2018/10/31 by Matthieu Martin, Fabio Nobile, Martin, Matthieu C. +1 · 2 citations
Decision Sciences · Mathematics · #Probabilistic and Robust Engineering Design #Probability and Risk Models #Statistical Methods and Inference
paper · pdf · doi:10.48550/arxiv.1810.13378
We consider an optimal control problem (OCP) for a partial differential\nequation (PDE) with random coefficients. The optimal control function is a\ndeterministic, distributed forcing term that minimizes an expected quadratic\nregularized loss functional. For the numerical approximation of this\nPDE-constrained OCP, we replace the expectation in the objective functional by\na suitable quadrature formula and, eventually, discretize the PDE by a Galerkin\nmethod. To practically solve such approximate OCP, we propose an importance\nsampling version the SAGA algorithm, a type of Stochastic Gradient algorithm\nwith a fixed-length memory term, which computes at each iteration the gradient\nof the loss functional in only one quadrature point, randomly chosen from a\npossibly non-uniform distribution. We provide a full error and complexity\nanalysis of the proposed numerical scheme. In particular we compare the\ncomplexity of the generalized SAGA algorithm with importance sampling, with\nthat of the Stochastic Gradient (SG) and the Conjugate Gradient (CG)\nalgorithms, applied to the same discretized OCP.We show that SAGA converges\nexponentially in the number of iterations as for a CG algorithm and has a\nsimilar asymptotic computational complexity, in terms of computational cost\nversus accuracy (proportional with the time required if no parallel computing\nis used). Moreover, it features good pre-asymptotic properties, as shown by our\nnumerical experiments, which makes it appealing in a limited budget context.\n