Kernels

This module provides kernels for the analytic continuation. In general, we have \(G_i = \sum_j K_{ij} H_j\). Note that for non-uniform \(\omega\)-grids, \(H = A \Delta\omega\).

At the moment, only the kernel for the continuation of \(G(\tau)\) is implemented, i.e., the TauKernel.

For any kernel the preblur version can be used by the PreblurKernel (see Continuation of metallic solutions using preblur).

class triqs_maxent.kernels.KernelSVD(K=None)[source]

Bases: object

A kernel object with a singular value decomposition

Parameters:

Karray: the kernel matrix

Attributes:

K: The actual kernel matrix
S: Get the vector of singular values of the kernel
U: Get the matrix of left-singular vectors of the kernel
V: Get the matrix of right-singular vectors of the kernel

Methods

`reduce_singular_space`([threshold])	Reduce the singular space
`svd`()	Perform the SVD if not yet performed

svd()[source]

Perform the SVD if not yet performed

Usually, this function does not need to be called by the user.

We have \(K = USV^\dagger\).

property U

Get the matrix of left-singular vectors of the kernel

If not performed already, the SVD is done.

property S

Get the vector of singular values of the kernel

If not performed already, the SVD is done.

property V

Get the matrix of right-singular vectors of the kernel

If not performed already, the SVD is done.

property K: The actual kernel matrix

reduce_singular_space(threshold=1e-14)[source]

Reduce the singular space

All singular values smaller than the threshold are dropped from S, U, V.

Parameters:

thresholdfloat: the threshold for dropping singular values

class triqs_maxent.kernels.Kernel[source]

Bases: KernelSVD

The kernel for the analytic continuation

Attributes:

K: The actual kernel matrix
K_delta: The kernel including a \(\Delta\omega\).
S: Get the vector of singular values of the kernel
U: Get the matrix of left-singular vectors of the kernel
V: Get the matrix of right-singular vectors of the kernel
data_variable

Methods

`parameter_change`()	Notify the kernel of a parameter change
`reduce_singular_space`([threshold])	Reduce the singular space
`svd`()	Perform the SVD if not yet performed
`transform`(T_)	multiply the kernel from the left with the matrix `T_`

property K_delta

The kernel including a \(\Delta\omega\).

Use this to get the reconstructed Green function as \(G_{rec} = K_{delta} A\). Note that it does not get rotated with transform(), i.e. it gives back the original \(G_{rec}\).

parameter_change()[source]

Notify the kernel of a parameter change

This should be called when, e.g., the data variable (e.g., \(\tau\)) or the omega-mesh is changed.

transform(T_)[source]

multiply the kernel from the left with the matrix T_

T_ is the absolute rotation with respect to the unrotated quantities

class triqs_maxent.kernels.DataKernel(data_variable, omega, K)[source]

Bases: Kernel

A kernel given by a matrix

Parameters:

data_variablearray: the array of the data variable, i.e. the variable that the input-data depends on; typically, one continues \(G(\tau)\), then this would correspond to the \(\tau\)-grid.
omegaOmegaMesh: the frequency mesh
Karray: the kernel matrix

Attributes:

K: The actual kernel matrix
K_delta: The kernel including a \(\Delta\omega\).
S: Get the vector of singular values of the kernel
U: Get the matrix of left-singular vectors of the kernel
V: Get the matrix of right-singular vectors of the kernel
data_variable

Methods

`parameter_change`()	Notify the kernel of a parameter change
`reduce_singular_space`([threshold])	Reduce the singular space
`svd`()	Perform the SVD if not yet performed
`transform`(T_)	multiply the kernel from the left with the matrix `T_`

class triqs_maxent.kernels.TauKernel(tau, omega, beta=None)[source]

Bases: Kernel

A kernel for continuing \(G(\tau)\)

This kernel is defined as

\[K(\tau, \omega) = - \frac{\exp(-\tau\omega)}{1 + \exp(\beta \omega)}.\]

