# Optimal Control of PDEs under Uncertainty

*An introduction with application to optimal shape design of structures*

*An introduction with application to optimal shape design of structures*

**Authors**: **Jesús Martínez-Frutos**** and ****Francisco Periago**

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**About this book:**

This book gives a direct and **comprehensive introduction** to the theoretical and numerical concepts in the emergent field of optimal control of partial differential equations (PDEs) under uncertainty. The main objective of the book is to provide**graduate students and researchers** with a smooth transition from optimal control of deterministic PDEs to optimal control of random PDEs. Coverage includes uncertainty modelling in control problems, variational formulation of PDEs with random inputs, robust and risk-averse formulations of optimal control problems, existence theory and numerical resolution methods. The exposition is focused on running the hole path starting from uncertainty modelling and ending in the practical implementation of numerical schemes for the numerical approximation of the considered problems. To this end, a selected number of illustrative examples are analysed in detail along the book. Computer codes, written in **MatLab**, for all these examples are provided.

**Table of Contents:**

**Table of Contents:**

- 1
**Introduction**. - 2Mathematical
**Preliminaries**. - 3
**Mathematical Analysis**of Optimal Control Problems Under Uncertainty. - 4Numerical Resolution of
**Robust Optimal Control**Problems. - 5Numerical Resolution of
**Risk Averse Optimal Control**Problems. - 6
**Structural Optimization**Under Uncertainty. - 7
**Miscellaneous**Topics and Open Problems.

**Multimedia material (online lectures)**

**Related works:**

Marín, Francisco J; Martínez-Frutos, Jesús; Periago, Francisco A polynomial chaos-based approach to risk-averse piezoelectric control of random vibrations of beams Journal Article International Journal for Numerical Methods in Engineering, 115 (6), pp. 738-755, 2018, ISSN: 1097-0207. @article{Marín2018b, title = {A polynomial chaos-based approach to risk-averse piezoelectric control of random vibrations of beams}, author = {Francisco J. Marín and Jesús Martínez-Frutos and Francisco Periago}, url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.5823 https://www.upct.es/mc3/files/FPE/NME-Jan-18-0063-R1.pdf }, issn = {1097-0207}, year = {2018}, date = {2018-08-10}, journal = {International Journal for Numerical Methods in Engineering}, volume = {115}, number = {6}, pages = {738-755}, abstract = {This paper proposes a risk-averse formulation for the problem of piezoelectric control of random vibrations of elastic structures. The proposed formulation, inspired by the notion of risk aversion in Economy, is applied to the piezoelectric control of a Bernoulli-Euler beam subjected to uncertainties in its input data. To address the high computational burden associated to the presence of random fields in the model and the discontinuities involved in the cost functional and its gradient, a combination of a non-intrusive anisotropic polynomial chaos approach for uncertainty propagation with a Monte Carlo sampling method is proposed. In a first part, the well-posedness of the control problem is established by proving the existence of optimal controls. In a second part, an adaptive gradient-based method is proposed for the numerical resolution of the problem. Several experiments illustrate the performance of the proposed approach and the significant differences that may occur between the classical deterministic formulation of the problem and its stochastic risk-averse counterpart.}, keywords = {}, pubstate = {published}, tppubtype = {article} } This paper proposes a risk-averse formulation for the problem of piezoelectric control of random vibrations of elastic structures. The proposed formulation, inspired by the notion of risk aversion in Economy, is applied to the piezoelectric control of a Bernoulli-Euler beam subjected to uncertainties in its input data. To address the high computational burden associated to the presence of random fields in the model and the discontinuities involved in the cost functional and its gradient, a combination of a non-intrusive anisotropic polynomial chaos approach for uncertainty propagation with a Monte Carlo sampling method is proposed. In a first part, the well-posedness of the control problem is established by proving the existence of optimal controls. In a second part, an adaptive gradient-based method is proposed for the numerical resolution of the problem. Several experiments illustrate the performance of the proposed approach and the significant differences that may occur between the classical deterministic formulation of the problem and its stochastic risk-averse counterpart. | |

