MomentGauge.Statistic.PolyGaugedStatistics#

Module Contents#

Classes#

PolyGaugedStatistics

The base class for store pre-defined gauged polynomial statistics.

Maxwellian_1D_gauged_stats

The 1D version of polynomial statistics for 35 moments with gauge transformation.

PolyGaugedStatistics_sr_sx_wx

The base class for store pre-defined gauged polynomial statistics with gauge parameter \(s_r\), \(s_x\), and \(w_x\)

ESBGK_1D_gauged_stats

The 1D version of polynomial statistics for ESBGK moments with gauge transformation.

M35_1D_gauged_stats

The 1D version of polynomial statistics for 35 moments with gauge transformation.

M35_P2_1D_gauged_stats

The 1D version of polynomial statistics for 35 moments with gauge transformation.
class MomentGauge.Statistic.PolyGaugedStatistics.PolyGaugedStatistics(base_statistics: MomentGauge.Statistic.PolyStatistics.PolyStatistics)#

Bases: MomentGauge.Statistic.PolyStatistics.PolyStatistics

The base class for store pre-defined gauged polynomial statistics.

The gauged statistics are transformation of ordinary polynomial statistics by a gauge transformation \(A_{ij}(\mathbf{g})\).

Specifically, given a set of polynomial statistics \(\phi_i(\mathbf{u})\), their gauged version is a set of statistics \(\phi_i(\mathbf{u}, \mathbf{g})\) admits extra gauge parameters \(\mathbf{g}\) such that

\begin{equation} \phi_i(\mathbf{u}, \mathbf{g}) = A_{ij}(\mathbf{g})\phi_j(\mathbf{u}) \end{equation}

Moreover we have gauge transformation between different gauge parameters

\begin{equation} \phi_i(\mathbf{u}, \mathbf{g}') = T_{ij}(\mathbf{g}',\mathbf{g})\phi_j(\mathbf{u}, \mathbf{g}) \end{equation}
suff_stats#

a list of moment functions [\(\phi_i\), i=0,cdots,N-1] in which each \(\phi_i\) is a polynomial function \(\phi_i\) ( u, \(*\) gauge_paras ) whose

Parameters:

u : float array of shape (3) - The 3D sample vector (\(u_x\), \(u_y\), \(u_z\))

\(*\) gauge_paras : - Arbitrary many extra parameters. The \(*\) refers to the unpacking operator in python.

Returns:

float – the moment value

Type:

list

abstract gauge(func)#

Convert the functions of \(\mathbf{u}\) into function of \(\mathbf{u}\) and gauge parameters

Parameters:

func (function) –

a polynomial function \(\phi\) ( u ) whose

Parameters:

u : float array of shape (3) - The 3D sample vector

Returns:

float – the moment value

Returns:

a polynomial function \(\phi\) ( u, \(*\) gauge_paras ) whose

Parameters:

u : float array of shape (3) - The 3D sample vector

\(*\) gauge_paras : - Arbitrary many extra parameters. The \(*\) refers to the unpacking operator in python.

Returns:

float – the moment value

Return type:

function

abstract gauge_transformation_matrix(gauge_para2=(), gauge_para1=())#

Compute the gauge transformation matrix \(T_{ij}(\mathbf{g}',\mathbf{g})\) between different gauge parameters

\begin{equation} \phi_i(\mathbf{u}, \mathbf{g}') = T_{ij}(\mathbf{g}',\mathbf{g})\phi_j(\mathbf{u}, \mathbf{g}) \end{equation}
Parameters:

  • gauge_para2 (tuple) – Tuple containing arbitrary many extra gauge parameters such as \(\mathbf{g}'\)

  • gauge_para1 (tuple) – Tuple containing arbitrary many extra gauge parameters such as \(\mathbf{g}\)

Returns:

the matrix \(T_{ij}(\mathbf{g}',\mathbf{g})\)

Return type:

float array of shape (N,N)

standard_gauge_paras(moments, gauge_para=())#

the standard gauge parameters \(\mathbf{g}\) prefered among all possible gauges.

