Difference between revisions of "Emgr"

Revision as of 15:59, 22 October 2020

Synopsis

emgr - Empirical Gramian Framework (Version 5.8). Empirical gramians can be computed for linear and nonlinear control systems for purposes of model order reduction, uncertainty quantification and system identification. Model reduction using empirical gramians can be applied to the state space, to the parameter space or to both through combined reduction. The emgr framework is a compact open source toolbox for gramian-based model reduction and compatible with OCTAVE and MATLAB.

Features

Application matrix for the empirical gramian framework

emgr encompasses seven types of gramians:

Empirical Controllability Gramian / Empirical Output Controllability Gramian
Empirical Observability Gramian / Empirical Average Observability Gramian
Empirical Cross Gramian / Empirical Non-Symmetric Cross Gramian
Empirical Linear Cross Gramian / Empirical Non-Symmetric Linear Cross Gramian
Empirical Sensitivity Gramian (parameter controllability)
Empirical Identifiability Gramian (parameter observability)
Empirical Joint Gramian (parameter observability)

emgr overview

Applications:

Model reduction
- Proper Orthogonal Decomposition (POD)
- Modified POD / Approximate Balancing
- Dominant Subspace Projection Model Reduction (DSPMR)
- Balanced Proper Orthogonal Decompsition (bPOD)
- Balanced Truncation
- Balanced Gains
Parameter reduction
Combined state and parameter reduction
Decentralized control
State sensitivity
Parameter sensitivity
Parameter identification
Nonlinearity quantification
Uncertainty quantification
System indices
System norms
Tau function

Meta Information

Code Meta Data
name:	Empirical Gramian Framework (emgr)
version:	5.8 (2020-05-01)
id:	10.5281/zenodo.3779889 (doi)
author:	Christian Himpe (0000-0003-2194-6754)
topic:	Science, Mathematics, Model Reduction
type:	Toolbox
license:	BSD-2-Clause (open-source)
repository:	https://github.com/gramian/emgr (git)
language:	Matlab
dependencies:	Octave >=5.2, Matlab >=2017b
systems:	Linux, Windows
website:	https://gramian.de
keywords:	Controllability, Observability, Cross Gramian, Model Reduction, Model Order Reduction

Cite

Please cite emgr as:

C. Himpe (2020). emgr - EMpirical GRamian Framework (Version 5.8) [Software]. Available from https://gramian.de . doi:10.5281/zenodo.3779889

 @MISC{emgr58,
  author = {C. Himpe},
  title = {{emgr -{EMpirical GRamian Framework} (Version~5.8)},
  howpublished = {\url{https://gramian.de}},
  year = {2020},
  doi = {10.5281/zenodo.3779889}
 }

References

C. Himpe. "Comparing (Empirical-Gramian-Based) Model Order Reduction Algorithms". arXiv, math.OC: 2002.12226, 2020.

P. Benner, C. Himpe. "Cross-Gramian-Based Dominant Subspaces". Advances in Computational Mathematics 45(5): 91, 2533--2553, 2019.

S. Grundel, C. Himpe, J. Saak. "On Empirical System Gramians". Proceedings in Applied Mathematics and Mechanics, 19: e201900006, 2019.

C. Himpe. "emgr - The Empirical Gramian Framework". Algorithms 11(7): 91, 2018.

C. Himpe, T. Leibner, S. Rave, J. Saak. "Fast Low-Rank Empirical Cross Gramians". Proceedings in Applied Mathematics and Mechanics, 17: 841--842, 2017.

C. Himpe. "Combined State and Parameter Reduction for Nonlinear Systems with an Application in Neuroscience". Westfälische Wilhelms Universität, Sierke Verlag Göttingen, 2017.

C. Himpe, M. Ohlberger. "A Note on the Cross Gramian for Non-Symmetric Systems". System Science and Control Engineering 4(1): 199--208, 2016.

C. Himpe, M. Ohlberger. "The Empirical Cross Gramian for Parametrized Nonlinear Systems". IFAC-PapersOnLine (8th Vienna International Conference on Mathematical Modelling), 48(1): 727--728, 2015.

C. Himpe, M. Ohlberger. "Model Reduction for Complex Hyperbolic Networks". Proceedings of the European Control Conference: 2739--2743, 2014.

C. Himpe, M. Ohlberger. "Cross-Gramian Based Combined State and Parameter Reduction for Large-Scale Control Systems". Mathematical Problems in Engineering, 2014: 843869, 2014.

C. Himpe, M. Ohlberger. "A Unified Software Framework for Empirical Gramians". Journal of Mathematics 2013: 365909, 2013.