Morales, Ociel; Periago, Francisco; Vallejo, José A Robust Optimal Design of Quantum Electronic Devices Journal Article Open Access Mathematical Problems in Engineering, 2018 , pp. 10, 2018, ISSN: 2331-8422. @article{Morales2017, title = {Robust Optimal Design of Quantum Electronic Devices}, author = {Ociel Morales and Francisco Periago and José A. Vallejo}, url = {https://www.hindawi.com/journals/mpe/2018/3095257/abs/ https://www.upct.es/mc3/files/FPE/3095257.pdf}, issn = {2331-8422}, year = {2018}, date = {2018-04-05}, journal = {Mathematical Problems in Engineering}, volume = {2018}, pages = {10}, abstract = {We consider the optimal design of a sequence of quantum barriers in order to manufacture an electronic device at the nanoscale such that the dependence of its transmission coefficient on the bias voltage is linear. The technique presented here is easily adaptable to other response characteristics. The transmission coefficient is computed using the Wentzel-Kramers-Brillouin (WKB) method, so we can explicitly compute the gradient of the objective function. In contrast with earlier treatments, manufacturing uncertainties are incorporated in the model through random variables and the optimal design problem is formulated in a probabilistic setting. As a measure of robustness, a weighted sum of the expectation and the variance of a least-squares performance metric is considered. Several simulations illustrate the proposed approach.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We consider the optimal design of a sequence of quantum barriers in order to manufacture an electronic device at the nanoscale such that the dependence of its transmission coefficient on the bias voltage is linear. The technique presented here is easily adaptable to other response characteristics. The transmission coefficient is computed using the Wentzel-Kramers-Brillouin (WKB) method, so we can explicitly compute the gradient of the objective function. In contrast with earlier treatments, manufacturing uncertainties are incorporated in the model through random variables and the optimal design problem is formulated in a probabilistic setting. As a measure of robustness, a weighted sum of the expectation and the variance of a least-squares performance metric is considered. Several simulations illustrate the proposed approach. | |

Martínez-Frutos, Jesús; Herrero-Pérez, David; Kessler, Mathieu; Periago, Francisco Risk-averse structural topology optimization under random fields using stochastic expansion methods Journal Article Computer Methods in Applied Mechanics and Engineering, 330 , pp. 180-206, 2018, ISSN: 0045-7825. @article{JMF_CMAME_2017, title = {Risk-averse structural topology optimization under random fields using stochastic expansion methods}, author = {Jesús Martínez-Frutos and David Herrero-Pérez and Mathieu Kessler and Francisco Periago}, url = {https://www.sciencedirect.com/science/article/pii/S0045782517306990 https://www.upct.es/mc3/files/JMF/riskaverse_cmame.pdf}, issn = {0045-7825}, year = {2018}, date = {2018-03-01}, journal = {Computer Methods in Applied Mechanics and Engineering}, volume = {330}, pages = {180-206}, abstract = {This work proposes a level-set based approach for solving risk-averse structural topology optimization problems considering random field loading and material uncertainty. The use of random fields increases the dimensionality of the stochastic domain, which poses several computational challenges related to the minimization of the Excess Probability as a measure of risk awareness. This problem is addressed both from the theoretical and numerical viewpoints. First, an existence result under a typical geometrical constraint on the set of admissible shapes is proved. Second, a level-set continuous approach to find the numerical solution of the problem is proposed. Since the considered cost functional has a discontinuous integrand, the numerical approximation of the functional and its sensitivity combine an adaptive anisotropic Polynomial Chaos (PC) approach with a Monte-Carlo (MC) sampling method for uncertainty propagation. Furthermore, to address the increment of dimensionality induced by the random field, an anisotropic sparse grid stochastic collocation method is used for the efficient computation of the PC coefficients. A key point is that the non-intrusive nature of such an approach facilitates the use of High Performance Computing (HPC) to alleviate the computational burden of the problem. Several numerical experiments including random field loading and material uncertainty are presented to show the feasibility of the proposal.}, keywords = {}, pubstate = {published}, tppubtype = {article} } This work proposes a level-set based approach for solving risk-averse structural topology optimization problems considering random field loading and material uncertainty. The use of random fields increases the dimensionality of the stochastic domain, which poses several computational challenges related to the minimization of the Excess Probability as a measure of risk awareness. This problem is addressed both from the theoretical and numerical viewpoints. First, an existence result under a typical geometrical constraint on the set of admissible shapes is proved. Second, a level-set continuous approach to find the numerical solution of the problem is proposed. Since the considered cost functional has a discontinuous integrand, the numerical approximation of the functional and its sensitivity combine an adaptive anisotropic Polynomial Chaos (PC) approach with a Monte-Carlo (MC) sampling method for uncertainty propagation. Furthermore, to address the increment of dimensionality induced by the random field, an anisotropic sparse grid stochastic collocation method is used for the efficient computation of the PC coefficients. A key point is that the non-intrusive nature of such an approach facilitates the use of High Performance Computing (HPC) to alleviate the computational burden of the problem. Several numerical experiments including random field loading and material uncertainty are presented to show the feasibility of the proposal. | |

