A Multilevel Stochastic Collocation Algorithm for Optimization of PDEs with Uncertain Coefficients

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

“…[ 47 , 51 ]. Sparse-tensor discretization has been used for optimal control problems in, for instance, [ 28 , 29 ].…”

Section: Introductionmentioning

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

“…The following assumption is essential for the tight approximation of the probability function p(u, x), for each x ∈ D, using the sandwich condition (38).…”

Section: The Approximation Functions and Their Propertiesmentioning

confidence: 99%

“…for τ ∈ (0, 1) and x ∈ D c . Consequently, property P3, inequalities (38) and equations (40), (41) imply the lower and upper approximations of p(•, x) given as…”

Section: The Approximation Functions and Their Propertiesmentioning

confidence: 99%

“…In [2,40,41] the authors consider mean-variance risk-avers objective functions for the optimization of elliptic PDE systems with random data. Various numerical methods are also investigated for the solution of elliptic PDE optimization problems with random coefficients, e.g., sparse grid stochastic collocation ( [38,39,43,58]), reduced basis ( [19]), finite element approximation [32,60], etc.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Chance constrained optimization of elliptic PDE systems with a smoothing convex approximation

et al. 2020

View full text Add to dashboard Cite

In this paper, we consider chance constrained optimization of elliptic partial differential equation (CCPDE) systems with random parameters and constrained state variables. We demonstrate that, under standard assumptions, CCPDE is a convex optimization problem. Since chance constrained optimization problem are generally nonsmooth and difficult to solve directly, we propose a smoothing inner-outer approximation method to generate a sequence of smooth approximate problems for the CCPDE. Thus, the optimal solution of the convex CCPDE is approximable through optimal solutions of the inner-outer approximation problems. A numerical example demonstrates the viability of the proposed approach.

show abstract

“…Numerous studies (Ganapathysubramanian and Zabaras 2008; Sankaran and Marsden 2011; Agarwal and Aluru 2009) have used adaptive grids using Smolyak algorithm. Present study is limited to isotropic grids only with node distribution based on Clenshaw-Curtis extrema (Kouri 2014). …”

Section: Stochastic Collocationmentioning

confidence: 99%

Dispersion modeling of thermal power plant emissions on stochastic space

Gorle

Sambana

2015

Theor Appl Climatol

View full text Add to dashboard Cite

This study aims to couple a deterministic atmospheric dispersion solver based on Gaussian model with a nonintrusive stochastic model to quantify the propagation of multiple uncertainties. The nonintrusive model is based on probabilistic collocation framework. The advantage of nonintrusive nature is to retain the existing deterministic plume dispersion model without missing the accuracy in extracting the statistics of stochastic solution. The developed model is applied to analyze the SO 2 emission released from coal firing unit in the second stage of the National Thermal Power Corporation (NTPC) in Dadri, India using Burban^conditions. The entire application is split into two cases, depending on the source of uncertainty. In case 1, the uncertainties in stack gas exit conditions are used to construct the stochastic space while in case 2, meteorological conditions are considered as the sources of uncertainty. Both cases develop 2D uncertain random space in which the uncertainty propagation is quantified in terms of plume rise and pollutant concentration distribution under slightly unstable atmospheric stability conditions. Starting with deterministic Gaussian plume model demonstration and its application, development of stochastic collocation model, convergence study, error analysis, and uncertainty quantification are presented in this paper.

show abstract

A Multilevel Stochastic Collocation Algorithm for Optimization of PDEs with Uncertain Coefficients

Cited by 34 publications

References 39 publications

Stochastic proximal gradient methods for nonconvex problems in Hilbert spaces

Stochastic proximal gradient methods for nonconvex problems in Hilbert spaces

Chance constrained optimization of elliptic PDE systems with a smoothing convex approximation

Dispersion modeling of thermal power plant emissions on stochastic space

Contact Info

Product

Resources

About