A Stochastic Gradient Method With Mesh Refinement for PDE-Constrained Optimization Under Uncertainty

Geiersbach, Caroline; Wollner, Winnifried

doi:10.1137/19m1263297

Cited by 32 publications

(36 citation statements)

References 37 publications

(26 reference statements)

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“…Additionally, there is no need to determine the Lipschitz constant for the gradient, which in the application depends on (among other things) the Poincaré constant and the lower bound on the random fields, and thus lead to a prohibitively small constant step size. This phenomenon has been demonstrated in [20].…”

Section: Resultsmentioning

confidence: 64%

“…Additionally, we allow for stochastic gradients subject to additive bias, which is not covered by existing results. This theory can be used to develop mesh refinement strategies in applications with PDEs [ 20 ].…”

Section: Asymptotic Convergence Resultsmentioning

confidence: 99%

“…In lieu of efficiency estimates, it would be desirable to have better termination conditions that do not rely on increased sampling as our heuristic did in the numerical experiments. Finally, it would be natural to investigate mesh refinement strategies as in [ 20 ]. For more involved choices of nonsmooth terms, the computation is also subject to numerical error and should be treated.…”

Section: Discussionmentioning

confidence: 99%

“…This is done to project the stochastic gradient onto the L 2 (D) space as in [20]. Hence, the last line of Algorithm 2 is given by the expression u n+1 ∶= prox t n h u n − t n P̂hG(u n , n ) .…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…We denote the mesh fineness with . The state and adjoint were discretized using piecewise linear finite elements, (where denotes the space of polynomials of degree up to i ), given by the set For the controls, we choose a discretization of by piecewise constants, given by the set We use the -projection defined for each by This is done to project the stochastic gradient onto the space as in [ 20 ]. Hence, the last line of Algorithm 2 is given by the expression For the computation of the proximity operator , we use the formula from [ 5 , Example 6.22], defined piecewise on each element of the mesh.…”

Section: Application To Pde-constrained Optimization Under Uncertaintmentioning

confidence: 99%

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Stochastic proximal gradient methods for nonconvex problems in Hilbert spaces

Geiersbach

Scarinci

2021

Comput Optim Appl

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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.

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Section: Resultsmentioning

confidence: 64%

Section: Asymptotic Convergence Resultsmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Numerical Experimentsmentioning

confidence: 99%

Section: Application To Pde-constrained Optimization Under Uncertaintmentioning

confidence: 99%

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Stochastic proximal gradient methods for nonconvex problems in Hilbert spaces

Geiersbach

Scarinci

2021

Comput Optim Appl

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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.

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