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Notebook[{
Cell[CellGroupData[{
Cell["Parametric Model Reduction Tutorial", "Title"],
Cell[BoxData[
RowBox[{\(Evgenii\ B . \ Rudnyi\), ",",
ButtonBox[\(\(\(http\)\(:\)\) // \(\(Evgenii . Rudnyi . Ru\)\(/\)\)\),
ButtonData:>{
URL[ "http://Evgenii.Rudnyi.Ru/"], None},
ButtonStyle->"Hyperlink"]}]], "Text"],
Cell[CellGroupData[{
Cell["Introduction", "Subtitle"],
Cell[TextData[{
"This notebook allows you to repeat computations described in the paper\n\n\
E. B. Rudnyi, L. H. Feng, M. Salleras, S. Marco, J. G. Korvink. Error \
Indicator to Automatically Generate Dynamic Compact Parametric Thermal \
Models. THERMINIC 2005, 11th International Workshop on Thermal Investigations \
of ICs and Systems, 27 - 30 September 2005, Belgirate, Lake Maggiore, Italy, \
p. 139 - 145. ",
ButtonBox["Preprint",
ButtonData:>{
URL[
"http://www.imtek.uni-freiburg.de/simulation/mor4ansys/pdf/\
rudnyi05therminic.pdf"], None},
ButtonStyle->"Hyperlink"],
" is at \
http://www.imtek.uni-freiburg.de/simulation/mor4ansys/pdf/rudnyi05therminic.\
pdf\n\nThe goal was to preserve three film coefficients for a simple thermal \
problem during model reduction process by means of multivariate expansion. \
The engineering problem is described in\n\nE. B. Rudnyi and J. G. Korvink. \
Boundary Condition Independent Thermal Model. In: Benner, P., Mehrmann, V., \
Sorensen, D. (eds) Dimension Reduction of Large-Scale Systems, Lecture Notes \
in Computational Science and Engineering (LNCSE). Springer-Verlag, \
Berlin/Heidelberg, Germany, v. 45, p. 345-348, 2005. ",
ButtonBox["Preprint",
ButtonData:>{
URL[ "http://www.imtek.uni-freiburg.de/simulation/benchmark/"], None},
ButtonStyle->"Hyperlink"],
" is at http://www.imtek.uni-freiburg.de/simulation/benchmark/\n\nThe \
system matrices are available from ",
ButtonBox["Oberwolfach Benchmark Collection",
ButtonData:>{
URL[ "http://www.imtek.uni-freiburg.de/simulation/benchmark/"], None},
ButtonStyle->"Hyperlink"],
" (for convenience they are included in the distribution along with this \
notebook)."
}], "Text"],
Cell["\<\
The original dynamic model after the discretization by the finite \
element method is as follows
E dx/dt + (K + ht Kt + hs Ks + hb Kb) x = Bu
The goal is to find a low-dimensional model in the similar form that \
approximates the dynamic properties of the original system in a wide range of \
film coefficients, ht, hs, hb.
It is assumed that you browse the papers above before starting the tutorial.