Marín, Francisco J; Martínez-Frutos, Jesús; Periago, Francisco Robust Averaged Control of Vibrations for the Bernoulli-Euler Beam Equation Journal Article Journal of Optimization Theory and Applications, 174 (2), pp. 428–454, 2017, ISSN: 1573-2878. Abstract | BibTeX | Altmetric | Links: @article{Marín2017, title = {Robust Averaged Control of Vibrations for the Bernoulli-Euler Beam Equation}, author = {Francisco J. Marín and Jesús Martínez-Frutos and Francisco Periago}, url = {https://doi.org/10.1007/s10957-017-1128-x https://www.upct.es/mc3/files/FPE/marin_17.pdf}, doi = {10.1007/s10957-017-1128-x}, issn = {1573-2878}, year = {2017}, date = {2017-08-01}, journal = {Journal of Optimization Theory and Applications}, volume = {174}, number = {2}, pages = {428--454}, abstract = {This paper proposes an approach for the robust averaged control of random vibrations for the Bernoulli--Euler beam equation under uncertainty in the flexural stiffness and in the initial conditions. The problem is formulated in the framework of optimal control theory and provides a functional setting, which is so general as to include different types of random variables and second-order random fields as sources of uncertainty. The second-order statistical moment of the random system response at the control time is incorporated in the cost functional as a measure of robustness. The numerical resolution method combines a classical descent method with an adaptive anisotropic stochastic collocation method for the numerical approximation of the statistics of interest. The direct and adjoint stochastic systems are uncoupled, which permits to exploit parallel computing architectures to solve the set of deterministic problem that arise from the stochastic collocation method. As a result, problems with a relative large number of random variables can be solved with a reasonable computational cost. Two numerical experiments illustrate both the performance of the proposed method and the significant differences that may occur when uncertainty is incorporated in this type of control problems.}, keywords = {}, pubstate = {published}, tppubtype = {article} } This paper proposes an approach for the robust averaged control of random vibrations for the Bernoulli--Euler beam equation under uncertainty in the flexural stiffness and in the initial conditions. The problem is formulated in the framework of optimal control theory and provides a functional setting, which is so general as to include different types of random variables and second-order random fields as sources of uncertainty. The second-order statistical moment of the random system response at the control time is incorporated in the cost functional as a measure of robustness. The numerical resolution method combines a classical descent method with an adaptive anisotropic stochastic collocation method for the numerical approximation of the statistics of interest. The direct and adjoint stochastic systems are uncoupled, which permits to exploit parallel computing architectures to solve the set of deterministic problem that arise from the stochastic collocation method. As a result, problems with a relative large number of random variables can be solved with a reasonable computational cost. Two numerical experiments illustrate both the performance of the proposed method and the significant differences that may occur when uncertainty is incorporated in this type of control problems. | |

