Recent changes to Numerics.DimRed

Discussion for Numerics.DimRed page

Burkhard Schmidt — Thu, 01 Jun 2017 13:40:29 -0000

Indices k (on N matrices) in last but one equation missing

Numerics.DimRed modified by Burkhard Schmidt

Burkhard Schmidt — Thu, 27 Apr 2017 13:43:14 -0000

--- v140
+++ v141
@@ -86,6 +86,13 @@
 [[img src="Error_3.gif" alt="||\Sigma-\hat{\Sigma}||^2_{\mathcal{H}_2}=tr\left(C_EW_EC_E^\ast\right)"]]

+Examples
+--------------
+
+ * [Controlled population transfer in an asymmetric double well](Demos.DoubleWell.DimReduce_1/)
+ * ... more coming soon ...
+ 
+ 
 References
 ----------

Numerics.DimRed modified by Burkhard Schmidt

Burkhard Schmidt — Tue, 25 Apr 2017 09:11:52 -0000

--- v139
+++ v140
@@ -68,7 +68,7 @@
 The need for efficient numerical treatment of control problems leads to the problem of model order reduction, i.e. the approximation of large-scale systems by significantly smaller ones. While model reduction of linear systems has been studied for several years now, and a well established theory including error bounds and structure-preserving properties fulfilled by a reduced-order model exists, the situation is less favorable for non-linear systems (which occur, e.g. in [control of quantum problems](Numerics.Control)). There, the mathematical techniques developed to linear systems are more difficult and much less general. Here we will deal with following two classes of methods, representing generalizations of their linear counterparts:

 * [Balanced truncation](Numerics.DimRed.BalTru)
-* [ℋ2 norm interpolation](Numerics.DimRed.H2Norm)
+* [ℋ2 optimal model reduction](Numerics.DimRed.H2Model)

 ℋ2 error norm
 --------------

Numerics.DimRed modified by Burkhard Schmidt

Burkhard Schmidt — Thu, 23 Jun 2016 07:05:38 -0000

--- v138
+++ v139
@@ -56,7 +56,7 @@

  [[img src="Lyapunov_3.gif" alt="\begin{array}{rcl}AX_0+X_0\hat{A}^\ast +B\hat{B}^\ast&=&0\\AX_j+X_j\hat{A}^\ast+\sum_{k=1}^m N_kX_{j-1}\hat{N}^\ast_k+B\hat{B}^\ast&=&0,\,j.gt.0\end{array}"]]

-(and similarly for the second (dual) equation for the observability Gramian). Convergence X_j→W_C is guaranteed if the eigenvalue of A with the largest (negative) real part is sufficiently separated from the imaginary axis \[5\]. While for linear control systems the Lyapunov equations can always be solved if matrix A is stable (see [here](Numerics.Control)), for the generalized Lyapunov equations we recall that a system is called stable when the system matrix A has only eigenvalues in the open left half complex plane (i.e., excluding the imaginary axis). Stability thus means that there are constants λ , a > 0 such that ||exp(At)|| ≤ λ exp(−at), where ||•|| is a suitable matrix norm. If moreover
+and similarly for the second (dual) equation for the observability Gramian. Convergence X_j→W_C is guaranteed if the eigenvalue of A with the largest (negative) real part is sufficiently separated from the imaginary axis \[5\]. While for linear control systems the Lyapunov equations can always be solved if matrix A is stable (see [here](Numerics.Control)), for the generalized Lyapunov equations we recall that a system is called stable when the system matrix A has only eigenvalues in the open left half complex plane (i.e., excluding the imaginary axis). Stability thus means that there are constants λ , a > 0 such that ||exp(At)|| ≤ λ exp(−at), where ||•|| is a suitable matrix norm. If moreover

 [[img src="WCO_4.gif" alt="\frac{\lambda^2}{2a}\sum_{k=1}^m||N_k||^2\langle 1"]]

@@ -103,4 +103,4 @@

 7. P. Benner, T. Breiten: [SIAM Journal on Matrix Analysis and Application, **33**, 859–885 (2012)](http://dx.doi.org/10.1137/110836742)

-For a review of the theory of linear control systems, see e.g. the excellent [lecture notes of Umea university](http://www.control.tfe.umu.se/Courses/Optimal_Control_for_Linear_Systems_2010) and/or the [books by Kemin Zhou](http://www.ece.lsu.edu/kemin).
+For a review of the theory of linear control systems, see e.g. the [lecture notes of Umea university](http://www.control.tfe.umu.se/Courses/Optimal_Control_for_Linear_Systems_2010) and/or the [books by Kemin Zhou](http://www.ece.lsu.edu/kemin).

