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On computational Poisson geometry II: Numerical methods

  • * Corresponding author: J. C. Ruíz–Pantaleón

    * Corresponding author: J. C. Ruíz–Pantaleón 

This work is partially supported by UNAM–DGAPA–PAPIIT grant IN104819

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  • We present twelve numerical methods for evaluation of objects and concepts from Poisson geometry. We describe how each method works with examples, and explain how it is executed in code. These include methods that evaluate Hamiltonian and modular vector fields, compute the image under the coboundary and trace operators, the Lie bracket of differential 1–forms, gauge transformations, and normal forms of Lie–Poisson structures on $ {\mathbf{R}^{{3}}} $. The complexity of each of our methods is calculated, and we include experimental verifications on examples in dimensions two and three.

    Mathematics Subject Classification: 53–04, 53D17, 65P10.

    Citation:

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  • Figure 2.1.  Left: Symplectic foliation of $ {\Pi_{\mathfrak{so}(3)}} $ in (5). Right: Modular vector field of $ \Pi $ in (15) relative to the Euclidean volume form on $ {\mathbf{R}^{{3}}} $. The color scale indicates the magnitude of the vectors

    Figure 2.2.  Left: Symplectic foliation of $ {\Pi_{\mathfrak{sl}(2)}} $ in (7). Right: Vector field W in (14), tangent to the symplectic foliation of $ {\Pi_{\mathfrak{sl}(2)}} $ on $ {\mathbf{R}^{{3}}_{x} \setminus \{x_{3}{\rm{–axis}}\}} $. The color scale indicates the magnitude of the vectors

    Figure 2.3.  Symplectic (open book) foliation of $ \Pi $ in (20)

    Figure 3.1.  Log–log graphs of the execution time in seconds versus the number of points in $ {10^{\kappa}} $ –point (irregular) meshes of the $\mathtt{NumPoissonGeometry}$ functions 1–10 in Table 3.3, for $ {\kappa = 3, \ldots, 7} $. In red, the fitted linear model used to predict the asymptotic behavior of the runtime for each function, with the corresponding determination coefficient (R–squared) indicated in each legend. We include a zoom-graph in each plot due to the accumulation of runtime values

    Figure 3.2.  Log–log graphs of the execution time in seconds versus the number of points in $ {10^{\kappa}} $ –point (irregular) meshes of the $\mathtt{NumPoissonGeometry}$ functions 1–5 and 8–11 in Table 3.5, for $ {\kappa = 3, \ldots, 7} $. In red, the fitted linear model used to predict the asymptotic behavior of the runtime for each function, with the corresponding determination coefficient (R–squared) indicated in each legend. We include a zoom–graph in each plot due to the accumulation of runtime values

    Figure 3.3.  Log–log graphs of the execution time in seconds versus the number of points in $ {10^{\kappa}} $ –point (irregular) meshes of the $\mathtt{NumPoissonGeometry}$ functions 6 and 7 in Table 3.5, for $ {\kappa = 3, \ldots, 7} $. In red, the fitted model used to predict the asymptotic behavior of the runtime for each function, with the corresponding determination coefficient (R–squared) indicated in each legend. We include a zoom–graph in each plot due to the accumulation of runtime values

    Figure 3.4.  Log-log graph of the execution time in seconds versus the number of points in a $ {10^{\kappa}} $-point (irregular) mesh of the $\mathtt{NumPoissonGeometry}$ function $\mathtt{num\_modular\_vf}$, for $ {\kappa = 4, \ldots, 7} $. In red, the fitted linear model used to predict the asymptotic behavior of the runtime, with the corresponding determination coefficient (R-squared) indicated in the legend

    Figure 3.5.  Log–log graph of the execution time in seconds versus the number of points in a $ {10^{\kappa}} $ –point (irregular) mesh of the $\mathtt{num\_flaschka\_ratiu\_bivector}$ method, for $ {\kappa = 3, \ldots, 7} $. In red, the fitted linear model used to predict the asymptotic behavior of the runtime, with the corresponding determination coefficient (R–squared) indicated in the legend. We include a zoom–graph due to the accumulation of runtime values