Parameters:

  • moments (float array of shape (N)) – The array containing moments of sufficient statistics given the gauge parameters \(\mathbf{g}\)

  • gauge_para (tuple) – Tuple containing arbitrary many extra gauge parameters such as \(\mathbf{g}\)

Returns:

the standard gauge parameters \(\mathbf{g}\) prefered among all possible gauges.

Return type:

float array

class MomentGauge.Statistic.PolyGaugedStatistics.Maxwellian_1D_gauged_stats#

Bases: PolyGaugedStatistics

The 1D version of polynomial statistics for 35 moments with gauge transformation.

suff_stats#

a list of moment functions [\(\phi_i,i=0,\cdots,2\)] in which each \(\phi_i\) is a polynomial function \(\phi_i\) ( u, gauge_paras ) whose

Parameters:

u : float array of shape (3) - The 3D sample vector (\(u_x\), \(u_y\), \(u_z\))

gauge_paras : float array of shape (3) - The array (\(s\),:math:w_x).

Returns:

float – the moment value

Specifically,

\(\phi_0\) (u ) = 1.

\(\phi_1\) (u ) = \(\bar{u}_x\)

\(\phi_2\) (u ) = \(\bar{u}_x^2 + \bar{u}_y^2 + \bar{u}_z^2\).

in which \(\bar{u}_x = \frac{u_x - w_x}{s}\), \(\bar{u}_y = \frac{u_y}{s}\), \(\bar{u}_z = \frac{u_z}{s}\)

Type:

list of length (9)

gauge(func)#

Convert the functions of \(\mathbf{u}\) into function of \(\mathbf{u}\) and gauge parameters (\(s\), \(w_x\)).

Parameters:

func (function) –

a polynomial function \(\phi\) ( u ) whose

Parameters:

u : float array of shape (3) - The 3D sample vector

Returns:

float – the moment value

Returns:

a polynomial function \(\phi\) ( u, gauge_paras ) whose

Parameters:

u : float array of shape (3) - The 3D sample vector

gauge_paras : float array of shape (3) - The array (\(s\), \(w_x\)).

Returns:

float – the moment value

Return type:

function

gauge_transformation_matrix(gauge_para2=(), gauge_para1=())#

Compute the gauge transformation matrix between different gauge parameters

\begin{equation} \phi_i(\mathbf{u}, (s', w_x') )= T_{ij}(s', w_x',s, w_x) \phi_j(\mathbf{u}, (s, w_x)); \quad i,j = 0, \cdots, 2 \end{equation}
Parameters:

  • gauge_para2 (tuple) – A tuple containing the array (\(s'\), \(w_x'\)) which is a float array of shape (2).

  • gauge_para1 – A tuple containing the array (\(s\), \(w_x\)) which is a float array of shape (2).

Returns:

the matrix \(T_{ij}(s', w_x',s, w_x)\)

Return type:

float array of shape (3,3)

standard_gauge_paras(moments, gauge_para=())#

Compute the Hermite gauge parameters

Parameters:

  • moments (float array of shape (3)) – The array containing moments of sufficient statistics \((M_0(s, w_x), M_1(s, w_x), M_2(s, w_x))\)

  • gauge_para (tuple) – A tuple containing the array (\(s\), \(w_x\)) which is a float array of shape (2).

Returns:

the Hermite gauge parameters (\(s\), \(w_x\))

Return type:

float array of shape (2)

class MomentGauge.Statistic.PolyGaugedStatistics.PolyGaugedStatistics_sr_sx_wx(base_statistics: MomentGauge.Statistic.PolyStatistics.PolyStatistics)#

Bases: PolyGaugedStatistics

The base class for store pre-defined gauged polynomial statistics with gauge parameter \(s_r\), \(s_x\), and \(w_x\)

The gauged statistics are transformation of ordinary polynomial statistics by a gauge transformation \(A_{ij}(\mathbf{g})\).