Martínez-Frutos, Jesús; Kessler, Mathieu; Münch, Arnaud; Periago, Francisco Robust optimal Robin boundary control for the transient heat equation with random input data Journal Article International Journal for Numerical Methods in Engineering, 108 (2), pp. 116–135, 2016, ISSN: 1097-0207. Abstract | BibTeX | Altmetric | Links: @article{NME:NME5210, title = {Robust optimal Robin boundary control for the transient heat equation with random input data}, author = {Jesús Martínez-Frutos and Mathieu Kessler and Arnaud Münch and Francisco Periago}, url = {http://dx.doi.org/10.1002/nme.5210 https://www.upct.es/mc3/files/FPE/ijnme_16.pdf}, doi = {10.1002/nme.5210}, issn = {1097-0207}, year = {2016}, date = {2016-03-02}, journal = {International Journal for Numerical Methods in Engineering}, volume = {108}, number = {2}, pages = {116--135}, abstract = {The problem of robust optimal Robin boundary control for a parabolic partial differential equation with uncertain input data is considered. As a measure of robustness, the variance of the random system response is included in two different cost functionals. Uncertainties in both the underlying state equation and the control variable are quantified through random fields. The paper is mainly concerned with the numerical resolution of the problem. To this end, a gradient-based method is proposed considering different functional costs to achieve the robustness of the system. An adaptive anisotropic sparse grid stochastic collocation method is used for the numerical resolution of the associated state and adjoint state equations. The different functional costs are analysed in terms of computational efficiency and its capability to provide robust solutions. Two numerical experiments illustrate the performance of the algorithm. Copyright © 2016 John Wiley & Sons, Ltd.}, keywords = {}, pubstate = {published}, tppubtype = {article} } The problem of robust optimal Robin boundary control for a parabolic partial differential equation with uncertain input data is considered. As a measure of robustness, the variance of the random system response is included in two different cost functionals. Uncertainties in both the underlying state equation and the control variable are quantified through random fields. The paper is mainly concerned with the numerical resolution of the problem. To this end, a gradient-based method is proposed considering different functional costs to achieve the robustness of the system. An adaptive anisotropic sparse grid stochastic collocation method is used for the numerical resolution of the associated state and adjoint state equations. The different functional costs are analysed in terms of computational efficiency and its capability to provide robust solutions. Two numerical experiments illustrate the performance of the algorithm. Copyright © 2016 John Wiley & Sons, Ltd. | |