Numerics.DimRed modified by Burkhard Schmidt

Burkhard Schmidt — Thu, 23 Jun 2016 06:57:31 -0000

--- v137
+++ v138
@@ -36,7 +36,7 @@

 Controllability and observability 
 ---------------------------------
-Two important properties of control systems are the [controllability](http://en.wikipedia.org/wiki/Controllability) (aka reachability) and the [observability](http://en.wikipedia.org/wiki/Observability) which can be regarded as dual aspects of the same problem. For the case of linear control systems, the corresponding [controllability Gramian W_C](http://en.wikipedia.org/wiki/Controllability_Gramian) (often referred to as P in the literature) and the [observability Gramian W_O](http://en.wikipedia.org/wiki/Observability_Gramian) (often referred to as Q in the literature) are defined as 
+Two important properties of control systems are the [controllability](http://en.wikipedia.org/wiki/Controllability) and the [observability](http://en.wikipedia.org/wiki/Observability) which can be regarded as dual aspects of the same problem. For the case of linear control systems, the corresponding [controllability Gramian W_C](http://en.wikipedia.org/wiki/Controllability_Gramian) (often referred to as P in the literature) and the [observability Gramian W_O](http://en.wikipedia.org/wiki/Observability_Gramian) (often referred to as Q in the literature) are defined as 

 [[img src="WCO_1.gif" alt="\begin{matrix}W_C&=&\int_0^\infty \exp(At)BB^\ast\exp(A^\ast t)dt\\W_O&=&\int_0^\infty \exp(A^\ast t)C^\ast C\exp(At)dt\end{matrix}"]]

Numerics.DimRed modified by Burkhard Schmidt

Burkhard Schmidt — Fri, 15 Apr 2016 07:08:04 -0000

--- v136
+++ v137
@@ -52,7 +52,7 @@

 [[img src="WCO_3.gif" alt="\begin{matrix}AW_C+W_C A^\ast +\sum_{k=1}^mN_kW_CN_k^\ast+BB^\ast&=&0\\A^\ast W_0+W_0A+\sum_{k=1}^m N_k^\ast W_0N_k+C^\ast C&=&0\end{matrix}"]] 

-Because direct methods for solving such equations have a numerical complexity of O(n⁶), alternative approaches become mandatory. For example, by mapping the Gramian matrices onto vectors, the generalized Lyapunov equations can be understood as systems of coupled linear equations which can be solved, e. g., by means of the [biconjugate gradient method](http://en.wikipedia.org/wiki/Biconjugate_gradient_method), preferentially with pre-conditioning \[3\]. Alternatively, the iterative methods \[4,5\] can be used instead which requires the solution of a standard Lyapunov equation in each step \[6\]
+Because direct methods for solving such equations have a numerical complexity of O(n⁶), alternative approaches become mandatory. For example, by mapping the Gramian matrices onto vectors, the generalized Lyapunov equations can be understood as systems of coupled linear equations which can be solved, e. g., by means of the [biconjugate gradient method](http://en.wikipedia.org/wiki/Biconjugate_gradient_method), preferentially with pre-conditioning \[3\]. Alternatively, an iterative method \[4,5\] can be used which requires the solution of a standard Lyapunov equation in each step \[6\]

  [[img src="Lyapunov_3.gif" alt="\begin{array}{rcl}AX_0+X_0\hat{A}^\ast +B\hat{B}^\ast&=&0\\AX_j+X_j\hat{A}^\ast+\sum_{k=1}^m N_kX_{j-1}\hat{N}^\ast_k+B\hat{B}^\ast&=&0,\,j.gt.0\end{array}"]]

