statescale.kernels#

SurrogateKernel(snapshots, point_data, ...)

A surrogate kernel.

GriddataKernel(snapshots, point_data, ...)

A griddata kernel.

Detailed API Description

class statescale.kernels.SurrogateKernel(snapshots, point_data, cell_data, field_data, **kwargs)[source]#

A surrogate kernel.

Parameters:

  • point_data (dict) – A dict of point data.

  • cell_data (dict) – A dict of cell data.

  • field_data (dict) – A dict of field data.

  • **kwargs –

    Additional keyword arguments for the calibration of the kernel.

    modes2-tuple of int or None, optional

    Mode-range for the surrogate model. Default is (2, 10). If None, the modes are chosen in such a way that cumsum(S) / S >= threshold of the singular values are included.

    thresholdfloat, optional

    Default threshold to evaluate the number of modes for the surrogate model. Default is 0.995.

Notes

The surrogate kernel is a snapshot-based proper orthogonal decomposition (POD) surrogate with interpolation of modal coefficients and is based on [1], with a selection of the maximum number of modes as outlined in [2].

Naming conventions#

Symbol

Description

x_si

Snapshots

d_s...

Time-dependent data at snapshots with arbitrary trailing axes

mean(d)_...

Mean over all snapshots of data

Δd_s...

Centered data at snapshots

U_...m

Unitary matrix of U S Vh = svd(Δd_…s)

α_sm

Factors at snapshots to obtain the centered data

x_ai

Signal

α_am

Factors for the signal to obtain the centered data

Δd_a...

Centered data for the signal

d_a...

Data for the signal (with arbitrary trailing axes)
Indices#

Index

Description

s

s-th snapshot

i

i-th vector component of snapshot / signal

m

j-th vector component of flattened data

m

m-th mode of surrogate model

a

a-th timestep of signal

First, the centered data at the snapshots is computed by subtracting the mean over all snapshots.

\[\Delta d_{sj} = d_{sj} - \operatorname{mean}(d)_j\]

Then, a singular value decomposition (SVD) of the centered data is performed to obtain the principal components (modes) of the data. The number of modes to be used in the surrogate model is determined based on the provided mode range and the threshold for the cumulative sum of singular values.

\[\Delta d_{js} = U_{jm}\ S_{mm}\ V^H_{ms}\]
\[ \begin{align}\begin{aligned}\alpha_{sm} &= \Delta d_{sj}\ U_{jm}\\\bar{\alpha}_{sm} &= \bar{d}_{sj}\ U_{jm}\end{aligned}\end{align} \]

Next, the factors at the signal points are obtained by interpolating the modal coefficients from the snapshots to the signal points using the provided upscale function.

\[ \begin{align}\begin{aligned}\alpha_{am} &= \text{upscale}(x_{si}, \alpha_{sm}, x_{ai})\\\beta_{am} &= \alpha_{am} + \bar{\alpha}_{am}\end{aligned}\end{align} \]

The factors at the snapshots are computed by projecting the data onto the selected modes.

\[d_{aj} = \beta_{am}\ U_{mj}\]

Finally, the kernel parameters are stored in SurrogateKernelParameters for later use in the surrogate model.

See also

statescale.SnapshotModel

A model with point-, cell- and field-data at snapshots and with methods to interpolate the data at points of interest.

statescale.kernels.GriddataKernel

A griddata kernel.

References

static evaluate(xi, upscale, kernel_parameters, indices=None, axis=None, **kwargs)[source]#
class statescale.kernels.GriddataKernel(snapshots, point_data, cell_data, field_data, **kwargs)[source]#

A griddata kernel.

Parameters:

  • point_data (dict) – A dict of point data.

  • cell_data (dict) – A dict of cell data.

  • field_data (dict) – A dict of field data.

  • **kwargs – Additional keyword arguments (not used).

See also

statescale.SnapshotModel

A model with point-, cell- and field-data at snapshots and with methods to interpolate the data at points of interest.

statescale.kernels.SurrogateKernel

A surrogate kernel.

static evaluate(xi, upscale, kernel_parameters, indices=None, axis=None, **kwargs)[source]#