    Table 1.1.  Our numerical methods, with their corresponding algorithms, and examples where they are used. The right column is an informal summary of the algorithmic complexities, computed and presented in detail in Section 3

    Method Algorithm Examples Complexity
    $\mathtt{num\_bivector\_field}$ 2.1 [16,7,31,36] O($ m^2 $)
    $\mathtt{num\_bivector\_to\_matrix}$ 2.2 [16,7,31,36] O($ m^2 $)
    $\mathtt{num\_hamiltonian\_vf}$ 2.3 [33,7,50,6,26] O($ m^2 $)
    $\mathtt{num\_poisson\_bracket}$ 2.4 [34,16,31,36] O($ m^2 $)
    $\mathtt{num\_sharp\_morphism}$ 2.5 [16,31,36] O($ m^2 $)
    $\mathtt{num\_coboundary\_operator}$* 2.6 [43,16,31,2,41] O($ 2^m $)
    $\mathtt{num\_modular\_vf}$* 2.7 [1,16,31,27,36,3,44] O($ 2^m $)
    $\mathtt{num\_curl\_operator}$* 2.8 [24,12,16,36] O($ 2^m $)
    $\mathtt{num\_one\_forms\_bracket}$* 2.9 [16,31,36,25] O($ m^2 $)
    $\mathtt{num\_gauge\_transformation}$ 2.10 [8,4,14] O($ m^7 $)
    $\mathtt{num\_linear\_normal\_form\_R3}$* 2.11 [39,43,23,16,48,7,36,14,41,20] O($ m $)
    $\mathtt{num\_flaschka\_ratiu\_bivector}$* 2.12 [24,12,22,49,17] O($ m^6 $)
     | Show Table
    DownLoad: CSV

    Table 3.1.  Worst–case time complexity of $\mathtt{NumPoissonGeometry}$ methods. In the second column: $ m $ denotes the dimension of $ {\mathbf{R}^{{m}}} $, $ k $ is the number of points in a mesh on $ {\mathbf{R}^{{m}}} $, we denote by $ {[\cdot]} $ the integer part function and by $ \mathrm{comb} $ a combination

    Method Time Complexity
    1. $\mathtt{num\_bivector\_field}$ $ { \mathscr{O}(m^2k|bivector|)} $
    2. $\mathtt{num\_bivector\_to\_matrix}$ $ { \mathscr{O}(m^{2}k|bivector|)} $
    3. $\mathtt{num\_hamiltonian\_vf}$ $ { \mathscr{O}(mk(m|bivector| + \mathrm{len}(ham\_function)))} $
    4. $\mathtt{num\_poisson\_bracket}$ $ { \mathscr{O}(mk(m|bivector| + \mathrm{len}(function\_1) + \mathrm{len}(function\_2)))} $
    5. $\mathtt{num\_sharp\_morphism}$ $ { \mathscr{O}(m(mk|bivector| + |one\_form|))} $
    6. $\mathtt{num\_coboundary\_operator}$ $ { \mathscr{O}(\mathrm{comb}(m, [m/2])|bivector|\mathrm{len}(function)(m^{5} + k))} $
    7. $\mathtt{num\_modular\_vf}$ $ { \mathscr{O}(\mathrm{comb}(m, [m/2])|bivector|\mathrm{len}(function)(m+k))} $
    8. $\mathtt{num\_curl\_operator}$ $ { \mathscr{O}(\mathrm{comb}(m, [m/2])|multivector|\mathrm{len}(function)(m+k))} $
    9. $\mathtt{num\_one\_forms\_bracket}$ $ { \mathscr{O}(m^2k|bivector||one\_form\_1||one\_form\_2|)} $
    10. $\mathtt{num\_gauge\_transformation}$ $ { \mathscr{O}(m^2k(m^5 + |bivector| + |two\_form|))} $
    11. $\mathtt{num\_linear\_normal\_form\_R3}$ $ { \mathscr{O}(k|bivector|)} $
    12. $\mathtt{num\_flaschka\_ratiu\_bivector}$ $ { \mathscr{O}(m^{6}k|bivector|)} $
     | Show Table
    DownLoad: CSV

