D6.3 Report on stochastic optimisation for simple problems

Ayoul-Guilmard, Quentin; Ganesh, Sundar; Nobile, Fabio; Rossi, Riccardo; Soriano, Cecilia

doi:10.23967/exaqute.2021.2.001

Cited by 2 publications

(1 citation statement)

References 7 publications

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“…The first one, called Stochastic Approximation (SA) method [52,Chapter 5.9], includes iterative methods that at each iteration draw new realizations independent from the previous ones. Examples of such approaches are the stochastic gradient method and its variants, which have been recently studied for OCPUU in [35,36,15,2]. In this manuscript, we adopt a second approach called Sample Average Approximation (SAA) method [52,Chapter 5.1], in which the original objective functional is replaced by an accurate approximation obtained discretizing once and for all the probability space using Stochastic Collocation Methods (SCMs), with Monte Carlo, Quasi-Monte Carlo [19], or Gaussian quadrature formulae.…”

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confidence: 99%

Preconditioners for robust optimal control problems under uncertainty

Nobile¹,

Vanzan²

2021

Preprint

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The discretization of robust quadratic optimal control problems under uncertainty using the finite element method and the stochastic collocation method leads to large saddle-point systems, which are fully coupled across the random realizations. Despite its relevance for numerous engineering problems, the solution of such systems is notoriusly challenging. In this manuscript, we study efficient preconditioners for all-at-once approches using both an algebraic and an operator preconditioning framework. We show in particular that for values of the regularization parameter not too small, the saddle-point system can be efficiently solved by preconditioning in parallel all the state and adjoint equations. For small values of the regularization parameter, robustness can be recovered by the additional solution of a small linear system, which however couples all realizations. A mean approximation and a Chebyshev semi-iterative method are investigated to solve this reduced system. Our analysis considers a random elliptic partial differential equation whose diffusion coefficient κ(x, ω) is modeled as an almost surely continuous and positive random field, though not necessarily uniformly bounded and coercive. We further provide estimates on the dependence of the preconditioned system on the variance of the random field. Such estimates involve either the first or second moment of the random variables 1/ min x∈D κ(x, ω) and max x∈D κ(x, ω), where D is the spatial domain. The theoretical results are confirmed by numerical experiments, and implementation details are further addressed.

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mentioning

confidence: 99%

Preconditioners for robust optimal control problems under uncertainty

Nobile¹,

Vanzan²

2021

Preprint

View full text Add to dashboard Cite

show abstract

Preconditioners for robust optimal control problems under uncertainty

Nobile

Vanzan

2022

Numerical Linear Algebra App

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The discretization of robust quadratic optimal control problems under uncertainty using the finite element method and the stochastic collocation method leads to large saddle-point systems, which are fully coupled across the random realizations. Despite its relevance for numerous engineering problems, the solution of such systems is notoriously challenging. In this manuscript, we study efficient preconditioners for all-at-once approaches using both an algebraic and an operator preconditioning framework. We show in particular that for values of the regularization parameter not too small, the saddle-point system can be efficiently solved by preconditioning in parallel all the state and adjoint equations.For small values of the regularization parameter, robustness can be recovered by the additional solution of a small linear system, which however couples all realizations. A mean approximation and a Chebyshev semi-iterative method are proposed to solve this reduced system. We consider a random elliptic partial differential equation whose diffusion coefficient 𝜅(x, 𝜔) is modeled as an almost surely continuous and positive random field, though not necessarily uniformly bounded and coercive. We further provide estimates of the dependence of the spectrum of the preconditioned system matrix on the statistical properties of the random field and on the discretization of the probability space. Such estimates involve either the first or second moment of the random variables 1∕ min x∈D 𝜅(x, 𝜔) and max x∈D 𝜅(x, 𝜔), where D is the spatial domain. The theoretical results are confirmed by numerical experiments, and implementation details are further addressed.

show abstract

D6.3 Report on stochastic optimisation for simple problems

Cited by 2 publications

References 7 publications

Preconditioners for robust optimal control problems under uncertainty

Preconditioners for robust optimal control problems under uncertainty

Preconditioners for robust optimal control problems under uncertainty

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