FitzHugh-Nagumo System - Revision history

Schuetz2 at 05:28, 17 June 2025

2025-06-17T05:28:45Z

← Older revision		Revision as of 05:28, 17 June 2025
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	title = {FitzHugh-Nagumo System},		title = {FitzHugh-Nagumo System},
	howpublished = {{MORwiki} -- Model Order Reduction Wiki},		howpublished = {{MORwiki} -- Model Order Reduction Wiki},
	url = <nowiki>{~~http~~://modelreduction.org/~~index.php~~/FitzHugh-Nagumo_System}</nowiki>,		url = <nowiki>{https://modelreduction.org/morwiki/FitzHugh-Nagumo_System}</nowiki>,
	year = 2018		year = 2018
	}		}

Himpe: Fix CrossRefs

2021-08-27T07:35:02Z

Fix CrossRefs

← Older revision		Revision as of 07:35, 27 August 2021
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	where <math>\epsilon=0.015,\;h=0.5,\;\gamma=2,\;g=0.05;\;i_0(t)=5\cdot		where <math>\epsilon=0.015,\;h=0.5,\;\gamma=2,\;g=0.05;\;i_0(t)=5\cdot
	10^4t^3 \exp(-15t)</math> is the external stimulus, and the variables <math>v</math> and <math>w</math> are the voltage and the recovery of the voltage, respectively. ~~<xr id="fig:fhn"/>~~ shows the typical limit cycle behaviour described above.		10^4t^3 \exp(-15t)</math> is the external stimulus, and the variables <math>v</math> and <math>w</math> are the voltage and the recovery of the voltage, respectively. Fig. 1 shows the typical limit cycle behaviour described above.

	==Reformulation as a quadratic-bilinear system==		==Reformulation as a quadratic-bilinear system==

Saak: /* Dimensions */

2020-03-12T09:32:21Z

Dimensions

← Older revision		Revision as of 09:32, 12 March 2020
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	System dimensions:		System dimensions:

	<math>E \in \mathbb{R}^{~~1536~~ \times ~~1536~~}</math>,		<math>E \in \mathbb{R}^{1\,536 \times 1\,536}</math>,
	<math>A \in \mathbb{R}^{~~1536~~ \times ~~1536~~}</math>,		<math>A \in \mathbb{R}^{1\,536 \times 1\,536}</math>,
	<math>H \in \mathbb{R}^{~~1536~~ \times ~~2359296~~}</math>,		<math>H \in \mathbb{R}^{1\,536 \times 2\,359\,296}</math>,
	<math>N_1 \in \mathbb{R}^{~~1536~~ \times ~~1536~~}</math>,		<math>N_1 \in \mathbb{R}^{1\,536 \times 1\,536}</math>,
	<math>N_2 \in \mathbb{R}^{~~1536~~ \times ~~1536~~}</math>,		<math>N_2 \in \mathbb{R}^{1\,536 \times 1\,536}</math>,
	<math>B \in \mathbb{R}^{~~1536~~ \times 2}</math>,		<math>B \in \mathbb{R}^{1\,536 \times 2}</math>,
	<math>C \in \mathbb{R}^{2 \times ~~1536~~}</math>.		<math>C \in \mathbb{R}^{2 \times 1\,536}</math>.

	==Citation==		==Citation==

Himpe: /* Dimensions */ add output

2018-11-23T09:06:41Z

Dimensions: add output

← Older revision		Revision as of 09:06, 23 November 2018
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	:<math>		:<math>
	\begin{align}		\begin{align}
	E\dot{x}(t) = A x(t) + H x(t) \otimes x(t) + \sum_{i=1}^2N_i x(t) u_i(t) + B u(t)		E\dot{x}(t) &= A x(t) + H x(t) \otimes x(t) + \sum_{i=1}^2N_i x(t) u_i(t) + B u(t) \\
			y(t) &= Cx(t)
	\end{align}		\end{align}
	</math>		</math>