From a computational viewpoint, several Krylov subspaces are computed and \
then merged to form a projection basis. Several points in the parameter space \
of the transfer function are employed to choose an optimal dimension for each \
Krylov subspace. \
\>", "Text"]
}, Open ]],
Cell[CellGroupData[{
Cell["Setting up", "Subtitle"],
Cell["\<\
Make the directory with system matrices current. You may need to \
change the next expression.\
\>", "Text"],
Cell[BoxData[
\(SetDirectory["\<.\>"]\)], "Input"],
Cell[TextData[{
"Load the functions. They are included for convenience but the latest \
versions as well as documentation are available from my ",
ButtonBox["site",
ButtonData:>{
URL[ "http://evgenii.rudnyi.ru/soft/Post4MOR.tar.gz"], None},
ButtonStyle->"Hyperlink"],
" at http://evgenii.rudnyi.ru/soft/Post4MOR.tar.gz. The content of the \
archive is also available ",
ButtonBox["online",
ButtonData:>{
URL[ "http://evgenii.rudnyi.ru/soft/Post4MOR/"], None},
ButtonStyle->"Hyperlink"],
" at http://evgenii.rudnyi.ru/soft/Post4MOR/. "
}], "Text"],
Cell[BoxData[
\(<< Post4MOR.m\)], "Input"],
Cell[BoxData[
\(<< ModelReduction.m\)], "Input"],
Cell["\<\
The first set of functions that is in the file Post4MOR.m is well \
documented. The goal of the second set was to add new functionality with the \
minimal development cost. As a result, my design was to reuse functions from \
Post4MOR.m as much as possible. Documentation for ModelReduction.m is in the \
beginning of the file.\
\>", "Text"],
Cell[TextData[{
"Turn off some especially annoying ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" warning messages."
}], "Text"],
Cell[BoxData[{
\(\(Off[General::"\"];\)\), "\[IndentingNewLine]",
\(\(Off[General::"\"];\)\)}], "Input"]
}, Open ]],
Cell[CellGroupData[{
Cell["Reading in and Preparing the Original System", "Subtitle"],
Cell["\<\
The model in the benchmark was written as
E dx/dt = (A - hbottom Abottom -htop Atop -hside Aside)x + Bu
I convert it to the form used in the paper
E dx/dt + (K + ht Kt + hs Ks + hb Kb) x = Bu
K = -A
Kt = Atop
Ks = Aside
Kb = Abottom
Also, I leave only one output to reduce the number of plots. However, the \
procedure will work for many outputs as well.\
\>", "Text"],
Cell[BoxData[{
\(\(matE\ = \
Import["\", "\"];\)\), "\[IndentingNewLine]",
\(\(matK\ = \ \(-Import["\", "\"]\);\)\), "\
\[IndentingNewLine]",
\(\(matB\ = \
Import["\", "\"];\)\), "\[IndentingNewLine]",
\(\(matKt\ = \
Import["\", "\"];\)\), "\[IndentingNewLine]",
\(\(matKs\ = \
Import["\", "\"];\)\), "\[IndentingNewLine]", \
\(\(matKb\ = \
Import["\", "\"];\)\), \
"\[IndentingNewLine]",
\(\(matC\ = \
Take[Import["\", "\"], 1,
Length[matK]];\)\)}], "Input"],
Cell["\<\
Function MakeParametricSystem takes as an input a list of system \
matrices but we also have to define three functions to convert this list \
(mat) to three system matrices of DynamicSystem for a given list of \
parameters (par).\
\>", "Text"],
Cell[BoxData[{
\(\(getM = Function[{mat, par}, Null];\)\), "\[IndentingNewLine]",
\(\(getE =
Function[{mat, par}, mat[\([1]\)]];\)\), "\[IndentingNewLine]",
\(\(getK =
Function[{mat, par},
mat[\([2]\)] + par[\([1]\)]*mat[\([3]\)] +
par[\([2]\)]*mat[\([4]\)] +
par[\([3]\)]*mat[\([5]\)]];\)\)}], "Input"],
Cell[BoxData[
\(\(psys =
MakeParametricSystem[{matE, matK, matKt, matKs, matKb}, matB,
matC, {"\"}, {getM, getE, getK}];\)\)], "Input"],
Cell["\<\
For better readability of the code below, I define several \
additional function to access system matrices from ParametricSystem.\
\>", \
"Text"],
Cell[BoxData[{
\(MatrixE[sys_ParametricSystem] :=
Matrix[sys, 1]\), "\[IndentingNewLine]",
\(MatrixK[sys_ParametricSystem] :=
Matrix[sys, 2]\), "\[IndentingNewLine]",
\(MatrixKt[sys_ParametricSystem] :=
