Adaptive discontinuous Galerkin approximation of optimal control problems governed by transient convection-diffusion equations

Yücel, Hamdullah; Stoll, Martin; Benner, Peter

doi:10.1553/etna_vol48s407

Cited by 2 publications

(6 citation statements)

References 29 publications

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“…Many widely-used preconditioned iterative methods for the solution of time-dependent PDE-constrained optimization problems of type ( 1)-( 2) involve a backward Euler discretization for the time variable. [5][6][7][8][9][10] Applying this scheme gives the following system of equations 1 :…”

Section: Backward Euler Discretizationmentioning

confidence: 99%

“…Many widely‐used preconditioned iterative methods for the solution of time‐dependent PDE‐constrained optimization problems of type (1)–(2) involve a backward Euler discretization for the time variable 5‐10 . Applying this scheme gives the following system of equations:

\{\begin{aligned} τ M {bold-italicy}^{n} + {false(L^{E} false)}^{⊤} {bold-italicp}^{n} prefix- M {bold-italicp}^{n + 1} = τ M {\overset{bold-italicy}{true}}^{n}, & n = 0, \dots, n_{t} prefix- 1, \\ {bold-italicp}^{n_{t}} = bold0, \\ {bold-italicy}^{0} = {bold-italicy}_{0}, \\ prefix- M {bold-italicy}^{n} + L^{E} {bold-italicy}^{n + 1} prefix- \frac{τ}{β} M {bold-italicp}^{n + 1} = τ {bold-italicf}^{n + 1}, & n = 0, \dots, n_{t} prefix- 1, \end{aligned}

where

L^{E} = τ L + M

y_{0}

is the discretization of the initial condition for y , and

f^{n} = {f_{i}^{n}}_{i = 1}^{n_{x}}, f_{i}^{n} =

…”

Section: First‐order Optimality Conditions and Discretizations In Timementioning

confidence: 99%

“…with Mp n t = 0 and My 0 = My 0 an appropriate discretization of the final and initial conditions on p and y, and f n defined as in (10). In matrix form, we write…”

Section: Crank-nicolson Discretization and Symmetrization Of The Systemmentioning

confidence: 99%

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Fast iterative solver for the optimal control of time‐dependent PDEs with Crank–Nicolson discretization in time

Leveque

Pearson

2021

Numerical Linear Algebra App

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In this article, we derive a new, fast, and robust preconditioned iterative solution strategy for the all‐at‐once solution of optimal control problems with time‐dependent PDEs as constraints, including the heat equation and the non‐steady convection–diffusion equation. After applying an optimize‐then‐discretize approach, one is faced with continuous first‐order optimality conditions consisting of a coupled system of PDEs. As opposed to most work in preconditioning the resulting discretized systems, where a (first‐order accurate) backward Euler method is used for the discretization of the time derivative, we employ a (second‐order accurate) Crank–Nicolson method in time. We apply a carefully tailored invertible transformation for symmetrizing the matrix, and then derive an optimal preconditioner for the saddle‐point system obtained. The key components of this preconditioner are an accurate mass matrix approximation, a good approximation of the Schur complement, and an appropriate multigrid process to apply this latter approximation—these are constructed using our work in transforming the matrix system. We prove the optimality of the approximation of the Schur complement through bounds on the eigenvalues, and test our solver against a widely‐used preconditioner for the linear system arising from a backward Euler discretization. These demonstrate the effectiveness and robustness of our solver with respect to mesh‐sizes, regularization parameter, and diffusion coefficient.

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Section: Backward Euler Discretizationmentioning

confidence: 99%

\{\begin{aligned} τ M {bold-italicy}^{n} + {false(L^{E} false)}^{⊤} {bold-italicp}^{n} prefix- M {bold-italicp}^{n + 1} = τ M {\overset{bold-italicy}{true}}^{n}, & n = 0, \dots, n_{t} prefix- 1, \\ {bold-italicp}^{n_{t}} = bold0, \\ {bold-italicy}^{0} = {bold-italicy}_{0}, \\ prefix- M {bold-italicy}^{n} + L^{E} {bold-italicy}^{n + 1} prefix- \frac{τ}{β} M {bold-italicp}^{n + 1} = τ {bold-italicf}^{n + 1}, & n = 0, \dots, n_{t} prefix- 1, \end{aligned}

where

L^{E} = τ L + M

y_{0}

is the discretization of the initial condition for y , and

f^{n} = {f_{i}^{n}}_{i = 1}^{n_{x}}, f_{i}^{n} =

…”

Section: First‐order Optimality Conditions and Discretizations In Timementioning

confidence: 99%

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Fast iterative solver for the optimal control of time‐dependent PDEs with Crank–Nicolson discretization in time

Leveque

Pearson

2021

Numerical Linear Algebra App

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“…Many widely-used preconditioned iterative methods for the solution of time-dependent PDE-constrained optimization problems of type (2.1)-(2.2) involve a backward Euler discretization for the time variable [9,32,33,42,44,47]. Applying this scheme gives the following system of equations 1 :…”

Section: Backward Euler Discretizationmentioning

confidence: 99%

“…When preconditioners are sought for certain time-dependent problems, it is typical to apply a (first-order accurate) backward Euler method in time, as this leads to particularly convenient structures within the matrix and facilitates effective preconditioning; see [33] for a mesh-and β-robust preconditioner for the heat control problem, and [9,32,42,44,47] for applications to different problems. However, the required discretization strategy results in slower convergence in time than space: if a method is second-order accurate in space, it is reasonable to choose τ = O(h 2 ), where τ is the time step and h is the mesh-size in space.…”

mentioning

confidence: 99%

Fast Iterative Solver for the Optimal Control of Time-Dependent PDEs with Crank-Nicolson Discretization in Time

Leveque¹,

Pearson²

2020

Preprint

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In this article, we derive a new, fast, and robust preconditioned iterative solution strategy for the all-at-once solution of optimal control problems with time-dependent PDEs as constraints, including the heat equation and the non-steady convection-diffusion equation. After applying an optimize-then-discretize approach, one is faced with continuous first-order optimality conditions consisting of a coupled system of PDEs. As opposed to most work in preconditioning the resulting discretized systems, where a (first-order accurate) backward Euler method is used for the discretization of the time derivative, we employ a (second-order accurate) Crank-Nicolson method in time. We apply a carefully tailored invertible transformation for symmetrizing the matrix, and then derive an optimal preconditioner for the saddle-point system obtained. The key components of this preconditioner are an accurate mass matrix approximation, a good approximation of the Schur complement, and an appropriate multigrid process to apply this latter approximation-these are constructed using our work in transforming the matrix system. We prove the optimality of the approximation of the Schur complement through bounds on the eigenvalues, and test our solver against a widely-used preconditioner for the linear system arising from a backward Euler discretization. These demonstrate the effectiveness and robustness of our solver with respect to mesh-sizes, regularization parameter, and diffusion coefficient.

show abstract

Adaptive discontinuous Galerkin approximation of optimal control problems governed by transient convection-diffusion equations

Cited by 2 publications

References 29 publications

Fast iterative solver for the optimal control of time‐dependent PDEs with Crank–Nicolson discretization in time

Fast iterative solver for the optimal control of time‐dependent PDEs with Crank–Nicolson discretization in time

Fast Iterative Solver for the Optimal Control of Time-Dependent PDEs with Crank-Nicolson Discretization in Time

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