Himpe: /* Dimensions */ fix dynamic system

2018-11-23T09:04:54Z

Dimensions: fix dynamic system

← Older revision		Revision as of 09:04, 23 November 2018
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	:<math>		:<math>
	\begin{align}		\begin{align}
	E\dot{x}(t) = A x(t) + H x(t) \otimes x(t) + N x(t) u(t) + B u(t)		E\dot{x}(t) = A x(t) + H x(t) \otimes x(t) + \sum_{i=1}^2N_i x(t) u_i(t) + B u(t)
	\end{align}		\end{align}
	</math>		</math>
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	<math>E \in \mathbb{R}^{1536 \times 1536}</math>,		<math>E \in \mathbb{R}^{1536 \times 1536}</math>,
	<math>A \in \mathbb{R}^{1536 \times 1536}</math>,		<math>A \in \mathbb{R}^{1536 \times 1536}</math>,
			<math>H \in \mathbb{R}^{1536 \times 2359296}</math>,
			<math>N_1 \in \mathbb{R}^{1536 \times 1536}</math>,
			<math>N_2 \in \mathbb{R}^{1536 \times 1536}</math>,
	<math>B \in \mathbb{R}^{1536 \times 2}</math>,		<math>B \in \mathbb{R}^{1536 \times 2}</math>,
	~~<math>H \in \mathbb{R}^{1536 \times 2359296}</math>,~~		<math>C \in \mathbb{R}^{2 \times 1536}</math>.
	~~<math>N_1 \in \mathbb{R}^{1536 \times 1536}</math>, <math>N_2 \in \mathbb{R}^{1536 \times 1536}</math>,~~ <math>C \in \mathbb{R}^{2 \times 1536}</math>.

	==Citation==		==Citation==

Himpe: /* Dimensions */

2018-11-23T09:00:43Z

Dimensions

← Older revision		Revision as of 09:00, 23 November 2018
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	==Dimensions==		==Dimensions==

	System structure:		System structure:
	:<math>		:<math>
	\begin{align}		\begin{align}
	E\dot{x}(t) = A x(t) + H x(t) \otimes x(t) + N x(t) u_(t) + B u(t)		E\dot{x}(t) = A x(t) + H x(t) \otimes x(t) + N x(t) u(t) + B u(t)
	\end{align}		\end{align}
	</math>		</math>

Pontes: /* Dimensions */

2018-05-28T15:27:54Z

Dimensions

← Older revision		Revision as of 15:27, 28 May 2018
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	<math>B \in \mathbb{R}^{1536 \times 2}</math>,		<math>B \in \mathbb{R}^{1536 \times 2}</math>,
	<math>H \in \mathbb{R}^{1536 \times 2359296}</math>,		<math>H \in \mathbb{R}^{1536 \times 2359296}</math>,
	<math>~~N_?~~ \in \mathbb{R}^{1536 \times 1536}</math>.		<math>N_1 \in \mathbb{R}^{1536 \times 1536}</math>, <math>N_2 \in \mathbb{R}^{1536 \times 1536}</math>, <math>C \in \mathbb{R}^{2 \times 1536}</math>.


	==Citation==		==Citation==

Pontes: /* Data */

2018-05-28T15:26:20Z

Data

← Older revision		Revision as of 15:26, 28 May 2018
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	[[Media:FitzHughNagumo.tar.gz\|FitzNagumo.tar.gz]].		[[Media:FitzHughNagumo.tar.gz\|FitzNagumo.tar.gz]].

	The archive contains the matrices <math>E</math>, <math>A</math>, <math>B</math>, <math>H</math> and <math>~~N_?~~</math>; <math>~~N_?~~</math> is a zero matrix.		The archive contains the matrices <math>E</math>, <math>A</math>, <math>B</math>, <math>H</math> and <math>N_1</math>; The matrix <math>N_2</math> is a zero matrix of appropiate size.

	For more information on the discretization details, see <ref name="chat10"/>.		For more information on the discretization details, see <ref name="chat10"/>.

			In addition, one can have an output
			:<math>
			y(t) = Cx(t),
			</math>
			where the matrix <math>C</math> is a <math>2\times N</math> matrix such that <math>C(1,1) = 1,
			C(2,1+k) = 1 </math> and the other elements are zero. This output <math>y(t)</math> corresponds to the left boundary of the limit cycles. For more information on the output, see <ref name="benner12"/>.