With this, we have

\[G(\tau) = \int d\omega\, K(\tau, \omega) A(\omega).\]

Parameters:

tauarray: the \(\tau\)-mesh where the data is given
omegaOmegaMesh: the \(\omega\)-mesh where the spectral function should be calculated
betafloat: the inverse temperature; if not given, it is taken to be the last tau-value

Attributes:

K: The actual kernel matrix
K_delta: The kernel including a \(\Delta\omega\).
S: Get the vector of singular values of the kernel
U: Get the matrix of left-singular vectors of the kernel
V: Get the matrix of right-singular vectors of the kernel
data_variable: \(\tau\)

Methods

`parameter_change`()	Notify the kernel of a parameter change
`reduce_singular_space`([threshold])	Reduce the singular space
`svd`()	Perform the SVD if not yet performed
`transform`(T_)	multiply the kernel from the left with the matrix `T_`

property data_variable: \(\tau\)

class triqs_maxent.kernels.IOmegaKernel(iomega, omega, beta=None)[source]

Bases: Kernel

A kernel for continuing \(G(i\omega)\)

This kernel is defined as

\[K(i\omega, \omega) = \frac{1}{i\omega - \omega}\]

With this, we have

\[G(i\omega) = \int d\omega\, K(i\omega, \omega) A(\omega).\]

Parameters:

iomegaarray: the \(iomega\)-mesh where the data is given (as a real array, i.e. it is internally multiplied by 1.0j)
omegaOmegaMesh: the \(\omega\)-mesh where the spectral function should be calculated
betafloat: the inverse temperature; if not given, it is taken from the difference of the first two \(i\omega\) values

Attributes:

K: The actual kernel matrix
K_delta: The kernel including a \(\Delta\omega\).
S: Get the vector of singular values of the kernel
U: Get the matrix of left-singular vectors of the kernel
V: Get the matrix of right-singular vectors of the kernel
data_variable: \(i\omega\)

Methods

`parameter_change`()	Notify the kernel of a parameter change
`reduce_singular_space`([threshold])	Reduce the singular space
`svd`()	Perform the SVD if not yet performed
`transform`(T_)	multiply the kernel from the left with the matrix `T_`

property data_variable: \(i\omega\)

class triqs_maxent.kernels.PreblurKernel(K, b)[source]

Bases: Kernel

A kernel for the preblur formalism

In the preblur formalism, the equation \(G = KA\) is replaced by \(G = KBH\), with a hidden image \(H\) and a blur matrix \(B\).

The dicretization of \(\omega\) has to be accounted for by including a \(\Delta\omega\). In fact, the total equation is \(G_i = \sum_{jk} K_{ij} \Delta\omega_j B_{jk} H_k\), where \(H_k\) already includes a \(\Delta\omega_k\).

Note that the PreblurKernel should always be used together with the PreblurA_of_H.

Parameters:

KKernel: the kernel to use up to the preblur matrix (e.g. a TauKernel instance)
bfloat: the blur parameter, i.e., the width of the Gaussian that the hidden image is convolved with to get the blurred spectral function

Attributes:

K: The actual kernel matrix
K_delta: The kernel including a \(\Delta\omega\).
S: Get the vector of singular values of the kernel
U: Get the matrix of left-singular vectors of the kernel
V: Get the matrix of right-singular vectors of the kernel
data_variable
omega

Methods

`parameter_change`()	Notify the kernel of a parameter change
`reduce_singular_space`([threshold])	Reduce the singular space
`svd`()	Perform the SVD if not yet performed
`transform`(T)	multiply the kernel from the left with the matrix `T_`

get_omega
set_omega

parameter_change()[source]

Notify the kernel of a parameter change

This should be called when, e.g., the data variable (e.g., \(\tau\)) or the omega-mesh is changed.

transform(T)[source]

multiply the kernel from the left with the matrix T_

T_ is the absolute rotation with respect to the unrotated quantities