has potential use as a benchmark for the application of generic Hamilton Jacobi equation solvers.…”