A linear implicit Euler method for the finite element discretization of a controlled stochastic heat equation
Benner, Peter ; Stillfjord, Tony LU
and Trautwein, Christoph
(2022)
In IMA Journal of Numerical Analysis
42(3).
p.2118-2150
- Abstract
We consider a numerical approximation of a linear quadratic control problem constrained by the stochastic heat equation with nonhomogeneous Neumann boundary conditions. This involves a combination of distributed and boundary control, as well as both distributed and boundary noise. We apply the finite element method for the spatial discretization and the linear implicit Euler method for the temporal discretization. Due to the low regularity induced by the boundary noise, convergence orders above 1/2 in space and 1/4 in time cannot be expected. We prove such optimal convergence orders for our full discretization when the distributed noise and the initial condition are sufficiently smooth. Under less smooth conditions the convergence order... (More)
We consider a numerical approximation of a linear quadratic control problem constrained by the stochastic heat equation with nonhomogeneous Neumann boundary conditions. This involves a combination of distributed and boundary control, as well as both distributed and boundary noise. We apply the finite element method for the spatial discretization and the linear implicit Euler method for the temporal discretization. Due to the low regularity induced by the boundary noise, convergence orders above 1/2 in space and 1/4 in time cannot be expected. We prove such optimal convergence orders for our full discretization when the distributed noise and the initial condition are sufficiently smooth. Under less smooth conditions the convergence order is further decreased. Our results only assume that the related (deterministic) differential Riccati equation can be approximated with a certain convergence order, which is easy to achieve in practice. We confirm these theoretical results through a numerical experiment in a two-dimensional domain.
(Less)
- author
- Benner, Peter
; Stillfjord, Tony
LU
and Trautwein, Christoph
- organization
- publishing date
- 2022
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Convergence analysis, Full discretization, Heat conduction, Optimal control, Stochastic partial differential equation
- in
- IMA Journal of Numerical Analysis
- volume
- 42
- issue
- 3
- pages
- 33 pages
- publisher
- Oxford University Press
- external identifiers
-
- scopus:85135636761
- ISSN
- 0272-4979
- DOI
- 10.1093/imanum/drab033
- language
- English
- LU publication?
- yes
- id
- 95b9e23b-4f86-4ea6-b36f-c5922534f36c
- date added to LUP
- 2022-09-12 12:17:55
- date last changed
- 2022-09-12 12:17:55
@article{95b9e23b-4f86-4ea6-b36f-c5922534f36c,
abstract = {{<p>We consider a numerical approximation of a linear quadratic control problem constrained by the stochastic heat equation with nonhomogeneous Neumann boundary conditions. This involves a combination of distributed and boundary control, as well as both distributed and boundary noise. We apply the finite element method for the spatial discretization and the linear implicit Euler method for the temporal discretization. Due to the low regularity induced by the boundary noise, convergence orders above 1/2 in space and 1/4 in time cannot be expected. We prove such optimal convergence orders for our full discretization when the distributed noise and the initial condition are sufficiently smooth. Under less smooth conditions the convergence order is further decreased. Our results only assume that the related (deterministic) differential Riccati equation can be approximated with a certain convergence order, which is easy to achieve in practice. We confirm these theoretical results through a numerical experiment in a two-dimensional domain. </p>}},
author = {{Benner, Peter and Stillfjord, Tony and Trautwein, Christoph}},
issn = {{0272-4979}},
keywords = {{Convergence analysis; Full discretization; Heat conduction; Optimal control; Stochastic partial differential equation}},
language = {{eng}},
number = {{3}},
pages = {{2118--2150}},
publisher = {{Oxford University Press}},
series = {{IMA Journal of Numerical Analysis}},
title = {{A linear implicit Euler method for the finite element discretization of a controlled stochastic heat equation}},
url = {{http://dx.doi.org/10.1093/imanum/drab033}},
doi = {{10.1093/imanum/drab033}},
volume = {{42}},
year = {{2022}},
}