Specifically, given a set of polynomial statistics \(\phi_i(\mathbf{u})\), their gauged version is a set of statistics \(\phi_i(\mathbf{u}, \mathbf{g})\) admits extra gauge parameters \(\mathbf{g}\) such that

\begin{equation} \phi_i(\mathbf{u}, \mathbf{g}) = A_{ij}(\mathbf{g})\phi_j(\mathbf{u}) = \phi_i(\bar{\mathbf{u}}) \end{equation}

such that \(\bar{\mathbf{u}}={ \bar{u}_x, \bar{u}_y, \bar{u}_z }\), \(\bar{u}_x = \frac{u_x - w_x}{s_x}\), \(\bar{u}_y = \frac{u_y}{s_r}\), \(\bar{u}_z = \frac{u_z}{s_r}\).

suff_stats#

a list of moment functions [\(\phi_i\), i=0,cdots,N-1] in which each \(\phi_i\) is a polynomial function \(\phi_i\) ( u, \(*\) gauge_paras ) whose

Parameters:

u : float array of shape (3) - The 3D sample vector (\(u_x\), \(u_y\), \(u_z\))

\(*\) gauge_paras : - Arbitrary many extra parameters. The \(*\) refers to the unpacking operator in python.

Returns:

float – the moment value

Type:

list

gauge(func)#

Convert the functions of \(\mathbf{u}\) into function of \(\mathbf{u}\) and gauge parameters (\(s_r\), \(s_x\), \(w_x\)).

Parameters:

func (function) –

a polynomial function \(\phi\) ( u ) whose

Parameters:

u : float array of shape (3) - The 3D sample vector

Returns:

float – the moment value

Returns:

a polynomial function \(\phi\) ( u, gauge_paras ) whose

Parameters:

u : float array of shape (3) - The 3D sample vector

gauge_paras : float array of shape (3) - The array (\(s_r\), \(s_x\), \(w_x\)).

Returns:

float – the moment value

Return type:

function

gauge_transformation_matrix(gauge_para2=(), gauge_para1=())#

Compute the gauge transformation matrix between different gauge parameters

\begin{equation} \phi_i(\mathbf{u}, (s_r', s_x', w_x') )= T_{ij}(s_r', s_x', w_x',s_r, s_x, w_x) \phi_j(\mathbf{u}, (s_r, s_x, w_x)); \quad i,j = 0, \cdots, M \end{equation}

in which M is the number of sufficient statistics

Parameters:

  • gauge_para2 (tuple) – A tuple containing the array (\(s_r'\), \(s_x'\), \(w_x'\)) which is a float array of shape (3).

  • gauge_para1 – A tuple containing the array (\(s_r\), \(s_x\), \(w_x\)) which is a float array of shape (3).

Returns:

the matrix \(T_{ij}(s_r', s_x', w_x',s_r, s_x, w_x)\)

Return type:

float array of shape (M,M)

standard_gauge_paras(moments, gauge_para=())#

Compute the Hermite gauge parameters

Parameters:

  • moments (float array of shape (9)) – The array containing moments of sufficient statistics \((M_0(s_r, s_x, w_x), \cdots, M_8(s_r, s_x, w_x))\)

  • gauge_para (tuple) – A tuple containing the array (\(s_r\), \(s_x\), \(w_x\)) which is a float array of shape (2).

Returns:

the Hermite gauge parameters (\(s_r\), \(s_x\), \(w_x\))

Return type:

float array of shape (3)

class MomentGauge.Statistic.PolyGaugedStatistics.ESBGK_1D_gauged_stats#

Bases: PolyGaugedStatistics_sr_sx_wx

The 1D version of polynomial statistics for ESBGK moments with gauge transformation.

suff_stats#

a list of moment functions [\(\phi_i,i=0,\cdots,3\)] in which each \(\phi_i\) is a polynomial function \(\phi_i\) ( u, gauge_paras ) whose

Parameters:

u : float array of shape (3) - The 3D sample vector (\(u_x\), \(u_y\), \(u_z\))

gauge_paras : float array of shape (3) - The array (\(s_r\), \(s_x\), \(w_x\)).

