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)

Bases: object

A kernel object with a singular value decomposition

Parameters:

Karray

the kernel matrix

svd()

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)

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

Bases: KernelSVD

The kernel for the analytic continuation

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()

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_)

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)

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

class triqs_maxent.kernels.TauKernel(tau, omega, beta=None)

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

property data_variable

\(\tau\)

class triqs_maxent.kernels.IOmegaKernel(iomega, omega, beta=None)

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

property data_variable

\(i\omega\)

class triqs_maxent.kernels.PreblurKernel(K, b)

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

parameter_change()

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)

multiply the kernel from the left with the matrix T_

T_ is the absolute rotation with respect to the unrotated quantities