PDE-Constrained Optimal Control Problems with Uncertain Parameters using SAGA

We consider an optimal control problem (OCP) for a partial differential equation (PDE) with random coefficients. The function is deterministic, distributed forcing term that minimizes expected quadratic regularized loss functional. For the numerical approximation of this PDE-constrained OCP, we replace expectation in objective functional by suitable quadrature formula and, eventually, discretize PDE Galerkin method. To practically solve such approximate propose importance sampling version SAGA algorithm [A. Defazio, F. Bach, and S. Lacoste-Julien, Advances Neural Information Processing Systems 27, Curran Associates, Red Hook, NY, 2014, pp. 1646--1654], type stochastic gradient fixed-length memory term, which computes at each iteration only one point, randomly chosen from possibly nonuniform distribution. provide full error complexity analysis proposed scheme. In particular compare generalized sampling, conjugate (CG) algorithms, applied to same discretized OCP. show converges exponentially number iterations as CG has similar asymptotic computational complexity, terms cost versus accuracy. Moreover, it features good preasymptotic properties, shown our experiments, makes appealing limited budget context.