Numerics.DimRed modified by Burkhard Schmidt

Burkhard Schmidt — Mon, 22 Feb 2016 09:44:34 -0000

--- v135
+++ v136
@@ -36,7 +36,7 @@

 Controllability and observability 
 ---------------------------------
-Ttwo important properties of control systems are the [controllability](http://en.wikipedia.org/wiki/Controllability) (aka reachability) and the [observability](http://en.wikipedia.org/wiki/Observability) which can be regarded as dual aspects of the same problem. For the case of linear control systems, the corresponding [controllability Gramian W_C](http://en.wikipedia.org/wiki/Controllability_Gramian) (often referred to as P in the literature) and the [observability Gramian W_O](http://en.wikipedia.org/wiki/Observability_Gramian) (often referred to as Q in the literature) are defined as 
+Two important properties of control systems are the [controllability](http://en.wikipedia.org/wiki/Controllability) (aka reachability) and the [observability](http://en.wikipedia.org/wiki/Observability) which can be regarded as dual aspects of the same problem. For the case of linear control systems, the corresponding [controllability Gramian W_C](http://en.wikipedia.org/wiki/Controllability_Gramian) (often referred to as P in the literature) and the [observability Gramian W_O](http://en.wikipedia.org/wiki/Observability_Gramian) (often referred to as Q in the literature) are defined as 

 [[img src="WCO_1.gif" alt="\begin{matrix}W_C&=&\int_0^\infty \exp(At)BB^\ast\exp(A^\ast t)dt\\W_O&=&\int_0^\infty \exp(A^\ast t)C^\ast C\exp(At)dt\end{matrix}"]]

Numerics.DimRed modified by Burkhard Schmidt

Burkhard Schmidt — Tue, 27 Jan 2015 11:01:22 -0000

--- v134
+++ v135
@@ -81,7 +81,7 @@

 [[img src="Error_2.gif" alt="A_EW_E+W_EA_E^\ast+\sum_{k=1}^mN_EW_EN_E^\ast+B_EB_E^\ast&=&0"]]

-and the resulting Gramian W_e can be used to obtain the error norm as
+and the resulting Gramian W_E can be used to obtain the error norm as

 [[img src="Error_3.gif" alt="||\Sigma-\hat{\Sigma}||^2_{\mathcal{H}_2}=tr\left(C_EW_EC_E^\ast\right)"]]

Numerics.DimRed modified by Burkhard Schmidt

Burkhard Schmidt — Thu, 22 Jan 2015 13:13:37 -0000

--- v133
+++ v134
@@ -4,7 +4,7 @@
 Introduction
 -------------

-Before discussing the dimension reduction itself, we shall first shed some light on the transfer function/matrix (and its H₂ norm). The transfer function/matrix describes how for every input/control u(t), the [linear/bilinear control systems](Numerics.Control) responds with an output/observation y(t). It is the task of dimension reduction to find lower-dimensional (reduced) systems which approximate this input-output behavior as closely as possible on any compact time interval \[0;T\]. We note that this problem is closely connected to the concepts of controllability and observability. 
+Before discussing the dimension reduction itself, we shall first shed some light on the transfer function/matrix (and its ℋ₂ norm). The transfer function/matrix describes how for every input/control u(t), the [linear/bilinear control systems](Numerics.Control) responds with an output/observation y(t). It is the task of dimension reduction to find lower-dimensional (reduced) systems which approximate this input-output behavior as closely as possible on any compact time interval \[0;T\]. We note that this problem is closely connected to the concepts of controllability and observability. 

 Transfer function/matrix and H₂ norm 
 -----------------------------------------------
@@ -52,7 +52,7 @@

 [[img src="WCO_3.gif" alt="\begin{matrix}AW_C+W_C A^\ast +\sum_{k=1}^mN_kW_CN_k^\ast+BB^\ast&=&0\\A^\ast W_0+W_0A+\sum_{k=1}^m N_k^\ast W_0N_k+C^\ast C&=&0\end{matrix}"]] 

-Because direct methods for solving such equations have a numerical complexity of O(n⁶), alternative approaches become mandatory. For example, by mapping the Gramian matrices onto vectors, the generalized Lyapunov equations can be understood as sy<tems of="" coupled="" linear="" equations="" which="" can="" be="" solved,="" e.="" g.,="" by="" means="" of="" the="" [biconjugate="" gradient="" method](http:="" en.wikipedia.org="" wiki="" Biconjugate_gradient_method),="" preferentially="" with="" pre-conditioning="" \[3\].="" Alternatively,="" the="" iterative="" methods="" \[4,5\]="" can="" be="" used="" instead="" which="" requires="" the="" solution="" of="" a="" standard="" Lyapunov="" equation="" in="" each="" step="" \[6\]="" +Because="" direct="" methods="" for="" solving="" such="" equations="" have="" a="" numerical="" complexity="" of="" O(n<sup="">6), alternative approaches become mandatory. For example, by mapping the Gramian matrices onto vectors, the generalized Lyapunov equations can be understood as systems of coupled linear equations which can be solved, e. g., by means of the [biconjugate gradient method](http://en.wikipedia.org/wiki/Biconjugate_gradient_method), preferentially with pre-conditioning \[3\]. Alternatively, the iterative methods \[4,5\] can be used instead which requires the solution of a standard Lyapunov equation in each step \[6\]

