Reduced Basis PMOR method - Revision history

Hessm at 07:58, 23 May 2013

2013-05-23T07:58:57Z

← Older revision		Revision as of 07:58, 23 May 2013
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	==Description==		==Description==

	The '''Reduced Basis Method'''<ref name="rozza08"/>, <ref name="grepl05"/> (RBM) we present here a [[Projection based MOR]] method applicable to static and time-dependent linear PDEs.		The '''Reduced Basis Method'''<ref name="rozza08"/>, <ref name="grepl05"/> (RBM) we present here is a [[Projection based MOR]] method, applicable to static and time-dependent linear PDEs.

	==Time-Independent PDEs==		==Time-Independent PDEs==

Hessm at 07:58, 23 May 2013

2013-05-23T07:58:38Z

← Older revision		Revision as of 07:58, 23 May 2013
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	==Description==		==Description==

	The '''Reduced Basis Method'''<ref name="rozza08"/>, <ref name="grepl05"/> (RBM) we present here is applicable to static and time-dependent linear PDEs.		The '''Reduced Basis Method'''<ref name="rozza08"/>, <ref name="grepl05"/> (RBM) we present here a [[Projection based MOR]] method applicable to static and time-dependent linear PDEs.

	==Time-Independent PDEs==		==Time-Independent PDEs==

Himpe: equation indent

2013-04-30T23:13:01Z

equation indent

← Older revision		Revision as of 23:13, 30 April 2013
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	The exact, infinite-dimensional formulation, indicated by the superscript e, is given by		The exact, infinite-dimensional formulation, indicated by the superscript e, is given by

			:<math>
	<math>
	\begin{cases}		\begin{cases}
	\text{For } \mu \in \mathcal{D} \subset \mathbb{R}^P, \text{ evaluate } \\		\text{For } \mu \in \mathcal{D} \subset \mathbb{R}^P, \text{ evaluate } \\
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	Through spatial discretization, e.g. finite element method, we consider the discretized system		Through spatial discretization, e.g. finite element method, we consider the discretized system

	<math>		:<math>
	\begin{cases}		\begin{cases}
	\text{For } \mu \in \mathcal{D} \subset \mathbb{R}^P, \text{ evaluate } \\		\text{For } \mu \in \mathcal{D} \subset \mathbb{R}^P, \text{ evaluate } \\
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	The essential assumption which allows the offline-online decomposition is that there exists an affine parameter dependence		The essential assumption which allows the offline-online decomposition is that there exists an affine parameter dependence

	<math>		:<math>
	a(w,v;\mu) = \sum_{q=1}^{Q^a} \Theta_a^q(\mu) a^q(w,v)		a(w,v;\mu) = \sum_{q=1}^{Q^a} \Theta_a^q(\mu) a^q(w,v)
	</math>		</math>

	<math>		:<math>
	f(v;\mu) = \sum_{q=1}^{Q^f} \Theta_f^{q}(\mu) f^q(v).		f(v;\mu) = \sum_{q=1}^{Q^f} \Theta_f^{q}(\mu) f^q(v).
	</math>		</math>
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	The Lagrange Reduced Basis space is established by iteratively choosing Lagrange parameter samples		The Lagrange Reduced Basis space is established by iteratively choosing Lagrange parameter samples

	<math>		:<math>
	S_N = \{\mu^1,...,\mu^N\}		S_N = \{\mu^1,...,\mu^N\}
	</math>		</math>
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	and considering the associated Lagrange RB spaces		and considering the associated Lagrange RB spaces

	<math>		:<math>
	V_N = \text{span}\{u(\mu^n), 1 \leq n \leq N \}		V_N = \text{span}\{u(\mu^n), 1 \leq n \leq N \}
	</math>		</math>
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	We then consider the galerkin projection onto the RB-space <math> V_N </math>		We then consider the galerkin projection onto the RB-space <math> V_N </math>

	<math>		:<math>
	\begin{cases}		\begin{cases}
	\text{For } \mu \in \mathcal{D} \subset \mathbb{R}^P, \text{ evaluate } \\		\text{For } \mu \in \mathcal{D} \subset \mathbb{R}^P, \text{ evaluate } \\
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	The exact, infinite-dimensional formulation, indicated by the superscript e, is given by		The exact, infinite-dimensional formulation, indicated by the superscript e, is given by

	<math>		:<math>
	\begin{cases}		\begin{cases}
	\text{For } \mu \in \mathcal{D} \subset \mathbb{R}^P, t^k \in [0,T] \text{ evaluate } \\		\text{For } \mu \in \mathcal{D} \subset \mathbb{R}^P, t^k \in [0,T] \text{ evaluate } \\
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	Assume a reference discretization form is given as follows,		Assume a reference discretization form is given as follows,

	<math>		:<math>
	\begin{cases}		\begin{cases}
	\text{For } \mu \in \mathcal{D} \subset \mathbb{R}^P, t^k \in [0,T] \text{ evaluate } \\		\text{For } \mu \in \mathcal{D} \subset \mathbb{R}^P, t^k \in [0,T] \text{ evaluate } \\
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	To apply the offline-online decomposition, we assume they are affine parameter-dependent, i.e.		To apply the offline-online decomposition, we assume they are affine parameter-dependent, i.e.

