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We consider a general class of nonsmooth optimal control problems withpartial differential equation (PDE) constraints, which are very challenging dueto its nonsmooth objective functionals and the resulting high-dimensional andill-conditioned systems after discretization. We focus on the application of aprimal-dual method, with which different types of variables can be treatedindividually and thus its main computation at each iteration only requiressolving two PDEs. Our target is to accelerate the primal-dual method witheither larger step sizes or operator learning techniques. For the acceleratedprimal-dual method with larger step sizes, its convergence can be still provedrigorously while it numerically accelerates the original primal-dual method ina simple and universal way. For the operator learning acceleration, weconstruct deep neural network surrogate models for the involved PDEs. Once aneural operator is learned, solving a PDE requires only a forward pass of theneural network, and the computational cost is thus substantially reduced. Theaccelerated primal-dual method with operator learning is mesh-free, numericallyefficient, and scalable to different types of PDEs. The accelerationeffectiveness of these two techniques is promisingly validated by somepreliminary numerical results.

Accelerated primal-dual methods with enlarged step sizes and operator learning for nonsmooth optimal control problems