Sparse sampling

Sparse sampling is a way to transition between the truncated IR representation of a propagator (useful for convergence analysis) and a sparse set of sampling points in imaginary time or Matsubara frequency. This is mediated by two classes:

• sparse_ir.TauSampling: sparse sampling in imaginary time, useful for, e.g., constructing Feynman diagrams with a spontaneous interation term.

• sparse_ir.MatsubaraSampling: sparse sampling in Matsubara frequencies, useful for, e.g., solving diagrammatic equations such as the Dyson equation.

All sampling classes contain sampling_points, which are the corresponding sampling points in time or frequency, and a method evaluate(), which allows one to go from coefficients to sampling points, and a method fit() to go back:

 ________________                   ___________________
|                |    evaluate()   |                   |
|     Basis      |---------------->|     Value on      |
|  coefficients  |<----------------|  sampling_points  |
|________________|      fit()      |___________________|

Warning

When storing data in sparse time/frequency, always store the sampling points together with the data. The exact location of the sampling points may be different from between platforms and/or between releases.

Sparse sampling transformers

class sparse_ir.TauSampling(basis, sampling_points=None, use_positive_taus=False)

Sparse sampling in imaginary time.

Allows the transformation between the IR basis and a set of sampling points in (scaled/unscaled) imaginary time.

evaluate(al, axis=0)

Transform basis coefficients to sampling points.

Parameters:

alarray_like

Basis coefficients

axisint, optional

Axis along which to transform

Returns:

ndarray

Values at sampling points

fit(ax, axis=0)

Fit basis coefficients from sampling point values.

property tau

Tau sampling points.

class sparse_ir.MatsubaraSampling(basis, sampling_points=None, positive_only=False)

Sparse sampling in Matsubara frequencies.

Allows the transformation between the IR basis and a set of sampling points in (scaled/unscaled) imaginary frequencies.

By setting positive_only=True, one assumes that functions to be fitted are symmetric in Matsubara frequency, i.e.:

Ghat(iv) == Ghat(-iv).conj()

or equivalently, that they are purely real in imaginary time. In this case, sparse sampling is performed over non-negative frequencies only, cutting away half of the necessary sampling space.

evaluate(al, axis=0)

Transform basis coefficients to sampling points.

Parameters:

alarray_like

Basis coefficients

axisint, optional

Axis along which to transform

Returns:

ndarray

Values at Matsubara frequencies (complex)

fit(ax, axis=0)

Fit basis coefficients from Matsubara frequency values.

property wn

Matsubara frequency indices.

Base classes

Note

The base classes for sampling are currently being refactored. Please refer to the concrete sampling classes above for the current API.