Martínez-Frutos, Jesús; Herrero-Pérez, David; Kessler, Mathieu; Periago, Francisco Robust shape optimization of continuous structures via the level set method Journal Article Computer Methods in Applied Mechanics and Engineering, 305 (Supplement C), pp. 271 - 291, 2016, ISSN: 0045-7825. Abstract | BibTeX | Altmetric | Links: @article{MARTINEZFRUTOS2016271, title = {Robust shape optimization of continuous structures via the level set method}, author = {Jesús Martínez-Frutos and David Herrero-Pérez and Mathieu Kessler and Francisco Periago}, url = {http://www.sciencedirect.com/science/article/pii/S0045782516300834 https://www.upct.es/mc3/files/FPE/cmame16.pdf}, doi = {https://doi.org/10.1016/j.cma.2016.03.003}, issn = {0045-7825}, year = {2016}, date = {2016-01-15}, journal = {Computer Methods in Applied Mechanics and Engineering}, volume = {305}, number = {Supplement C}, pages = {271 - 291}, abstract = {Abstract This work proposes a stochastic shape optimization method for continuous structures using the level-set method. Such a method aims to minimize the expected compliance and its variance as measures of the structural robustness. The behavior of continuous structures is modeled by linear elasticity equations with uncertain loading and material. This uncertainty can be modeled using random variables with different probability distributions as well as random fields. The proper problem formulation is ensured by the proof of the existence colorrev of solution under certain geometrical constraints on the set of admissible shapes. The proposed method addresses the stochastic linear elasticity problem in its weak form obtaining the explicit expressions for the continuous shape derivatives. Some numerical examples are presented to show the effectiveness of the proposed approach.}, keywords = {}, pubstate = {published}, tppubtype = {article} } Abstract This work proposes a stochastic shape optimization method for continuous structures using the level-set method. Such a method aims to minimize the expected compliance and its variance as measures of the structural robustness. The behavior of continuous structures is modeled by linear elasticity equations with uncertain loading and material. This uncertainty can be modeled using random variables with different probability distributions as well as random fields. The proper problem formulation is ensured by the proof of the existence colorrev of solution under certain geometrical constraints on the set of admissible shapes. The proposed method addresses the stochastic linear elasticity problem in its weak form obtaining the explicit expressions for the continuous shape derivatives. Some numerical examples are presented to show the effectiveness of the proposed approach. | |

Martínez-Frutos, Jesús; Kessler, Mathieu; Periago, Francisco Robust optimal shape design for an elliptic PDE with uncertainty in its input data Journal Article ESAIM: COCV, 21 (4), pp. 901-923, 2015. Abstract | BibTeX | Altmetric | Links: @article{refId0, title = {Robust optimal shape design for an elliptic PDE with uncertainty in its input data}, author = {Jesús Martínez-Frutos and Mathieu Kessler and Francisco Periago}, url = {https://doi.org/10.1051/cocv/2014049 https://www.upct.es/mc3/files/FPE/ESAIM_COCV_vfinal.pdf}, doi = {10.1051/cocv/2014049}, year = {2015}, date = {2015-01-01}, journal = {ESAIM: COCV}, volume = {21}, number = {4}, pages = {901-923}, abstract = {We consider a shape optimization problem for an elliptic partial differential equation with uncertainty in its input data. The design variable enters the lower-order term of the state equation and is modeled through the characteristic function of a measurable subset of the spatial domain. As usual, a measure constraint is imposed on the design variable. In order to compute a robust optimal shape, the objective function involves a weighted sum of both the mean and the variance of the compliance. Since the optimization problem is not convex, a full relaxation of it is first obtained. The relaxed problem is then solved numerically by using a gradient-based optimization algorithm. To this end, the adjoint method is used to compute the continuous gradient of the cost function. Since the variance enters the cost function, the underlying adjoint equation is non-local in the probabilistic space. Both the direct and adjoint equations are solved numerically by using a sparse grid stochastic collocation method. Three numerical experiments in 2D illustrate the theoretical results and show the computational issues which arise when uncertainty is quantified through random fields.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We consider a shape optimization problem for an elliptic partial differential equation with uncertainty in its input data. The design variable enters the lower-order term of the state equation and is modeled through the characteristic function of a measurable subset of the spatial domain. As usual, a measure constraint is imposed on the design variable. In order to compute a robust optimal shape, the objective function involves a weighted sum of both the mean and the variance of the compliance. Since the optimization problem is not convex, a full relaxation of it is first obtained. The relaxed problem is then solved numerically by using a gradient-based optimization algorithm. To this end, the adjoint method is used to compute the continuous gradient of the cost function. Since the variance enters the cost function, the underlying adjoint equation is non-local in the probabilistic space. Both the direct and adjoint equations are solved numerically by using a sparse grid stochastic collocation method. Three numerical experiments in 2D illustrate the theoretical results and show the computational issues which arise when uncertainty is quantified through random fields. |