    Table 3.2.  Input data used for the time performance tests of functions 1-10 in Table 3.3

    Function Input Function Input
    1 $ {\Pi_{0}} $ 6 $ {\Pi_{0}} $, $ {W = x_{2} \frac{\partial}{\partial x_{1}} - x_{1} \frac{\partial}{\partial x_{2}}} $
    2 $ {\Pi_{0}} $ 7 $ {\Pi_{0}} $, $ {f=1} $
    3 $ {\Pi_{0}} $, $ {h = x_{1}^{2} + x_{2}^{2}} $ 8 $ {\Pi_{0}} $, $ {f=1} $
    4 $ {\Pi_{0}} $, $ {f = x_{1}^{2} + x_{2}^{2}} $, $ {g=x_{1} + x_{2}} $ 9 $ {\Pi_{0}} $, $ {\alpha = x_{1} \mathrm{d}{x_{1}} + x_{2} \mathrm{d}{x_{2}}} $, $ {\beta = \mathrm{d}{x_{1}} + \mathrm{d}{x_{2}}} $
    5 $ {\Pi_{0}} $, $ {\alpha = x_{1} \mathrm{d}{x_{1}} + x_{2} \mathrm{d}{x_{2}}} $ 10 $ {\Pi_{0}} $, $ {\lambda = \mathrm{d}{x_{1}} \wedge \mathrm{d}{x_{2}}} $
     | Show Table
    DownLoad: CSV

    Table 3.3.  Summary of the time performance of $\mathtt{NumPoissonGeometry}$ functions, in dimension two

    Function Points in mesh/Processing time (in seconds)
    $ 10^{3} $ $ 10^{4} $ $ 10^{5} $ $ 10^{6} $ $ 10^{7} $
    1. $\mathtt{num\_bivector\_field}$ 0.004 $ \pm $ 0.689 0.038 $ \pm $ 0.009 0.356 $ \pm $ 0.002 3.545 $ \pm $ 0.026 35.711 $ \pm $ 0.164
    2. $\mathtt{num\_bivector\_to\_matrix}$ 0.006 $ \pm $ 3.633 0.046 $ \pm $ 0.001 0.438 $ \pm $ 0.004 4.442 $ \pm $ 0.037 45.155 $ \pm $ 1.466
    3. $\mathtt{num\_hamiltonian\_vf}$ 0.014 $ \pm $ 0.001 0.112 $ \pm $ 0.006 1.096 $ \pm $ 0.021 10.867 $ \pm $ 0.044 108.460 $ \pm $ 0.726
    4. $\mathtt{num\_poisson\_bracket}$ 0.021 $ \pm $ 0.006 0.169 $ \pm $ 0.001 1.652 $ \pm $ 0.008 16.721 $ \pm $ 0.049 168.110 $ \pm $ 1.637
    5. $\mathtt{num\_sharp\_morphism}$ 0.014 $ \pm $ 0.658 0.111 $ \pm $ 0.001 1.068 $ \pm $ 0.007 10.725 $ \pm $ 0.142 107.275 $ \pm $ 0.667
    6. $\mathtt{num\_coboundary\_operator}$ 0.001 $ \pm $ 0.087 0.008 $ \pm $ 0.001 0.084 $ \pm $ 0.006 0.848 $ \pm $ 0.011 8.638 $ \pm $ 0.045
    7. $\mathtt{num\_modular\_vf}$ 0.004 $ \pm $ 0.754 0.030 $ \pm $ 0.009 0.280 $ \pm $ 0.001 2.805 $ \pm $ 0.016 28.057 $ \pm $ 0.107
    8. $\mathtt{num\_curl\_operator}$ 0.022 $ \pm $ 0.009 0.196 $ \pm $ 0.024 1.923 $ \pm $ 0.004 18.487 $ \pm $ 0.136 182.774 $ \pm $ 1.260
    9. $\mathtt{num\_one\_forms\_bracket}$ 0.058 $ \pm $ 0.006 0.420 $ \pm $ 0.007 4.278 $ \pm $ 0.027 43.257 $ \pm $ 0.071 434.450 $ \pm $ 0.589
    10. $\mathtt{num\_gauge\_transformation}$ 0.051 $ \pm $ 0.001 0.446 $ \pm $ 0.010 4.380 $ \pm $ 0.016 43.606 $ \pm $ 0.212 434.704 $ \pm $ 1.234
     | Show Table
    DownLoad: CSV