Returns:

float – the moment value

Specifically,

\begin{equation} \{\phi_i(\mathbf{u}, (s_r, s_x, w_x) ),i=0,\cdots,3\} = \left\{1,\bar{u}_x,\bar{u}_x^2,\bar{u}_r^2,\right\} \end{equation}

in which \(\bar{u}_x = \frac{u_x - w_x}{s_x}\), \(\bar{u}_r = \frac{u_r}{s_r}\), \(u_r = \sqrt{u_y^2+u_z^2}\)

Type:

list of length (4)

gauge(func)#

Convert the functions of \(\mathbf{u}\) into function of \(\mathbf{u}\) and gauge parameters (\(s_r\), \(s_x\), \(w_x\)).

Parameters:

func (function) –

a polynomial function \(\phi\) ( u ) whose

Parameters:

u : float array of shape (3) - The 3D sample vector

Returns:

float – the moment value

Returns:

a polynomial function \(\phi\) ( u, gauge_paras ) whose

Parameters:

u : float array of shape (3) - The 3D sample vector

gauge_paras : float array of shape (3) - The array (\(s_r\), \(s_x\), \(w_x\)).

Returns:

float – the moment value

Return type:

function

gauge_transformation_matrix(gauge_para2=(), gauge_para1=())#

Compute the gauge transformation matrix between different gauge parameters

\begin{equation} \phi_i(\mathbf{u}, (s_r', s_x', w_x') )= T_{ij}(s_r', s_x', w_x',s_r, s_x, w_x) \phi_j(\mathbf{u}, (s_r, s_x, w_x)); \quad i,j = 0, \cdots, M \end{equation}

in which M is the number of sufficient statistics

Parameters:

  • gauge_para2 (tuple) – A tuple containing the array (\(s_r'\), \(s_x'\), \(w_x'\)) which is a float array of shape (3).

  • gauge_para1 – A tuple containing the array (\(s_r\), \(s_x\), \(w_x\)) which is a float array of shape (3).

Returns:

the matrix \(T_{ij}(s_r', s_x', w_x',s_r, s_x, w_x)\)

Return type:

float array of shape (M,M)

standard_gauge_paras(moments, gauge_para=())#

Compute the Hermite gauge parameters

Parameters:

  • moments (float array of shape (9)) – The array containing moments of sufficient statistics \((M_0(s_r, s_x, w_x), \cdots, M_8(s_r, s_x, w_x))\)

  • gauge_para (tuple) – A tuple containing the array (\(s_r\), \(s_x\), \(w_x\)) which is a float array of shape (2).

Returns:

the Hermite gauge parameters (\(s_r\), \(s_x\), \(w_x\))

Return type:

float array of shape (3)

class MomentGauge.Statistic.PolyGaugedStatistics.M35_1D_gauged_stats#

Bases: PolyGaugedStatistics_sr_sx_wx

The 1D version of polynomial statistics for 35 moments with gauge transformation.

suff_stats#

a list of moment functions [\(\phi_i,i=0,\cdots,8\)] in which each \(\phi_i\) is a polynomial function \(\phi_i\) ( u, gauge_paras ) whose

Parameters:

u : float array of shape (3) - The 3D sample vector (\(u_x\), \(u_y\), \(u_z\))

gauge_paras : float array of shape (3) - The array (\(s_r\), \(s_x\), \(w_x\)).