Matrix[sys, 3]\), "\[IndentingNewLine]",
\(MatrixKs[sys_ParametricSystem] :=
Matrix[sys, 4]\), "\[IndentingNewLine]",
\(MatrixKb[sys_ParametricSystem] := Matrix[sys, 5]\)}], "Input"]
}, Open ]],
Cell[CellGroupData[{
Cell["\<\
Computing Transfer Function for Several Points in the Parameter \
Space\
\>", "Subtitle"],
Cell[TextData[{
"The question how to choose the points to check the local error is open. In \
the paper, the points along diagonals in the parameter space were used with \
an assumption that provided the approximation error is small for these points \
it is also small for internal points. \n\nWe compute\n\nH{s = 100, ht = 10, \
hb = 10, hs = 10}\nH{s = 100, ht = ",
Cell[BoxData[
\(TraditionalForm\`10\^6\)]],
", hb = 10, hs = 10}\nH{s = 100, ht = ",
Cell[BoxData[
\(TraditionalForm\`10\^6\)]],
", hb = ",
Cell[BoxData[
\(TraditionalForm\`10\^6\)]],
", hs = 10}\nH{s = 100, ht = ",
Cell[BoxData[
\(TraditionalForm\`10\^6\)]],
", hb = 10, hs = ",
Cell[BoxData[
\(TraditionalForm\`10\^6\)]],
"}"
}], "Text"],
Cell[TextData[{
"YSeries returns a matrix with the number of rows equal to the number of \
outputs and the number of columns equal to the number of simulation points \
(frequencies). As ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" stores matrices by rows and we need a column, I use Transpose. First \
takes the first row from the matrix."
}], "Text"],
Cell[BoxData[{
\(\(maxfreq = 100;\)\), "\[IndentingNewLine]",
\(\(p1 = {10, 10, 10};\)\), "\[IndentingNewLine]",
\(\(exact1\ =
First[Transpose[
YSeries[\[IndentingNewLine]HarmonicSolution[{maxfreq},
MakeDynamicSystem[psys,
p1]]\[IndentingNewLine]]]];\)\), "\[IndentingNewLine]",
\(\(p2 = {1*^6, 10, 10};\)\), "\[IndentingNewLine]",
\(\(exact2\ = \
First[Transpose[
YSeries[\[IndentingNewLine]HarmonicSolution[{maxfreq},
MakeDynamicSystem[psys,
p2]]\[IndentingNewLine]]]];\)\), "\[IndentingNewLine]",
\(\(p3 = {1*^6, 1*^6, 10};\)\), "\[IndentingNewLine]",
\(\(exact3\ = \
First[Transpose[
YSeries[\[IndentingNewLine]HarmonicSolution[{maxfreq},
MakeDynamicSystem[psys,
p3]]\[IndentingNewLine]]]];\)\), "\[IndentingNewLine]",
\(\(p4 = {1*^6, 1*^6, 1*^6};\)\), "\[IndentingNewLine]",
\(\(exact4\ = \
First[Transpose[
YSeries[\[IndentingNewLine]HarmonicSolution[{maxfreq},
MakeDynamicSystem[psys,
p4]]\[IndentingNewLine]]]];\)\)}], "Input"],
Cell["\<\
It could be good to take more points but in the general case, the \
computations with the original system are quite expensive.\
\>", "Text"]
}, Open ]],
Cell[CellGroupData[{
Cell["Preparing Common Things", "Subtitle"],
Cell["\<\
An expansion point for the Laplace variable is zero, as this allows \
us to preserve the stationary solution. However, the expansion point for each \
parameter is 10. As a result, it is necessary to convert MatrixK to a new \
matrix.\
\>", "Text"],
Cell["Transformation to hb0 = 10, hs0 = 10, hb0 = 10.", "Text"],
Cell[BoxData[
\(matKnew\ = \
MatrixK[psys]\ + \
10. *\((MatrixKt[psys] + MatrixKs[psys] +
MatrixKb[psys])\)\)], "Input"],
Cell["\<\
Note that the dimension of the original model is equal to \
4257.\
\>", "Text"],
Cell["\<\
Matrix is factorized only once. Later on this factor is used in \
order to quickly solve a system of equations matKnew x = b.\
\>", "Text"],
Cell[BoxData[
\(\(invKnew =
LinearSolve[matKnew, Method -> "\"];\)\)], "Input"],
Cell["\<\
The starting vector will be the same for all Krylov \
subspaces.\
\>", "Text"],
Cell[BoxData[
\(\(start = invKnew[MatrixB[psys] . {1}];\)\)], "Input"],
Cell[TextData[{
"I multiply the input matrix by {1} in order to convert it from a matrix to \
a vector. In ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" a matrix consisting from a single column is different from a vector: a \
matrix 3x1 is {{1}, {2}, {3}} and a vector is {1, 2, 3}."
