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Volume 27, Issue 3
A Distributed Optimal Control Problem with Averaged Stochastic Gradient Descent

Qi Sun & Qiang Du

DOI: 10.4208/cicp.OA-2018-0295

Commun. Comput. Phys., 27 (2020), pp. 753-774.

Published online: 2020-02

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  • Abstract

In this work, we study a distributed optimal control problem, in which the governing system is given by second-order elliptic equations with log-normal coefficients. To lessen the curse of dimensionality that originates from the representation of stochastic coefficients, the Monte Carlo finite element method is adopted for numerical discretization where a large number of sampled constraints are involved. For the solution of such a large-scale optimization problem, stochastic gradient descent method is widely used but has slow convergence asymptotically due to its inherent variance. To remedy this problem, we adopt an averaged stochastic gradient descent method which performs stably even with the use of relatively large step sizes and small batch sizes. Numerical experiments are carried out to validate our theoretical findings.

  • Keywords

PDE-constrained elliptic control, high-dimensional random inputs, Monte Carlo finite element, stochastic gradient descent.

  • AMS Subject Headings

35R60, 78M31, 49J20

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

sunqi@csrc.ac.cn (Qi Sun)

qd2125@columbia.edu (Qiang Du)

  • BibTex
  • RIS
  • TXT
@Article{CiCP-27-753, author = {Sun , Qi and Du , Qiang}, title = {A Distributed Optimal Control Problem with Averaged Stochastic Gradient Descent}, journal = {Communications in Computational Physics}, year = {2020}, volume = {27}, number = {3}, pages = {753--774}, abstract = {

In this work, we study a distributed optimal control problem, in which the governing system is given by second-order elliptic equations with log-normal coefficients. To lessen the curse of dimensionality that originates from the representation of stochastic coefficients, the Monte Carlo finite element method is adopted for numerical discretization where a large number of sampled constraints are involved. For the solution of such a large-scale optimization problem, stochastic gradient descent method is widely used but has slow convergence asymptotically due to its inherent variance. To remedy this problem, we adopt an averaged stochastic gradient descent method which performs stably even with the use of relatively large step sizes and small batch sizes. Numerical experiments are carried out to validate our theoretical findings.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2018-0295}, url = {http://global-sci.org/intro/article_detail/cicp/13928.html} }
TY - JOUR T1 - A Distributed Optimal Control Problem with Averaged Stochastic Gradient Descent AU - Sun , Qi AU - Du , Qiang JO - Communications in Computational Physics VL - 3 SP - 753 EP - 774 PY - 2020 DA - 2020/02 SN - 27 DO - http://doi.org/10.4208/cicp.OA-2018-0295 UR - https://global-sci.org/intro/article_detail/cicp/13928.html KW - PDE-constrained elliptic control, high-dimensional random inputs, Monte Carlo finite element, stochastic gradient descent. AB -

In this work, we study a distributed optimal control problem, in which the governing system is given by second-order elliptic equations with log-normal coefficients. To lessen the curse of dimensionality that originates from the representation of stochastic coefficients, the Monte Carlo finite element method is adopted for numerical discretization where a large number of sampled constraints are involved. For the solution of such a large-scale optimization problem, stochastic gradient descent method is widely used but has slow convergence asymptotically due to its inherent variance. To remedy this problem, we adopt an averaged stochastic gradient descent method which performs stably even with the use of relatively large step sizes and small batch sizes. Numerical experiments are carried out to validate our theoretical findings.

Sun , Qi and Du , Qiang. (2020). A Distributed Optimal Control Problem with Averaged Stochastic Gradient Descent. Communications in Computational Physics. 27 (3). 753-774. doi:10.4208/cicp.OA-2018-0295
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