Returns:

float – the moment value

Specifically,

\begin{equation} \{\phi_i(\mathbf{u}, (s_r, s_x, w_x) ),i=0,\cdots,8\} = \left\{1,\bar{u}_x,\frac{\bar{u}_x^2-1}{\sqrt{2}},\frac{\bar{u}_r^2}{2} -1,\frac{\bar{u}_x^3-3\bar{u}_x}{\sqrt{6}},\frac{\bar{u}_x^4-6\bar{u}_x^2+3}{2 \sqrt{6}},\frac{1}{8} \bar{u}_r^4-\bar{u}_r^2+1,\frac{1}{2} \bar{u}_x (\bar{u}_r^2-1),\frac{( \bar{u}_x^2 -1)( \bar{u}_r^2-2)}{2 \sqrt{2}}\right\} \end{equation}

in which \(\bar{u}_x = \frac{u_x - w_x}{s_x}\), \(\bar{u}_r = \frac{u_r}{s_r}\), \(u_r = \sqrt{u_y^2+u_z^2}\)

Type:

list of length (9)

conservative_decomposition(moments, gauge_para=())#

Decompose the moments as the summation of the conserved part and the non-conserved part

Parameters:

  • moments (float array of shape (9)) – The array containing moments of sufficient statistics \((M_0(s_r, s_x, w_x), \cdots, M_8(s_r, s_x, w_x))\)

  • gauge_para (tuple) – A tuple containing the array (\(s_r\), \(s_x\), \(w_x\)) which is a float array of shape (2).

Returns:

the conservative part of the moments. The non conservative part is moments - conservative part.

Return type:

float array of shape (9)

gauge(func)#

Convert the functions of \(\mathbf{u}\) into function of \(\mathbf{u}\) and gauge parameters (\(s_r\), \(s_x\), \(w_x\)).

Parameters:

func (function) –

a polynomial function \(\phi\) ( u ) whose

Parameters:

u : float array of shape (3) - The 3D sample vector

Returns:

float – the moment value

Returns:

a polynomial function \(\phi\) ( u, gauge_paras ) whose

Parameters:

u : float array of shape (3) - The 3D sample vector

gauge_paras : float array of shape (3) - The array (\(s_r\), \(s_x\), \(w_x\)).

Returns:

float – the moment value

Return type:

function

gauge_transformation_matrix(gauge_para2=(), gauge_para1=())#

Compute the gauge transformation matrix between different gauge parameters

\begin{equation} \phi_i(\mathbf{u}, (s_r', s_x', w_x') )= T_{ij}(s_r', s_x', w_x',s_r, s_x, w_x) \phi_j(\mathbf{u}, (s_r, s_x, w_x)); \quad i,j = 0, \cdots, M \end{equation}

in which M is the number of sufficient statistics

Parameters:

  • gauge_para2 (tuple) – A tuple containing the array (\(s_r'\), \(s_x'\), \(w_x'\)) which is a float array of shape (3).

  • gauge_para1 – A tuple containing the array (\(s_r\), \(s_x\), \(w_x\)) which is a float array of shape (3).

Returns:

the matrix \(T_{ij}(s_r', s_x', w_x',s_r, s_x, w_x)\)

Return type:

float array of shape (M,M)

standard_gauge_paras(moments, gauge_para=())#

Compute the Hermite gauge parameters

Parameters:

  • moments (float array of shape (9)) – The array containing moments of sufficient statistics \((M_0(s_r, s_x, w_x), \cdots, M_8(s_r, s_x, w_x))\)

  • gauge_para (tuple) – A tuple containing the array (\(s_r\), \(s_x\), \(w_x\)) which is a float array of shape (2).

Returns:

the Hermite gauge parameters (\(s_r\), \(s_x\), \(w_x\))

Return type:

float array of shape (3)

class MomentGauge.Statistic.PolyGaugedStatistics.M35_P2_1D_gauged_stats#

Bases: PolyGaugedStatistics_sr_sx_wx

The 1D version of polynomial statistics for 35 moments with gauge transformation.

suff_stats#

a list of moment functions [\(\phi_i,i=0,\cdots,8\)] in which each \(\phi_i\) is a polynomial function \(\phi_i\) ( u, gauge_paras ) whose

Parameters:

u : float array of shape (3) - The 3D sample vector (\(u_x\), \(u_y\), \(u_z\))

gauge_paras : float array of shape (3) - The array (\(s_r\), \(s_x\), \(w_x\)).