}], "Text"],
Cell["\<\
We will make a similar plots several times and below there are \
options to make it look nicer.\
\>", "Text"],
Cell[BoxData[
\(\(plotoptions = {SymbolShape \[Rule] {PlotSymbol[Triangle,
Filled \[Rule] False], PlotSymbol[Box, Filled \[Rule] False],
PlotSymbol[Diamond, Filled \[Rule] False],
PlotSymbol[Star, Filled \[Rule] False]},
SymbolStyle \[Rule] {RGBColor[1, 0, 0], RGBColor[0, 1, 0],
RGBColor[0, 0, 1], RGBColor[0, 1, 1]},
FrameLabel \[Rule] {"\", "\<\>"},
PlotLegend \[Rule] {"\<10,10,10\>", "\<1e6,10,10\>", \
"\<1e6,1e6,10\>", "\<1e6,1e6,1e6\>"}, LegendBorder \[Rule] {},
LegendPosition \[Rule] {\(-0.7\), \(-0.4\)},
LegendSize \[Rule] {0.8, 0.28},
Evaluate[Imtek`Post4MOR`Private`defaultPlotOptions]};\)\)], "Input"]
}, Open ]],
Cell[CellGroupData[{
Cell["Generating Krylov Subspace Along the Laplace variable", "Subtitle"],
Cell[TextData[{
"The function to compute the next vector is as follows \n\nnext = Knew^(-1) \
MatrixE prev \n\nand it is defined first. ArnoldiProcess uses this function \
and the starting vector defined above in order to compute the orthogonalized \
Krylov subspace \n\nKrylov{Knew^(-1) MatrixE v, Knew^(-1) f}\n\nThe \
projection matrix is returned in the transposed form as ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" stores matrices by rows. The maximum dimension 30 is taken based on \
previous experience."
}], "Text"],
Cell[BoxData[{
\(\(fun = Function[{prev}, invKnew[MatrixE[psys] . prev]];\)\), "\n",
\(\(V = ArnoldiProcess[fun, start, 30];\)\), "\n",
\(Dimensions[V]\)}], "Input"],
Cell["Now we project a parametric system onto the basis V", "Text"],
Cell[BoxData[
\(\(pred = ProjectSystem[psys, V];\)\)], "Input"],
Cell["\<\
to make a reduced parametric system. Then we compute the relative \
error between the original system and the reduced system for the four \
different parameter sets defined above as a function of the system dimension. \
MakeDynamicSystem converts ParametricSystem to DynamicSystem for given set of \
parameters, FrequencyConvergence runs HarmonicSimulation for maxfreq for \
reduced systems with different dimensions from 1 to Length[MatrixK[red]]. \
Local error computes the relative error between results from reduced systems \
and the original system.\
\>", "Text"],
Cell[BoxData[{
\(\(red1 = MakeDynamicSystem[pred, p1];\)\), "\[IndentingNewLine]",
\(\(er1 =
LocalError[FrequencyConvergence[maxfreq, red1],
exact1];\)\), "\[IndentingNewLine]",
\(\(red2 = MakeDynamicSystem[pred, p2];\)\), "\[IndentingNewLine]",
\(\(er2 =
LocalError[FrequencyConvergence[maxfreq, red2],
exact2];\)\), "\[IndentingNewLine]",
\(\(red3 = MakeDynamicSystem[pred, p3];\)\), "\[IndentingNewLine]",
\(\(er3 =
LocalError[FrequencyConvergence[maxfreq, red3],
exact3];\)\), "\[IndentingNewLine]",
\(\(red4 = MakeDynamicSystem[pred, p4];\)\), "\[IndentingNewLine]",
\(\(er4 =
LocalError[FrequencyConvergence[maxfreq, red4],
exact4];\)\)}], "Input"],
Cell[BoxData[
\(\(MultipleListPlot[{er1, er2, er3, er4},
Evaluate[plotoptions]];\)\)], "Input"],
Cell["\<\
We see that the relative error for the set p1 decreases while the \
relative error for other sets (p2, p3, p4) stays high. This can be easily \
explained because we generate vectors for the Laplace variable and they do \
not approximate the system behavior well when film coefficients change \
considerably.
