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Person# Matthieu Claude Martin

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Related research domains (19)

Monte Carlo method

Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use rando

Optimal control

Optimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has nu

Complexity

Complexity characterises the behaviour of a system or model whose components interact in multiple ways and follow local rules, leading to non-linearity, randomness, collective dynamics, hierarchy, and

Related publications (6)

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Sebastian Krumscheid, Matthieu Claude Martin, Fabio Nobile

We consider the numerical approximation of an optimal control problem for an elliptic Partial Differential Equation (PDE) with random coefficients. Specifically, the control function is a deterministic, distributed forcing term that minimizes the expected squared L2 misfit between the state (i.e. solution to the PDE) and a target function, subject to a regularization for well posedness. For the numerical treatment of this risk-averse Optimal Control Problem (OCP) we consider a Finite Element discretization of the underlying PDE, a Monte Carlo sampling method, and gradient-type iterations to obtain the approximate optimal control. We provide full error and complexity analyses of the proposed numerical schemes. In particular we investigate the complexity of a conjugate gradient method applied to the fully discretized OCP (so called Sample Average Approximation), in which the Finite Element discretization and Monte Carlo sample are chosen in advance and kept fixed over the iterations. This is compared with a Stochastic Gradient method on a fixed or varying Finite Element discretization, in which the expectation in the computation of the steepest descent direction is approximated by Monte Carlo estimators, independent across iterations, with small sample sizes. We show in particular that the second strategy results in an improved computational complexity. The theoretical error estimates and complexity results are confirmed by numerical experiments.

2021This thesis work focuses on optimal control of partial differential equations (PDEs) with uncertain parameters, treated as a random variables. In particular, we assume that the random parameters are not observable and look for a deterministic control which is robust with respect to the randomness.
The theoretical framework is based on adjoint calculus to compute the gradient of the objective functional. Unlike the deterministic case, we have a set of PDEs indexed by some uncertain parameters and the objective functional we consider includes some risk measure (e.g. expectation, variance, quantile (Value at Risk), ...) to take care of all realizations of our uncertain parameters. Introducing the regular class of coherent risk measure, results of convergence and regularity of the optimal control have been derived.

Matthieu Claude Martin, Fabio Nobile, Panagiotis Tsilifis

In this paper, we present a multilevel Monte Carlo (MLMC) version of the Stochastic Gradient (SG) method for optimization under uncertainty, in order to tackle Optimal Control Problems (OCP) where the constraints are described in the form of PDEs with random parameters. The deterministic control acts as a distributed forcing term in the random PDE and the objective function is an expected quadratic loss. We use a Stochastic Gradient approach to compute the optimal control, where the steepest descent direction of the expected loss, at each iteration, is replaced by independent MLMC estimators with increasing accuracy and computational cost. The refinement strategy is chosen a-priori such that the bias and, possibly, the variance of the MLMC estimator decays as a function of the iteration counter. Detailed convergence and complexity analyses of the proposed strategy are presented and asymptotically optimal decay rates are identified such that the total computational work is minimized. We also present and analyze an alternative version of the multilevel SG algorithm that uses a randomized MLMC estimator at each iteration. Our methodology is validated through a numerical example.