	==Dimensions==		==Dimensions==

Himpe: /* Data */ data contents

2018-05-14T14:40:45Z

Data: data contents

← Older revision		Revision as of 14:40, 14 May 2018
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	In <ref name="chat10"/>, the previous system of coupled nonlinear PDEs is spatially discretized by means of a finite difference scheme with <math>k=512 </math> nodes for each PDE. Hence, one obtains a nonlinear (cubic) system of ODEs with state dimension <math>2\cdot k=1024</math>. As discussed above, the nonlinear system can be rewritten as a quadratic-bilinear system of dimension <math>3\cdot k=1536</math>, and all matrices of the quadratic-bilinear formulation discretized are in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format. The matrix name is used as an extension of the matrix file and can be found at:		In <ref name="chat10"/>, the previous system of coupled nonlinear PDEs is spatially discretized by means of a finite difference scheme with <math>k=512 </math> nodes for each PDE. Hence, one obtains a nonlinear (cubic) system of ODEs with state dimension <math>2\cdot k=1024</math>. As discussed above, the nonlinear system can be rewritten as a quadratic-bilinear system of dimension <math>3\cdot k=1536</math>, and all matrices of the quadratic-bilinear formulation discretized are in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format. The matrix name is used as an extension of the matrix file and can be found at:

	[[Media:FitzHughNagumo.tar.gz\|FitzNagumo.tar.gz]].		[[Media:FitzHughNagumo.tar.gz\|FitzNagumo.tar.gz]].

			The archive contains the matrices <math>E</math>, <math>A</math>, <math>B</math>, <math>H</math> and <math>N_?</math>; <math>N_?</math> is a zero matrix.

	For more information on the discretization details, see <ref name="chat10"/>.		For more information on the discretization details, see <ref name="chat10"/>.

Himpe: Various fixes.

2018-05-14T14:34:10Z

Various fixes.

← Older revision		Revision as of 14:34, 14 May 2018
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	===Model Equations===		===Model Equations===
	Here, we present the setting from <ref name="chat10"~~>S. Chaturantabut and D.C. Sorensen, "<span class="plainlinks">[http:~~/~~/dx.doi.org/10.1137/090766498 Nonlinear Model Reduction via Discrete Empirical Interpolation]</span>", SIAM J. Sci. Comput., 32 (2010), pp. 2737-2764.</ref~~>, where the dynamics of the system is governed by the following coupled nonlinear PDEs:		Here, we present the setting from <ref name="chat10"/>, where the dynamics of the system is governed by the following coupled nonlinear PDEs:

	:<math>		:<math>
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	</math>		</math>

