Hidden Convexity: a computational approach

See Uditnarayan Kouskiya's blog:

Hidden convexity in the heat, linear transport, and Euler's rigid body equations: A computational approach

A finite element based computational scheme is developed and employed to assess a duality based
variational approach to the solution of the linear heat and transport PDE in one space dimension
and time, and the nonlinear system of ODEs of Euler for the rotation of a rigid body about
a fixed point. The formulation turns initial-(boundary) value problems into degenerate elliptic
boundary value problems in (space)-time domains representing the Euler-Lagrange equations of
suitably designed dual functionals in each of the above problems. We demonstrate reasonable
success in approximating solutions of this range of parabolic, hyperbolic, and ODE primal
problems, which includes energy dissipation as well as conservation, by a unified dual strategy
lending itself to a variational formulation. The scheme naturally associates a family of dual
solutions to a unique primal solution; such ‘gauge invariance’ is demonstrated in our computed
solutions of the heat and transport equations, including the case of a transient dual solution
corresponding to a steady primal solution of the heat equation. Primal evolution problems with
causality are shown to be correctly approximated by non-causal dual problems.