  [[img src="Lyapunov_3.gif" alt="\begin{array}{rcl}AX_0+X_0\hat{A}^\ast +B\hat{B}^\ast&=&0\\AX_j+X_j\hat{A}^\ast+\sum_{k=1}^m N_kX_{j-1}\hat{N}^\ast_k+B\hat{B}^\ast&=&0,\,j.gt.0\end{array}"]]

@@ -73,7 +73,7 @@
 ℋ2 error norm
 --------------

-The ℋ2 error norm is obtained in the following way \[7\]: First we have to set up norm of the error system which is defined as follows:
+The ℋ2 error norm is obtained in the following way \[3,7\]: First we have to set up norm of the error system which is defined as follows:

 [[img src="Error_1.gif" alt="A_E=\begin{bmatrix}A&0\\0&\hat{A}\end{bmatrix},\,N_{k,E}=\begin{bmatrix}N_k&0\\0&\hat{N}_k\end{bmatrix},\,B_E=\begin{bmatrix}B\\\hat{B}\end{bmatrix},\,C_E=\begin{bmatrix}C&-\hat{C}\end{bmatrix}"]]

Numerics.DimRed modified by Burkhard Schmidt

Burkhard Schmidt — Thu, 22 Jan 2015 13:11:10 -0000

--- v132
+++ v133
@@ -52,15 +52,15 @@

 [[img src="WCO_3.gif" alt="\begin{matrix}AW_C+W_C A^\ast +\sum_{k=1}^mN_kW_CN_k^\ast+BB^\ast&=&0\\A^\ast W_0+W_0A+\sum_{k=1}^m N_k^\ast W_0N_k+C^\ast C&=&0\end{matrix}"]] 

-While direct methods for solving such equations have a numerical complexity of O(n⁶), the iterative methods \[3,4\] can be used instead which requires the solution of a standard Lyapunov equation in each step \[5\]
+Because direct methods for solving such equations have a numerical complexity of O(n⁶), alternative approaches become mandatory. For example, by mapping the Gramian matrices onto vectors, the generalized Lyapunov equations can be understood as sy<tems of="" coupled="" linear="" equations="" which="" can="" be="" solved,="" e.="" g.,="" by="" means="" of="" the="" [biconjugate="" gradient="" method](http:="" en.wikipedia.org="" wiki="" Biconjugate_gradient_method),="" preferentially="" with="" pre-conditioning="" \[3\].="" Alternatively,="" the="" iterative="" methods="" \[4,5\]="" can="" be="" used="" instead="" which="" requires="" the="" solution="" of="" a="" standard="" Lyapunov="" equation="" in="" each="" step="" \[6\]="" [[img="" src="Lyapunov_3.gif" alt="\begin{array}{rcl}AX_0+X_0\hat{A}^\ast +B\hat{B}^\ast&amp;=&amp;0\\AX_j+X_j\hat{A}^\ast+\sum_{k=1}^m N_kX_{j-1}\hat{N}^\ast_k+B\hat{B}^\ast&amp;=&amp;0,\,j.gt.0\end{array}" ]]="" -(and="" similarly="" for="" the="" second="" (dual)="" equation="" for="" the="" observability="" Gramian).="" Convergence="" X<sub="">j→W_C is guaranteed if the eigenvalue of A with the largest (negative) real part is sufficiently separated from the imaginary axis \[4\]. While for linear control systems the Lyapunov equations can always be solved if matrix A is stable (see [here](Numerics.Control)), for the generalized Lyapunov equations we recall that a system is called stable when the system matrix A has only eigenvalues in the open left half complex plane (i.e., excluding the imaginary axis). Stability thus means that there are constants λ , a > 0 such that ||exp(At)|| ≤ λ exp(−at), where ||•|| is a suitable matrix norm. If moreover
+(and similarly for the second (dual) equation for the observability Gramian). Convergence X_j→W_C is guaranteed if the eigenvalue of A with the largest (negative) real part is sufficiently separated from the imaginary axis \[5\]. While for linear control systems the Lyapunov equations can always be solved if matrix A is stable (see [here](Numerics.Control)), for the generalized Lyapunov equations we recall that a system is called stable when the system matrix A has only eigenvalues in the open left half complex plane (i.e., excluding the imaginary axis). Stability thus means that there are constants λ , a > 0 such that ||exp(At)|| ≤ λ exp(−at), where ||•|| is a suitable matrix norm. If moreover