    Table 3.4.  Input data used for the time performance tests of functions 1–11 in Table 3.5

    Function Input
    1 $ \phantom{-}\Pi_{\mathfrak{sl}(2)} $
    2 $ \phantom{-}\Pi_{\mathfrak{sl}(2)} $
    3 $ \phantom{-}\Pi_{\mathfrak{sl}(2)} $, $ {h = x_{1}^{2} + x_{2}^{2} - x_{3}^{2}} $
    4 $ \phantom{-}\Pi_{\mathfrak{sl}(2)} $, $ {f = x_{1}^{2} + x_{2}^{2} - x_{3}^{2}} $, $ {g=x_{1} + x_{2} + x_{3}} $
    5 $ \phantom{-}\Pi_{\mathfrak{sl}(2)} $, $ {\alpha = x_{1} \mathrm{d}{x_{1}} + x_{2} \mathrm{d}{x_{2}} - x_{3} \mathrm{d}{x_{3}}} $
    6 $ \phantom{-}\Pi_{\mathfrak{sl}(2)} $, $ {W = e^{{-1}/{(x_1^2 + x_2^2 - x_3^2)^2}} [ {x_1x_{3}}/(x_1^2 + x_2^2)\frac{\partial}{\partial{x_{1}}} + {x_2x_{3}}/(x_1^2 + x_2^2)\frac{\partial}{\partial{x_{2}}} + \frac{\partial}{\partial{x_{3}}} ]} $
    7 $ \phantom{-}\Pi_{\mathfrak{sl}(2)} $, $ {f=1} $
    8 $ \phantom{-}\Pi_{\mathfrak{sl}(2)} $, $ {f=1} $
    9 $ \phantom{-}\Pi_{\mathfrak{sl}(2)} $, $ {\alpha = x_{1} \mathrm{d}{x_{1}} + x_{2} \mathrm{d}{x_{2}} - x_{3} \mathrm{d}{x_{3}}} $, $ {\beta = \mathrm{d}{x_{1}} + \mathrm{d}{x_{2}} + \mathrm{d}{x_{3}}} $
    10 $ \phantom{-}\Pi_{\mathfrak{sl}(2)} $, $ {\lambda = (x_{2}-x_{1}) \mathrm{d}{x_{1}} \wedge \mathrm{d}{x_{2}} + (x_{3}-x_{1}) \mathrm{d}{x_{1}} \wedge \mathrm{d}{x_{3}} + (x_{2}-x_{3}) \mathrm{d}{x_{2}} \wedge \mathrm{d}{x_{3}}} $
    11 $ -\Pi_{\mathfrak{sl}(2)} $
     | Show Table
    DownLoad: CSV

    Table 3.5.  Summary of the time performance of $\mathtt{NumPoissonGeometry}$ functions in dimension 3