	<math>		:<math>
	m(w,v;\mu) = \sum_{q=1}^{Q_m} \Theta_m^q(\mu,t) m^q(w,v)		m(w,v;\mu) = \sum_{q=1}^{Q_m} \Theta_m^q(\mu,t) m^q(w,v)
	</math>		</math>

	<math>		:<math>
	a(w,v;\mu) = \sum_{q=1}^{Q_a} \Theta_a^q(\mu,t) a^q(w,v)		a(w,v;\mu) = \sum_{q=1}^{Q_a} \Theta_a^q(\mu,t) a^q(w,v)
	</math>		</math>

	<math>		:<math>
	f(v;\mu) = \sum_{q=1}^{Q_f} \Theta_f^{q}(\mu,t) f^q(v).		f(v;\mu) = \sum_{q=1}^{Q_f} \Theta_f^{q}(\mu,t) f^q(v).
	</math>		</math>
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	The Lagrange Reduced Basis space <math> V_N </math> is usually established by POD-Greedy algorithm <ref name="haasdonk08"/>. Then the input-output response can be presented as follows, through Galerkin projection,		The Lagrange Reduced Basis space <math> V_N </math> is usually established by POD-Greedy algorithm <ref name="haasdonk08"/>. Then the input-output response can be presented as follows, through Galerkin projection,

	<math>		:<math>
	\begin{cases}		\begin{cases}
	\text{For } \mu \in \mathcal{D} \subset \mathbb{R}^P, t^k \in [0,T] \text{ evaluate } \\		\text{For } \mu \in \mathcal{D} \subset \mathbb{R}^P, t^k \in [0,T] \text{ evaluate } \\

Himpe: references with cite, reference link

2013-04-24T12:13:55Z

references with cite, reference link

← Older revision		Revision as of 12:13, 24 April 2013
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	==Description==		==Description==

	The Reduced Basis Method (RBM) we present here is applicable to static and time-dependent linear PDEs.		The '''Reduced Basis Method'''<ref name="rozza08"/>, <ref name="grepl05"/> (RBM) we present here is applicable to static and time-dependent linear PDEs.

	==Time-Independent PDEs==		==Time-Independent PDEs==
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	</math>		</math>

	The Lagrange Reduced Basis space <math> V_N </math> is usually established by POD-Greedy algorithm ~~[3]~~. Then the input-output response can be presented as follows, through Galerkin projection,		The Lagrange Reduced Basis space <math> V_N </math> is usually established by POD-Greedy algorithm <ref name="haasdonk08"/>. Then the input-output response can be presented as follows, through Galerkin projection,

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

	Note that the assumption of affine form can be relaxed in practice, then the empirical interpolation method ~~[1]~~ can be exploited for		Note that the assumption of affine form can be relaxed in practice, then the empirical interpolation method <ref name="barrault04"/> can be exploited for
	offline-online decomposition.		offline-online decomposition.

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	==References==		==References==
	~~[1] M. Barrault, Y. Maday, N.C. Nguyen, and A.T. Patera,~~
	~~An 'empirical interpolation' method: application~~
	~~to efficient reduced-basis discretization of partial differential equations,~~
	~~C. R. Acad. Sci. Paris Series I, 339 (2004), 667-672.~~

	~~[2] M. Grepl,~~		<references>
	~~Reduced--basis approximations and posteriori error estimation for parabolic partial differential equations,~~
	~~PhD thesis, MIT, 2005.~~

	~~[3] B~~. ~~Haasdonk~~ and M. ~~Ohlberger~~,		<ref name="barrault04"> M. Barrault, Y. Maday, N.C. Nguyen, and A.T. Patera, "<span class="plainlinks">[http://dx.doi.org/10.1016/j.crma.2004.08.006 An 'empirical interpolation' method: application to efficient reduced-basis discretization of partial differential equations]</span>", C. R. Acad. Sci. Paris Series I, 339 (2004), 667-672.</ref>
	~~Reduced~~ basis ~~method for finite volume approximations~~ of ~~parameterized linear evolution~~ equations,
	~~Mathematical Modeling and Numerical Analysis~~, 42 (~~2008~~), ~~277~~-~~302~~.