In the paper, the local approximation error of 1% was considered to be good \
enough and the dimension 28 was chosen as the optimal for this \
subspace.\
\>", "Text"],
Cell[BoxData[{
\(\(V\ = \ Take[V, \ 28];\)\), "\[IndentingNewLine]",
\(Dimensions[V]\)}], "Input"]
}, Open ]],
Cell[CellGroupData[{
Cell["\<\
Generating Krylov Subspace Along the parameter ht (film coefficient \
at the top).\
\>", "Subtitle"],
Cell["\<\
The function to compute the next vector now is as follows
next = Knew^(-1) MatrixKt prev
The maximum dimension 30 is taken rather arbitrary.\
\>", "Text"],
Cell[BoxData[{
\(\(fun = Function[{prev}, invKnew[MatrixKt[psys] . prev]];\)\), "\n",
\(\(W = ArnoldiProcess[fun, start, 30];\)\), "\n",
\(Dimensions[W]\)}], "Input"],
Cell["We merge subspaces V and W", "Text"],
Cell[BoxData[{
\(\(V\ = \ Orthogonalize[V, W];\)\), "\[IndentingNewLine]",
\(Dimensions[V]\)}], "Input"],
Cell["\<\
One vector is deflated as the starting vector was the same for both \
subspaces.\
\>", "Text"],
Cell["\<\
Now we project a parametric system onto the new basis V and compute \
relative errors again\
\>", "Text"],
Cell[BoxData[
\(\(pred = ProjectSystem[psys, V];\)\)], "Input"],
Cell[BoxData[{
\(\(red1 = MakeDynamicSystem[pred, p1];\)\), "\[IndentingNewLine]",
\(\(er1 =
LocalError[FrequencyConvergence[maxfreq, red1],
exact1];\)\), "\[IndentingNewLine]",
\(\(red2 = MakeDynamicSystem[pred, p2];\)\), "\[IndentingNewLine]",
\(\(er2 =
LocalError[FrequencyConvergence[maxfreq, red2],
exact2];\)\), "\[IndentingNewLine]",
\(\(red3 = MakeDynamicSystem[pred, p3];\)\), "\[IndentingNewLine]",
\(\(er3 =
LocalError[FrequencyConvergence[maxfreq, red3],
exact3];\)\), "\[IndentingNewLine]",
\(\(red4 = MakeDynamicSystem[pred, p4];\)\), "\[IndentingNewLine]",
\(\(er4 =
LocalError[FrequencyConvergence[maxfreq, red4],
exact4];\)\)}], "Input"],
Cell[BoxData[
\(\(MultipleListPlot[{er1, er2, er3, er4},
Evaluate[plotoptions]];\)\)], "Input"],
Cell["\<\
We see that now the local error defined for the film coefficient ht \
goes down. Surprisingly, the local error for other two film coefficients goes \
down simultaneously. This can be explained that in our case study the film \
coefficient at the top plays the major role.
