Controlled Heating - Revision history

FelixMueller at 14:11, 23 February 2016

2016-02-23T14:11:42Z

← Older revision		Revision as of 14:11, 23 February 2016
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	[[Category: MIOCP]]		[[Category: MIOCP]]
	[[Category:PDE model]]		[[Category:PDE model]]
			[[Category: Elliptic]]

FelixMueller at 17:34, 27 January 2016

2016-01-27T17:34:14Z

← Older revision		Revision as of 17:34, 27 January 2016
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	* [[:Category: Casadi \| Casadi code]] using FEniCS at [[Controlled Heating (FEniCS/Casadi)]]		* [[:Category: Casadi \| Casadi code]] using FEniCS at [[Controlled Heating (FEniCS/Casadi)]]

			[[Category: MIOCP]]
	[[Category:PDE model]]		[[Category:PDE model]]

MirkoHahn: /* Reference solution */

2016-01-12T21:56:59Z

Reference solution

← Older revision		Revision as of 21:56, 12 January 2016
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	= Reference solution =		= Reference solution =

	The reference solution was generated using finite element discretizations. The mesh was generated using FEniCS' mesh generation component with a resolution setting of <math>100</math>. The function space was constructed using first-degree Lagrangian elements. The optimization problem was solved using CasADi's IPOPT interface. The exact code used to solve the problem alongside solution data can be found under [[Controlled Heating (FEniCS/Casadi)]].		The reference solution was generated using finite element discretizations. The mesh was generated using FEniCS' mesh generation component with a resolution setting of <math>100</math>. The function space was constructed using first-degree Lagrangian elements. The optimization problem was solved using CasADi's IPOPT interface. The exact code used to solve the problem alongside solution data can be found under [[Controlled Heating (FEniCS/Casadi)]]. The optimal objective function value is <math>0.0145336</math>.

	<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">		<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">

MirkoHahn at 21:04, 12 January 2016

2016-01-12T21:04:23Z

← Older revision		Revision as of 21:04, 12 January 2016
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	Image:controlled_heating_error_color.png\| False color 2D plot of the error function.		Image:controlled_heating_error_color.png\| False color 2D plot of the error function.
	</gallery>		</gallery>

			= Source Code =

			Model descriptions are available in

			* [[:Category: Casadi \| Casadi code]] using FEniCS at [[Controlled Heating (FEniCS/Casadi)]]

	[[Category:PDE model]]		[[Category:PDE model]]

MirkoHahn: /* Reference solution */

2016-01-12T21:03:26Z

Reference solution

← Older revision		Revision as of 21:03, 12 January 2016
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	= Reference solution =		= Reference solution =

	''(~~There are currently no reference solutions for this problem~~)''		The reference solution was generated using finite element discretizations. The mesh was generated using FEniCS' mesh generation component with a resolution setting of <math>100</math>. The function space was constructed using first-degree Lagrangian elements. The optimization problem was solved using CasADi's IPOPT interface. The exact code used to solve the problem alongside solution data can be found under [[Controlled Heating (FEniCS/Casadi)]].

			<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
			Image:controlled_heating_solution_3d.png\| False color 3D surface plot of the solution.
			Image:controlled_heating_solution_color.png\| False color 2D plot of the solution.
			Image:controlled_heating_control_3d.png\| False color 3D surface plot of the control function.
			Image:controlled_heating_control_color.png\| False color 2D plot of the control function.
			Image:controlled_heating_error_3d.png\| False color 3D surface plot of the error function.
			Image:controlled_heating_error_color.png\| False color 2D plot of the error function.
			</gallery>

	[[Category:PDE model]]		[[Category:PDE model]]

MirkoHahn: /* Parameters */

2016-01-12T20:43:03Z

Parameters

← Older revision		Revision as of 20:43, 12 January 2016
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	:<math>		:<math>
	u_d(x) = \\|x\\|^2		u_d(x) = \\|x\\|^2
			</math>

			Additionally, let

			:<math>
			(C_1, C_2) := (-10, 10).
	</math>		</math>

MirkoHahn: /* Problem Statement */

2016-01-12T20:42:03Z

Problem Statement

← Older revision		Revision as of 20:42, 12 January 2016
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	</math>		</math>

	Here, <math>f</math> denotes the source term. Given a reference function <math>u_d</math>, the optimization problem is given by:		Here, <math>f</math> denotes the source term. We demand that <math>C_1 \leq f(x) \leq C_2</math> in <math>\Omega</math>. Given a reference function <math>u_d</math>, the optimization problem is given by:

	:<math>		:<math>
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	\text{s.t.} & \begin{array}[t]{rll}		\text{s.t.} & \begin{array}[t]{rll}
	- \Delta u(x) &= f(x) \qquad & \forall x \in \Omega \\[1.5ex]		- \Delta u(x) &= f(x) \qquad & \forall x \in \Omega \\[1.5ex]
	\frac{\partial u}{\partial \nu}(x) &= -u(x) \qquad & \forall x \in \partial \Omega		\frac{\partial u}{\partial \nu}(x) &= -u(x) \qquad & \forall x \in \partial \Omega \\
			f(x) &\in [C_1, C_2] \qquad & \forall x \in \Omega
	\end{array}		\end{array}
	\end{array}		\end{array}
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	\min\limits_{u,f} & \int\limits_\Omega (u - u_d)^2 \, \mathrm{d}x \\		\min\limits_{u,f} & \int\limits_\Omega (u - u_d)^2 \, \mathrm{d}x \\
	\text{s.t.} & \begin{array}[t]{rll}		\text{s.t.} & \begin{array}[t]{rll}
	\int\limits_\Omega \left\langle \nabla u, \nabla v \right\rangle \,\mathrm{d}x + \int\limits_{\partial \Omega} u \cdot v \,\mathrm{d}s(x) &= \int\limits_\Omega f \cdot v \,\mathrm{d}x \qquad &\forall v \in V		\int\limits_\Omega \left\langle \nabla u, \nabla v \right\rangle \,\mathrm{d}x + \int\limits_{\partial \Omega} u \cdot v \,\mathrm{d}s(x) &= \int\limits_\Omega f \cdot v \,\mathrm{d}x \qquad &\forall v \in V \\[1.5ex]
			f(x) &\in [C_1, C_2] \qquad & \forall x \in \Omega
	\end{array}		\end{array}
	\end{array}		\end{array}

MirkoHahn: Created page with "The controlled heating problem is a simple problem from the domain of continuous PDE constrained optimal control. The goal is to parametrize the wikipedia:Heat equation|heat..."

2016-01-11T19:11:43Z

Created page with "The controlled heating problem is a simple problem from the domain of continuous PDE constrained optimal control. The goal is to parametrize the wikipedia:Heat equation|heat..."

New page

The controlled heating problem is a simple problem from the domain of continuous PDE constrained optimal control. The goal is to parametrize the [[wikipedia:Heat equation|heat equation]] under Robin-type boundary conditions such that the stationary solution tracks a predefined reference function. This is equivalent to solving [[wikipedia:Poisson equation|Poisson's problem]] under Robin-type boundary equations. The problem is therefore closely related to the [[Source Inversion|source inversion problem]]. As opposed to the source inversion problem however, the controlled heating problem does not require the source term to be a linear combination of elementary source terms. The source term can be an arbitrary function which is usually chosen from the same function space as the temperature function.

= Problem Statement =

Let <math>\Omega \subset \mathbb{R}^n</math> be a contiguous domain and let <math>\nu \colon \partial \Omega \to \mathbb{R}^n</math> be an outer unit normal of <math>\Omega</math>. The Poisson problem with simple Robin-type boundary conditions then takes the following form:

:<math>
\begin{array}{rll}
- \Delta u &= f \qquad & \text{in } \Omega \\
\frac{\partial u}{\partial \nu} &= -u \qquad & \text{on } \partial \Omega
\end{array}
</math>

Here, <math>f</math> denotes the source term. Given a reference function <math>u_d</math>, the optimization problem is given by:

:<math>
\begin{array}[t]{rl}
\min\limits_{u,f} & \int\limits_\Omega (u - u_d)^2 \, \mathrm{d}x \\
\text{s.t.} & \begin{array}[t]{rll}
- \Delta u(x) &= f(x) \qquad & \forall x \in \Omega \\[1.5ex]
\frac{\partial u}{\partial \nu}(x) &= -u(x) \qquad & \forall x \in \partial \Omega
\end{array}
\end{array}
</math>

== Weak formulation ==

Some PDE discretization techniques (such as [[wikipedia:Finite Element Method|finite element methods]]) require the use of [[wikipedia:Weak solution|weak formulations]] of the original problem. The weak formulation of the Poisson problem with Robin-type boundary conditions as described above is obtained using [[wikipedia:Green's identities|Green's identities]]:

:<math>
\int\limits_\Omega \left\langle \nabla u, \nabla v \right\rangle \,\mathrm{d}x + \int\limits_{\partial \Omega} u \cdot v \,\mathrm{d}s(x) = \int\limits_\Omega f \cdot v \,\mathrm{d}x \qquad \forall v \in V
</math>

Here, <math>V</math> is a suitable space of test functions. The optimization problem then takes the following form:

:<math>
\begin{array}[t]{rl}
\min\limits_{u,f} & \int\limits_\Omega (u - u_d)^2 \, \mathrm{d}x \\
\text{s.t.} & \begin{array}[t]{rll}
\int\limits_\Omega \left\langle \nabla u, \nabla v \right\rangle \,\mathrm{d}x + \int\limits_{\partial \Omega} u \cdot v \,\mathrm{d}s(x) &= \int\limits_\Omega f \cdot v \,\mathrm{d}x \qquad &\forall v \in V
\end{array}
\end{array}
</math>

= Parameters =

For testing purposes, we consider <math>\Omega = B(0, 1) = \left\lbrace x \in \mathbb{R}^n \colon \|x\| < 1 \right\rbrace</math> with <math>n = 2</math>. The reference function is given by:

:<math>
u_d(x) = \|x\|^2
</math>

= Reference solution =

''(There are currently no reference solutions for this problem)''

[[Category:PDE model]]