	with <math>E,A,N_j \in \mathbb R^{n\times n}, j \in\{1,2\}, \ H \in \mathbb R^{n\times n^2} </math> and <math> B \in \mathbb R^{n\times 2}</math>, and the input function is <math>u(t)=[i_0(t),1]</math>. The idea relies on artificially introducing a new state variable defined as <math>z(t)=v(t)^2</math> and subsequently computing the dynamics of the new variable, i.e., specifying <math>\dot{z}(t).</math> The technique goes back to <ref name="gu11">C. Gu, "<span class="plainlinks">[http://dx.doi.org/10.1109/TCAD.2011.2142184 QLMOR: A Projection-Based Nonlinear Model Order Reduction Approach Using Quadratic-Linear Representation of Nonlinear Systems]</span>", IEEE T. Comput. Aid. D., 30 (2011), pp. 1307-1320.</~~ref~~>, where it is successfully applied to several smooth nonlinear control-affine systems. As discussed in <ref name="benner12"~~>P. Benner and T. Breiten, "<span class="plainlinks">[http:/~~/~~www.mpi-magdeburg.mpg.de/preprints/2012/12/ Two-Sided Moment Matching Methods for Nonlinear Model Reduction]</span>", 2012, Preprint MPIMD/12-12.</ref~~>, introducing <math>z</math> as an addictional variable yields a quadratic-bilinear control of dimension <math> n = 3\cdot k</math>, where <math>k</math> denotes the number of discretization nodes for each PDE, with state vector <math>x = [v,w,z]^T.</math> The increase of the state dimension has the advantage of reducing the nonlinearity from cubic to quadratic. This, however, opens up the possibility to reduce the system by model reduction schemes developed for quadratic-bilinear systems such as balanced truncation <ref name="morBenG17"/>, or interpolation-based approaches, e.g., given in <ref name="benner12"/><ref name="gu11"/>.		with <math>E,A,N_j \in \mathbb R^{n\times n}, j \in\{1,2\}, \ H \in \mathbb R^{n\times n^2} </math> and <math> B \in \mathbb R^{n\times 2}</math>, and the input function is <math>u(t)=[i_0(t),1]</math>. The idea relies on artificially introducing a new state variable defined as <math>z(t)=v(t)^2</math> and subsequently computing the dynamics of the new variable, i.e., specifying <math>\dot{z}(t).</math> The technique goes back to <ref name="gu11"/>, where it is successfully applied to several smooth nonlinear control-affine systems. As discussed in <ref name="benner12"/>, introducing <math>z</math> as an addictional variable yields a quadratic-bilinear control of dimension <math> n = 3\cdot k</math>, where <math>k</math> denotes the number of discretization nodes for each PDE, with state vector <math>x = [v,w,z]^T.</math> The increase of the state dimension has the advantage of reducing the nonlinearity from cubic to quadratic. This, however, opens up the possibility to reduce the system by model reduction schemes developed for quadratic-bilinear systems such as balanced truncation <ref name="morBenG17"/>, or interpolation-based approaches, e.g., given in <ref name="benner12"/> and <ref name="gu11"/>.

	==Data==		==Data==
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	For more information on the discretization details, see <ref name="chat10"/>.		For more information on the discretization details, see <ref name="chat10"/>.

			==Dimensions==
			System structure:
			:<math>
			\begin{align}
			E\dot{x}(t) = A x(t) + H x(t) \otimes x(t) + N x(t) u_(t) + B u(t)
			\end{align}
			</math>

			System dimensions:

			<math>E \in \mathbb{R}^{1536 \times 1536}</math>,
			<math>A \in \mathbb{R}^{1536 \times 1536}</math>,
			<math>B \in \mathbb{R}^{1536 \times 2}</math>,
			<math>H \in \mathbb{R}^{1536 \times 2359296}</math>,
			<math>N_? \in \mathbb{R}^{1536 \times 1536}</math>.


	==Citation==		==Citation==
Line 74:		Line 91:

	<references>		<references>
	<ref name="morBenG17">P. Benner and P. Goyal, "<span class="plainlinks">[https://arxiv.org/abs/1705.00160 Balanced Truncation Model Order Reduction for Quadratic-Bilinear Systems]</span>", arXiv e-prints, 2017</ref>
			<ref name="chat10">S. Chaturantabut and D.C. Sorensen, "<span class="plainlinks">[https://doi.org/10.1137/090766498 Nonlinear Model Reduction via Discrete Empirical Interpolation]</span>", SIAM J. Sci. Comput., 32: 2737--2764, 2010.</ref>

			<ref name="gu11">C. Gu, "<span class="plainlinks">[https://doi.org/10.1109/TCAD.2011.2142184 QLMOR: A Projection-Based Nonlinear Model Order Reduction Approach Using Quadratic-Linear Representation of Nonlinear Systems]</span>", IEEE T. Comput. Aid. D., 30: 1307--1320, 2011.</ref>

			<ref name="benner12">P. Benner and T. Breiten, "<span class="plainlinks">[https://doi.org/10.1137/14097255X Two-Sided Moment Matching Methods for Nonlinear Model Reduction]</span>", SIAM J. Sci. Comput., 37(2): B239--B260, 2015.</ref>

			<ref name="morBenG17">P. Benner and P. Goyal, "<span class="plainlinks">[https://arxiv.org/abs/1705.00160 Balanced Truncation Model Order Reduction for Quadratic-Bilinear Systems]</span>", arXiv e-prints, 2017</ref>

	</references>		</references>