Based on the convergence plot, we take the dimension of the reduced system of \
41.\
\>", "Text"],
Cell[BoxData[{
\(\(V\ = \ Take[V, \ 41];\)\), "\[IndentingNewLine]",
\(Dimensions[V]\)}], "Input"]
}, Open ]],
Cell[CellGroupData[{
Cell["\<\
Generating Krylov Subspace Along the parameter hs (film coefficient \
at the side).\
\>", "Subtitle"],
Cell["\<\
The function to compute the next vector now is as follows
next = Knew^(-1) MatrixKs prev
The maximum dimension 30 is taken rather arbitrary.\
\>", "Text"],
Cell[BoxData[{
\(\(fun = Function[{prev}, invKnew[MatrixKs[psys] . prev]];\)\), "\n",
\(\(W = ArnoldiProcess[fun, start, 30];\)\), "\n",
\(Dimensions[W]\)}], "Input"],
Cell["We merge subspaces V and W", "Text"],
Cell[BoxData[{
\(\(V\ = \ Orthogonalize[V, W];\)\), "\[IndentingNewLine]",
\(Dimensions[V]\)}], "Input"],
Cell["\<\
One vector is deflated as the starting vector was the same for both \
subspaces.\
\>", "Text"],
Cell["\<\
Now we project a parametric system onto the new basis V and compute \
relative errors again\
\>", "Text"],
Cell[BoxData[
\(\(pred = ProjectSystem[psys, V];\)\)], "Input"],
Cell[BoxData[{
\(\(red1 = MakeDynamicSystem[pred, p1];\)\), "\[IndentingNewLine]",
\(\(er1 =
LocalError[FrequencyConvergence[maxfreq, red1],
exact1];\)\), "\[IndentingNewLine]",
\(\(red2 = MakeDynamicSystem[pred, p2];\)\), "\[IndentingNewLine]",
\(\(er2 =
LocalError[FrequencyConvergence[maxfreq, red2],
exact2];\)\), "\[IndentingNewLine]",
\(\(red3 = MakeDynamicSystem[pred, p3];\)\), "\[IndentingNewLine]",
\(\(er3 =
LocalError[FrequencyConvergence[maxfreq, red3],
exact3];\)\), "\[IndentingNewLine]",
\(\(red4 = MakeDynamicSystem[pred, p4];\)\), "\[IndentingNewLine]",
\(\(er4 =
LocalError[FrequencyConvergence[maxfreq, red4],
exact4];\)\)}], "Input"],
Cell[BoxData[
\(\(MultipleListPlot[{er1, er2, er3, er4},
Evaluate[plotoptions]];\)\)], "Input"],
Cell["\<\
We can see that the new subspace does not reduce the local error \
and we leave the dimension of the reduced system of 41.\
\>", "Text"],
Cell[BoxData[{
\(\(V\ = \ Take[V, \ 41];\)\), "\[IndentingNewLine]",
\(Dimensions[V]\)}], "Input"],
Cell["\<\
The expansion along hb (the file coefficient at the bottom) \
produces the same result, that is, no further reduction in the local error. \
As a result, the basis of the dimension 41 is considered to be the optimal \
one. The optimal parametric reduced is system is\
\>", "Text"],
Cell[BoxData[
\(\(pred = ProjectSystem[psys, V];\)\)], "Input"]
}, Open ]],
Cell[CellGroupData[{
Cell["Comparing the Reduced and Original systems", "Subtitle"],
Cell["\<\
The procedure above is empirical by its nature. There were two main \
assumptions made:
1) One can ignore mixed moments.
2) The relative error at a few border points in the parameter space is enough \
to choose the dimension of the reduced model.
