A multiscale method for semi-linear elliptic equations with localized uncertainties and non-linearities

Nouy, Anthony; Pled, Florent

doi:10.1051/m2an/2018025

Cited by 9 publications

(7 citation statements)

References 79 publications

(96 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We will focus on a semilinear diffusion-reaction equation with uncertainties, which describes transport phenomena at equilibrium and is motivated by [ 41 ]. We assume that there exist random fields and , which are the diffusion and reaction coefficients, respectively.…”

Section: Application To Pde-constrained Optimization Under Uncertaintmentioning

confidence: 99%

Stochastic proximal gradient methods for nonconvex problems in Hilbert spaces

Geiersbach

Scarinci

2021

Comput Optim Appl

View full text Add to dashboard Cite

For finite-dimensional problems, stochastic approximation methods have long been used to solve stochastic optimization problems. Their application to infinite-dimensional problems is less understood, particularly for nonconvex objectives. This paper presents convergence results for the stochastic proximal gradient method applied to Hilbert spaces, motivated by optimization problems with partial differential equation (PDE) constraints with random inputs and coefficients. We study stochastic algorithms for nonconvex and nonsmooth problems, where the nonsmooth part is convex and the nonconvex part is the expectation, which is assumed to have a Lipschitz continuous gradient. The optimization variable is an element of a Hilbert space. We show almost sure convergence of strong limit points of the random sequence generated by the algorithm to stationary points. We demonstrate the stochastic proximal gradient algorithm on a tracking-type functional with a $$L^1$$ L 1 -penalty term constrained by a semilinear PDE and box constraints, where input terms and coefficients are subject to uncertainty. We verify conditions for ensuring convergence of the algorithm and show a simulation.

show abstract

Section: Application To Pde-constrained Optimization Under Uncertaintmentioning

confidence: 99%

Stochastic proximal gradient methods for nonconvex problems in Hilbert spaces

Geiersbach

Scarinci

2021

Comput Optim Appl

View full text Add to dashboard Cite

show abstract

“…for the H 1 0 (D)-seminorm. We will focus on a semilinear diffusion-reaction equation with uncertainties, which describes transport phenomena at equilibrium and is motivated by [39]. We assume there that exist random fields a : D × Ω → R and r : D × Ω → R, which are the diffusion and reaction coefficients, respectively.…”

Section: Model Problemmentioning

confidence: 99%

Stochastic Proximal Gradient Methods for Nonconvex Problems in Hilbert Spaces

Geiersbach,

Scarinci

2020

Preprint

View full text Add to dashboard Cite

For finite-dimensional problems, stochastic approximation methods have long been used to solve stochastic optimization problems. Their application to infinite dimensional problems is less understood, particularly for nonconvex objectives. This paper presents convergence results for the stochastic proximal gradient method applied to Hilbert spaces, motivated by optimization problems with partial differential equation (PDE) constraints with random inputs and coefficients. We study stochastic algorithms for nonconvex and nonsmooth problems, where the nonsmooth part is convex and the nonconvex part is the expectation, which is assumed to have a Lipschitz continuous gradient. The optimization variable is an element of a Hilbert space. We show almost sure convergence of strong limit points of the random sequence generated by the algorithm to stationary points. We demonstrate the stochastic proximal gradient algorithm on a tracking-type functional with a L 1 -penalty term constrained by a semilinear PDE and box constraints, where input terms and coefficients are subject to uncertainty. We verify conditions for ensuring convergence of the algorithm and show a simulation.

show abstract

“…The local and auxiliary analysis are computed in parallel thus the use of the auxiliary model does not add computational time to the general algorithm. The algorithm is a stationary iteration; its convergence can be proved under very general hypothesis (see [28] for a proof with weak hypothesis and [17] for the registration of the method amongst Schwarz alternating methods for which many convergence results exist). In the general case, relaxation may have to be used to ensure convergence by modifying (13) as follows: P i+1 = P i + ωr i , with ω small enough.…”

Section: Iterationsmentioning

confidence: 99%

Space/time global/local noninvasive coupling strategy: Application to viscoplastic structures

Blanchard

Allix

Gosselet

et al. 2019

Finite Elements in Analysis and Design

View full text Add to dashboard Cite

The purpose of this paper is to extend the non-invasive global/local iterative coupling technique [15] to the case of large structures undergoing nonlinear time-dependent evolutions at all scales. It appears that, due to the use of legacy codes, the use of different time grids at the global and local levels is mandatory in order to reach a satisfying level of precision. In this paper two strategies are proposed and compared for elastoviscoplastic models.The questions of the precision and performance of those schemes with respect to a monolithic approach is addressed. The methods are first exposed on a 2D example and then applied on a 3D part of industrial complexity.

show abstract

A multiscale method for semi-linear elliptic equations with localized uncertainties and non-linearities

Cited by 9 publications

References 79 publications

Stochastic proximal gradient methods for nonconvex problems in Hilbert spaces

Stochastic proximal gradient methods for nonconvex problems in Hilbert spaces

Stochastic Proximal Gradient Methods for Nonconvex Problems in Hilbert Spaces

Space/time global/local noninvasive coupling strategy: Application to viscoplastic structures

Contact Info

Product

Resources

About