Links

Official website: https://gramian.de

Oberwolfach References on Mathematical Software: Entry

SIAM DSweb Dynamical Systems Software: Entry

Contact

Christian Himpe

@@ Line 17: / Line 17: @@
 '''emgr''' encompasses seven types of gramians:
-* Empirical [[:Wikipedia:Controllability_Gramian|Controllability Gramian]]
+* Empirical [[:Wikipedia:Controllability_Gramian|Controllability Gramian]] / Empirical Output Controllability Gramian
-* Empirical [[:Wikipedia:Observability_Gramian|Observability Gramian]]
+* Empirical [[:Wikipedia:Observability_Gramian|Observability Gramian]] / Empirical Average Observability Gramian
-* Empirical [[:Wikipedia:Cross_Gramian|Cross Gramian]] (including an Empirical Non-Symmetric Cross Gramian)
+* Empirical [[:Wikipedia:Cross_Gramian|Cross Gramian]] / Empirical Non-Symmetric Cross Gramian
-* Empirical Linear [[:Wikipedia:Cross_Gramian|Cross Gramian]]
+* Empirical Linear Cross Gramian / Empirical Non-Symmetric Linear Cross Gramian
 * Empirical Sensitivity Gramian (parameter controllability)
 * Empirical Identifiability Gramian (parameter observability)
@@ Line 27: / Line 27: @@
 [[File:emgr-flyer.pdf|thumb|right|emgr overview]]
+=== Applications: ===
-applicable to:
+* Model reduction
-* Linear + Nonlinear Control Systems
+** Proper Orthogonal Decomposition (POD)
-* First + Second Order Control Systems
+** Modified POD / Approximate Balancing
-* Parametrized | Parametric Systems
+** Dominant Subspace Projection Model Reduction (DSPMR)
-* Time Invariant + Varying Systems
+** Balanced Proper Orthogonal Decompsition (bPOD)
-* Discretized [[:Wikipedia:Partial_differential_equation|PDEs]]
+** Balanced Truncation
+** Balanced Gains
-and with sample code for:
+* Parameter reduction
+* Combined state and parameter reduction
-* [[Balanced Truncation]] and balancing related methods
+* Decentralized control
-* Combined State and Parameter Reduction
+* State sensitivity
-* Parameter Identification + Sensitivity Analysis
-* Parameter Reduction + Robust Reduction
+* Parameter sensitivity
+* Parameter identification
-* Optimal Sensor + Actuator Placement
+* Nonlinearity quantification
-* Decentralized Control
+* Uncertainty quantification
-* Nonlinearity Quantification
+* System indices
+* System norms
+* Tau function
 == Meta Information ==
@@ Line 89: / Line 92: @@
 |Controllability, Observability, Cross Gramian, Model Reduction, Model Order Reduction
 |}
+== Cite ==
+Please cite '''emgr''' as:
+C. Himpe (2020). emgr - EMpirical GRamian Framework (Version 5.8) [Software].
+Available from https://gramian.de . doi:10.5281/zenodo.3779889
+  @MISC{emgr58,
+   author = {C. Himpe},
+   title = {{emgr -{EMpirical GRamian Framework} (Version~5.8)},
+   howpublished = {\url{https://gramian.de}},
+   year = {2020},
+   doi = {10.5281/zenodo.3779889}
+  }
 == References ==
+* [[User:Himpe|C. Himpe]]. "<span class="plainlinks">[https://arxiv.org/abs/2002.12226 Comparing (Empirical-Gramian-Based) Model Order Reduction Algorithms]</span>". arXiv, math.OC: 2002.12226, 2020.
+* P. Benner, [[User:Himpe|C. Himpe]]. "<span class="plainlinks">[https://doi.org/10.1007/s10444-019-09724-7 Cross-Gramian-Based Dominant Subspaces]</span>". Advances in Computational Mathematics 45(5): 91, 2533--2553, 2019.
 * S. Grundel, [[User:Himpe|C. Himpe]], J. Saak. "<span class="plainlinks">[https://doi.org/10.1002/pamm.201900006 On Empirical System Gramians]</span>". Proceedings in Applied Mathematics and Mechanics, 19: e201900006, 2019.
@@ Line 116: / Line 138: @@
 * Oberwolfach References on Mathematical Software: [http://orms.mfo.de/project?id=345 Entry]
-* DSweb Dynamical Systems Software: [https://dsweb.siam.org/Software/emgr-empirical-gramian-framework Entry]
+* SIAM DSweb Dynamical Systems Software: [https://dsweb.siam.org/Software/emgr-empirical-gramian-framework Entry]
 == Contact ==