Returns:

float – the moment value

Specifically,

\begin{equation} \{\phi_i(\mathbf{u}, (s_r, s_x, w_x) ),i=0,\cdots,10\} = \left\{1,\bar{u}_x,\frac{\bar{u}_x^2-1}{\sqrt{2}},\frac{\bar{u}_r^2}{2} -1,\frac{\bar{u}_x^3-3\bar{u}_x}{\sqrt{6}},\frac{\bar{u}_x^4-6\bar{u}_x^2+3}{2 \sqrt{6}},\frac{1}{8} \bar{u}_r^4-\bar{u}_r^2+1,\frac{1}{2} \bar{u}_x (\bar{u}_r^2-1),\frac{( \bar{u}_x^2 -1)( \bar{u}_r^2-2)}{2 \sqrt{2}} , \frac{\bar{u}_x^5}{2 \sqrt{30}}-\sqrt{\frac{5}{6}} \bar{u}_x^3+\frac{1}{2} \sqrt{\frac{15}{2}} \bar{u}_x, \frac{\bar{u}_x^6}{12 \sqrt{5}}-\frac{\sqrt{5} \bar{u}_x^4}{4}+\frac{3 \sqrt{5} \bar{u}_x^2}{4}-\frac{\sqrt{5}}{4} \right\} \end{equation}

in which \(\bar{u}_x = \frac{u_x - w_x}{s_x}\), \(\bar{u}_r = \frac{u_r}{s_r}\), \(u_r = \sqrt{u_y^2+u_z^2}\)

Type:

list of length (11)

gauge(func)#

Convert the functions of \(\mathbf{u}\) into function of \(\mathbf{u}\) and gauge parameters (\(s_r\), \(s_x\), \(w_x\)).

Parameters:

func (function) –

a polynomial function \(\phi\) ( u ) whose

Parameters:

u : float array of shape (3) - The 3D sample vector

Returns:

float – the moment value

Returns:

a polynomial function \(\phi\) ( u, gauge_paras ) whose

Parameters:

u : float array of shape (3) - The 3D sample vector

gauge_paras : float array of shape (3) - The array (\(s_r\), \(s_x\), \(w_x\)).

Returns:

float – the moment value

Return type:

function

gauge_transformation_matrix(gauge_para2=(), gauge_para1=())#

Compute the gauge transformation matrix between different gauge parameters

\begin{equation} \phi_i(\mathbf{u}, (s_r', s_x', w_x') )= T_{ij}(s_r', s_x', w_x',s_r, s_x, w_x) \phi_j(\mathbf{u}, (s_r, s_x, w_x)); \quad i,j = 0, \cdots, M \end{equation}

in which M is the number of sufficient statistics

Parameters:

  • gauge_para2 (tuple) – A tuple containing the array (\(s_r'\), \(s_x'\), \(w_x'\)) which is a float array of shape (3).

  • gauge_para1 – A tuple containing the array (\(s_r\), \(s_x\), \(w_x\)) which is a float array of shape (3).

Returns:

the matrix \(T_{ij}(s_r', s_x', w_x',s_r, s_x, w_x)\)

Return type:

float array of shape (M,M)

standard_gauge_paras(moments, gauge_para=())#

Compute the Hermite gauge parameters

Parameters:

  • moments (float array of shape (9)) – The array containing moments of sufficient statistics \((M_0(s_r, s_x, w_x), \cdots, M_8(s_r, s_x, w_x))\)

  • gauge_para (tuple) – A tuple containing the array (\(s_r\), \(s_x\), \(w_x\)) which is a float array of shape (2).

Returns:

the Hermite gauge parameters (\(s_r\), \(s_x\), \(w_x\))

Return type:

float array of shape (3)