 [[img src="WCO_4.gif" alt="\frac{\lambda^2}{2a}\sum_{k=1}^m||N_k||^2\langle 1"]]

-then controllability and observability Gramians exist. This can be achieved by a suitable scaling u→ηu, N→N/η, B →B/η with real η>1 which leaves the equations of motion invariant but, clearly, not the Gramians. Hence, by increasing η, we drive the system to its linear counterpart. For the limit η→∞, the system matrix N vanishes and we obtain a linear system. For this reason, η should not be chosen too large. Note that also the [shift of the spectrum of A](Numerics.Control) may serve to render the generalized Lyapunov equations solvable \[5\].
+then controllability and observability Gramians exist. This can be achieved by a suitable scaling u→ηu, N→N/η, B →B/η with real η>1 which leaves the equations of motion invariant but, clearly, not the Gramians. Hence, by increasing η, we drive the system to its linear counterpart. For the limit η→∞, the system matrix N vanishes and we obtain a linear system. For this reason, η should not be chosen too large. Note that also the [shift of the spectrum of A](Numerics.Control) may serve to render the generalized Lyapunov equations solvable \[6\].

 Dimension reduction
 -------------------
@@ -73,7 +73,7 @@
 ℋ2 error norm
 --------------

-The ℋ2 error norm is obtained in the following way \[6\]: First we have to set up norm of the error system which is defined as follows:
+The ℋ2 error norm is obtained in the following way \[7\]: First we have to set up norm of the error system which is defined as follows:

 [[img src="Error_1.gif" alt="A_E=\begin{bmatrix}A&0\\0&\hat{A}\end{bmatrix},\,N_{k,E}=\begin{bmatrix}N_k&0\\0&\hat{N}_k\end{bmatrix},\,B_E=\begin{bmatrix}B\\\hat{B}\end{bmatrix},\,C_E=\begin{bmatrix}C&-\hat{C}\end{bmatrix}"]]

@@ -93,12 +93,14 @@

 2. Z. Bai, D. Skoogh: [Lin. Alg. Appl. **415**, 406 (2006)](http://dx.doi.org/10.1016/j.laa.2005.04.032)

-3. E. Wachspress: [Appl. Math. Lett. **1**, 87 (1988)](http://dx.doi.org/10.1016/0893-9659(88)90183-8)
+3. T. Breiten, private communications (2014).

-4. T. Damm: [Numer. Lin. Alg. Appl. **15**, 853 (2008)](http://dx.doi.org/10.1137/110836742)
+4. E. Wachspress: [Appl. Math. Lett. **1**, 87 (1988)](http://dx.doi.org/10.1016/0893-9659(88)90183-8)

-5. B. Schäfer-Bung, C. Hartmann, B. Schmidt, and Ch. Schütte: [J. Chem. Phys. **135**, 014112 (2011)](http://dx.doi.org/10.1063/1.3605243)
+5. T. Damm: [Numer. Lin. Alg. Appl. **15**, 853 (2008)](http://dx.doi.org/10.1137/110836742)

-1. P. Benner, T. Breiten: [SIAM Journal on Matrix Analysis and Application, **33**, 859–885 (2012)](http://dx.doi.org/10.1137/110836742)
+6. B. Schäfer-Bung, C. Hartmann, B. Schmidt, and Ch. Schütte: [J. Chem. Phys. **135**, 014112 (2011)](http://dx.doi.org/10.1063/1.3605243)
+
+7. P. Benner, T. Breiten: [SIAM Journal on Matrix Analysis and Application, **33**, 859–885 (2012)](http://dx.doi.org/10.1137/110836742)

 For a review of the theory of linear control systems, see e.g. the excellent [lecture notes of Umea university](http://www.control.tfe.umu.se/Courses/Optimal_Control_for_Linear_Systems_2010) and/or the [books by Kemin Zhou](http://www.ece.lsu.edu/kemin).