Stochastic optimization methods for the simultaneous control of parameter-dependent systems

We address the application of stochastic optimization methods for the simultaneous control of parameter-dependent systems. In particular, we focus on the classical Stochastic Gradient Descent (SGD) approach of Robbins and Monro, and on the recently developed Continuous Stochastic Gradient (CSG) algorithm. We consider the problem of computing simultaneous controls through the minimization of a cost functional defined as the superposition of individual costs for each realization of the system. We compare the performances of these stochastic approaches, in terms of their computational complexity, with those of the more classical Gradient Descent (GD) and Conjugate Gradient (CG) algorithms, and we discuss the advantages and disadvantages of each methodology. In agreement with well-established results in the machine learning context, we show how the SGD and CSG algorithms can significantly reduce the computational burden when treating control problems depending on a large amount of parameters. This is corroborated by numerical experiments

Efficient Gaussian Sampling for Solving Large-Scale Inverse Problems using MCMC Methods

The resolution of many large-scale inverse problems using MCMC methods requires a step of drawing samples from a high dimensional Gaussian distribution. While direct Gaussian sampling techniques, such as those based on Cholesky factorization, induce an excessive numerical complexity and memory requirement, sequential coordinate sampling methods present a low rate of convergence. Based on the reversible jump Markov chain framework, this paper proposes an efficient Gaussian sampling algorithm having a reduced computation cost and memory usage. The main feature of the algorithm is to perform an approximate resolution of a linear system with a truncation level adjusted using a self-tuning adaptive scheme allowing to achieve the minimal computation cost. The connection between this algorithm and some existing strategies is discussed and its efficiency is illustrated on a linear inverse problem of image resolution enhancement.Comment: 20 pages, 10 figures, under review for journal publicatio