On computational Poisson geometry II: Numerical methods

Miguel Ángel Evangelista-Alvarado; José Crispín Ruíz-Pantaleón; Pablo Suárez-Serrato

doi:10.3934/jcd.2021012

Article Contents

2021, Volume 8, Issue 3: 273-307. Doi: 10.3934/jcd.2021012

This issue Previous Article Structure preserving discretization of time-reparametrized Hamiltonian systems with application to nonholonomic mechanics Next Article On polynomial forms of nonlinear functional differential equations

On computational Poisson geometry II: Numerical methods

Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, Ciudad Universitaria, Coyoacán, 04510, Mexico City, Mexico

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

Received: September 30, 2020

Revised: February 28, 2021

Early access: May 2021

Published: July 2021

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

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Abstract

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.

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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

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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

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

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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

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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

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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

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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

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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

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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 $)

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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\|)} $

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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}}} $

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

Function	Points in mesh/Processing time (in seconds)
Function	$ 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

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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)} $

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

Function	Points in mesh/Processing time (in seconds)
Function	$ 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

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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)
Function	$ 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

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