    Function Points in mesh/Processing time (in seconds)
    $ 10^{3} $ $ 10^{4} $ $ 10^{5} $ $ 10^{6} $ $ 10^{7} $
    1. $\mathtt{num\_bivector\_field}$ 0.009 $ \pm $ 0.009 0.051 $ \pm $ 0.004 0.496 $ \pm $ 0.002 4.984 $ \pm $ 0.023 49.565 $ \pm $ 0.222
    2. $\mathtt{num\_bivector\_to\_matrix}$ 0.008 $ \pm $ 3.164 0.057 $ \pm $ 0.002 0.553 $ \pm $ 0.019 5.442 $ \pm $ 0.023 55.249 $ \pm $ 1.690
    3. $\mathtt{num\_hamiltonian\_vf}$ 0.017 $ \pm $ 0.002 0.129 $ \pm $ 0.001 1.263 $ \pm $ 0.022 12.518 $ \pm $ 0.064 126.091 $ \pm $ 0.583
    4. $\mathtt{num\_poisson\_bracket}$ 0.036 $ \pm $ 0.001 0.299 $ \pm $ 0.010 2.936 $ \pm $ 0.067 29.600 $ \pm $ 0.933 292.625 $ \pm $ 6.094
    5. $\mathtt{num\_sharp\_morphism}$ 0.017 $ \pm $ 0.006 0.128 $ \pm $ 0.005 1.252 $ \pm $ 0.005 12.384 $ \pm $ 0.038 124.851 $ \pm $ 1.809
    6. $\mathtt{num\_coboundary\_operator}$ 1.589 $ \pm $ 0.016 1.705 $ \pm $ 0.029 2.815 $ \pm $ 0.032 12.972 $ \pm $ 0.166 111.034 $ \pm $ 1.365
    7. $\mathtt{num\_modular\_vf}$ 0.050 $ \pm $ 0.001 0.103 $ \pm $ 0.004 0.645 $ \pm $ 0.006 6.025 $ \pm $ 0.013 59.652 $ \pm $ 0.146
    8. $\mathtt{num\_curl\_operator}$ 0.019 $ \pm $ 0.010 0.129 $ \pm $ 0.027 1.199 $ \pm $ 0.032 10.911 $ \pm $ 0.181 105.841 $ \pm $ 1.230
    9. $\mathtt{num\_one\_forms\_bracket}$ 0.093 $ \pm $ 0.001 0.738 $ \pm $ 0.007 7.285 $ \pm $ 0.159 72.802 $ \pm $ 1.474 724.514 $ \pm $ 13.594
    10. $\mathtt{num\_gauge\_transformation}$ 0.051 $ \pm $ 0.001 0.445 $ \pm $ 0.010 4.395 $ \pm $ 0.013 43.794 $ \pm $ 0.173 437.326 $ \pm $ 0.824
    11. $\mathtt{num\_linear\_normal\_form\_R3}$ 0.016 $ \pm $ 0.438 0.061 $ \pm $ 0.002 0.504 $ \pm $ 0.012 4.903 $ \pm $ 0.017 48.786 $ \pm $ 0.219
     | Show Table
    DownLoad: CSV

    Table 3.6.  Mean time in seconds (with standard deviation) it takes to evaluate the $\mathtt{num\_flaschka\_ratiu\_bivector}$ method on a irregular mesh on $ \mathbf{R}^{{4}} $ with $ 10^{\kappa} $ points, computed by taking twenty-five samples, for $ {\kappa = 3, \ldots, 7} $

    Function Points in mesh/Processing time (in seconds)
    $ 10^{3} $ $ 10^{4} $ $ 10^{5} $ $ 10^{6} $ $ 10^{7} $
    $\mathtt{num\_flaschka\_ratiu\_bivector}$ 0.0158 $ \pm $ 0.105 0.057 $ \pm $ 0.003 0.505 $ \pm $ 0.003 4.993 $ \pm $ 0.029 49.563 $ \pm $ 0.207
     | Show Table
    DownLoad: CSV
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