	~~[4] G~~. ~~Rozza~~, ~~D.B.P~~. ~~Huynh, A~~.T. ~~Patera~~		<ref name="grepl05">M. Grepl, "<span class="plainlinks">[http://hdl.handle.net/1721.1/32387 Reduced--basis approximations and posteriori error estimation for parabolic partial differential equations]</span>" PhD thesis, MIT, 2005.</ref>
	Reduced ~~Basis Approximation~~ and ~~a Posteriori Error Estimation~~ for ~~Affinely Parametrized Elliptic Coercive Partial Differential Equations~~,
	~~Arch Comput Methods Eng (2008) 15: 229–275~~.

			<ref name="haasdonk08"> B. Haasdonk and M. Ohlberger, "<span class="plainlinks">[http://www.agh.ians.uni-stuttgart.de/publications/2008/HO08b Reduced basis method for finite volume approximations of parameterized linear evolution equations]</span>", Mathematical Modeling and Numerical Analysis, 42 (2008), 277-302.</ref>

			<ref name="rozza08">G. Rozza, D.B.P. Huynh, A.T. Patera, "<span class="plainlinks">[http://dx.doi.org/10.1007/s11831-008-9019-9 Reduced Basis Approximation and a Posteriori Error Estimation for Affinely Parametrized Elliptic Coercive Partial Differential Equations]</span>", Arch Comput Methods Eng (2008) 15: 229–275.</ref>

			</references>

	==Contact==		==Contact==

Grundel at 09:19, 19 April 2013

2013-04-19T09:19:40Z

← Older revision		Revision as of 09:19, 19 April 2013
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	[[Category:method]]		[[Category:method]]
	[[Category:parametric ~~method~~]]		[[Category:parametric]]

	==Description==		==Description==

Himpe: /* Contact information */

2013-03-25T11:44:11Z

Contact information

← Older revision		Revision as of 11:44, 25 March 2013
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	==Contact ~~information~~==		==Contact==

	'' [[User:hessm\|Martin Hess]]''		'' [[User:hessm\|Martin Hess]]''

	'' [[User:Zhangy\|Yongjin Zhang]]''		'' [[User:Zhangy\|Yongjin Zhang]]''

Himpe at 11:43, 25 March 2013

2013-03-25T11:43:54Z

← Older revision		Revision as of 11:43, 25 March 2013
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	[[Category:method]]		[[Category:method]]
	[[Category:parametric method]]		[[Category:parametric method]]

			==Description==

	The Reduced Basis Method (RBM) we present here is applicable to static and time-dependent linear PDEs.		The Reduced Basis Method (RBM) we present here is applicable to static and time-dependent linear PDEs.
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			==Contact information==
	Contact information:

	'' [[User:hessm\|Martin Hess]]''		'' [[User:hessm\|Martin Hess]]''

	'' [[User:Zhangy\|Yongjin Zhang]]''		'' [[User:Zhangy\|Yongjin Zhang]]''

Feng: /* References */

2012-12-20T14:07:17Z

References

← Older revision		Revision as of 14:07, 20 December 2012
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	==References==		==References==
	[1] M. Barrault, Y. Maday, N. Nguyen, and A. Patera,		[1] M. Barrault, Y. Maday, N.C. Nguyen, and A.T. Patera,
	An 'empirical interpolation' method: application		An 'empirical interpolation' method: application
	to efficient reduced-basis discretization of partial differential equations,		to efficient reduced-basis discretization of partial differential equations,

Zhangy: /* Time-Dependent PDEs */

2012-12-20T08:30:49Z

Time-Dependent PDEs

← Older revision		Revision as of 08:30, 20 December 2012
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	offline-online decomposition.		offline-online decomposition.

	This method has been used for [[~~batch chromatography~~\|~~Batch_Chromatography~~]], where the empirical interpolation method was used for treating the nonaffinity.		This method has been used for [[Batch_Chromatography\|Batch Chromatography]], where the empirical interpolation method was used for treating the nonaffinity.

	==References==		==References==

Zhangy: /* Time-Dependent PDEs */

2012-12-20T08:29:40Z

Time-Dependent PDEs

← Older revision		Revision as of 08:29, 20 December 2012
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	offline-online decomposition.		offline-online decomposition.

	This method has been used for [[batch chromatography\|~~Batch Chromatography~~]], where the empirical interpolation method was used for treating the nonaffinity.		This method has been used for [[batch chromatography\|Batch_Chromatography]], where the empirical interpolation method was used for treating the nonaffinity.

	==References==		==References==