In our experience, this was working okay for the problem in question but it \
is unclear whether this will work for other systems. Unfortunately, \
mathematical proofs to estimate errors in the case of parametric model \
reduction are missing.
Below, an empirical check of the quality of the reduced system is \
performed.\
\>", "Text"],
Cell["\<\
Function that compares the static, harmonic response and transient \
solution of the original and reduced systems. We compute relative error in \
per cent as follows Norm[full-red]/Norm[full]*100 for each output and then \
the maximum values is taken to the final table.\
\>", "Text"],
Cell[BoxData[{
\(error[r2_SimulationResult, r1_SimulationResult] :=
MapThread[
Norm[#1 - #2]/Norm[#2] &, {YSeries[r2],
YSeries[r1]}]\[IndentingNewLine]\), "\[IndentingNewLine]",
\(compareSystems[psys_ParametricSystem, pred_ParametricSystem, par_List] :=
Module[{sys, red, r1, r2, relS, relT, relH,
time = Table[10^i, {i, \(-5\), 4, 0.1}],
freq = Table[10^i, {i, \(-4\), 3, 0.5}]}, \[IndentingNewLine]Print[
par]; \[IndentingNewLine]sys =
MakeDynamicSystem[psys, par]; \[IndentingNewLine]red =
MakeDynamicSystem[pred, par]; \[IndentingNewLine]r1 =
StationarySolution[red]; \[IndentingNewLine]r2 =
StationarySolution[sys]; \[IndentingNewLine]relS =
Abs[r1 - r2]/
r2*100; \[IndentingNewLine]Print["\", relS]; \[IndentingNewLine]r1 =
AnsysTransientSolution[time, red]; \[IndentingNewLine]r2 =
AnsysTransientSolution[time,
sys]; \[IndentingNewLine]PlotResult[{r2, r1},
PlotStyle \[Rule] {RGBColor[1, 0, 0], RGBColor[0, 1, 0]},
FunctionX \[Rule] Log10,
CommonTitle -> "\"]; \
\[IndentingNewLine]PlotResult[
Difference[r2, r1,
ErrorFunction \[Rule] \((Log10[Abs[#1 - #2]/Abs[#1]] &)\)],
PlotStyle \[Rule] {RGBColor[0, 1, 0]}, FunctionX \[Rule] Log10,
CommonTitle -> "\"]; \
\[IndentingNewLine]relT =
error[r2,
r1]*100; \[IndentingNewLine]Print["\", relT]; \[IndentingNewLine]r1 = HarmonicSolution[freq, red];
r2 = HarmonicSolution[freq, sys]; \[IndentingNewLine]PlotResult[{r2,
r1}, FunctionY \[Rule] \((Log10[Abs[#]] &)\),
FunctionX \[Rule] Log10,
PlotStyle \[Rule] {RGBColor[1, 0, 0], RGBColor[0, 1, 0]},
CommonTitle -> "\"];
PlotResult[
Difference[r2, r1,
ErrorFunction \[Rule] \((Log10[Abs[#1 - #2]/Abs[#1]] &)\)],
FunctionX \[Rule] Log10, PlotStyle \[Rule] {RGBColor[0, 1, 0]},
CommonTitle -> "\"];
relH =
error[r2,
r1]*100; \[IndentingNewLine]Print["\",
relH]; \[IndentingNewLine]Flatten[{par, Max[relS], Max[relT],
Max[relH]}]]\)}], "Input"],
Cell["\<\
Below there is a check the function above for some parameter \
values. It takes some time as it makes simulation for the original system as \
well. First plot contains two curves: a red line for the original system and \
a green line for the reduced system. As the difference is small, one can see \
red just a little bit. The second plot gives the relative difference between \
the two curves.\
\>", "Text"],
Cell[BoxData[
\(compareSystems[psys, pred, {1, 100, 1000}]\)], "Input"],
Cell["\<\
Now we make a loop over different combinations of parameters. \
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We can see that the low-dimensional system of dimension 41 does \
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