dedalus.core.transforms

Spectral transform classes.

Module Contents

class BallRadialTransform(grid_shape, coeff_size, axis, ell_maps, regindex, regtotal, k, alpha, dtype=np.complex128, dealias_before_converting=None)

Abstract base class for all transforms.

backward(cdata, gdata, axis)

backward_reduced(cdata, gdata)

forward(gdata, cdata, axis)

forward_reduced(gdata, cdata)

N3c

N3g

alpha

dealias_before_converting = None

ell_maps

intertwiner

k

regindex

regtotal

class ComplexFFT(grid_size, coeff_size)

Abstract base class for complex-to-complex FFTs.

resize_coeffs(data_in, data_out, axis, rescale): Resize and rescale coefficients in standard FFT format by intermediate padding/truncation.

class ComplexFourierMMT(grid_size, coeff_size)

Complex-to-complex Fourier MMT.

backward_matrix(): Build backward transform matrix.

forward_matrix(): Build forward transform matrix.

class ComplexFourierTransform(grid_size, coeff_size)

Abstract base class for complex-to-complex Fourier transforms.

Parameters:

grid_size (int) – Grid size (N) along transform dimension.
coeff_size (int) – Coefficient size (M) along transform dimension.

Notes

Let KN = (N - 1) // 2 be the maximum fully resolved (non-Nyquist) mode on the grid. Let KM = (M - 1) // 2 be the maximum retained mode in coeff space. Then K = min(KN, KM) is the maximum wavenumber used in the transforms. A unit-amplitude normalization is used.

Forward transform:

if abs(k) <= K:: F(k) = (1/N) sum_{x=0}^{N-1} f(x) exp(-2 pi i k x / N)
else:: F(k) = 0

Backward transform:

f(x) = sum_{k=-K}^{K} F(k) exp(2 pi i k x / N)

Coefficient ordering:

If M is odd, the ordering is [0, 1, 2, …, KM, KM+1, -KM, -KM+1, …, -1], where the Nyquist mode k = KM + 1 is zeroed in both directions. If M is even, the ordering is [0, 1, 2, …, KM, -KM, -KM+1, …, -1].

KM

KN

Kmax

M

N

property wavenumbers: One-dimensional global wavenumber array.

class CosineMMT(grid_size, coeff_size)

Cosine MMT.

backward_matrix(): Build backward transform matrix.

forward_matrix(): Build forward transform matrix.

class CosineTransform(grid_size, coeff_size)

Abstract base class for cosine transforms.

Parameters:

grid_size (int) – Grid size (N) along transform dimension.
coeff_size (int) – Coefficient size (M) along transform dimension.

Notes

Let KN = (N - 1) be the maximum (Nyquist) mode on the grid. Let KM = (M - 1) be the maximum retained mode in coeff space. Then K = min(KN, KM) is the maximum wavenumber used in the transforms. A unit-amplitude normalization is used.

Forward transform:

if k == 0:: a(k) = (1/N) sum_{x=0}^{N-1} f(x)
elif k <= K:: a(k) = (2/N) sum_{x=0}^{N-1} f(x) cos(pi k x / N)
else:: a(k) = 0

Backward transform:

f(x) = sum_{k=0}^{K} a(k) cos(pi k x / N)

Coefficient ordering:

The cosine coefficients are ordered simply as [a(0), a(1), a(2), …, a(KM)]

KM

KN

Kmax

M

N

property wavenumbers: One-dimensional global wavenumber array.

class DiskRadialTransform(grid_shape, basis_shape, axis, m_maps, s, k, alpha, dtype=np.complex128, dealias_before_converting=None)

backward_reduced(cdata, gdata)

forward_reduced(gdata, cdata)

Nmax

Nphi

alpha

dealias_before_converting = None

k

m_maps

s

class FFTPACKRealFFT(grid_size, coeff_size)

Real-to-real FFT using scipy.fftpack.

backward(cdata, gdata, axis): Apply backward transform along specified axis.

forward(gdata, cdata, axis): Apply forward transform along specified axis.

class FFTWBase(*args, rigor=None, **kw)

Abstract base class for FFTW transforms.

rigor = None

class FFTWComplexFFT(*args, rigor=None, **kw)

Complex-to-complex FFT using FFTW.

backward(cdata, gdata, axis): Apply backward transform along specified axis.

forward(gdata, cdata, axis): Apply forward transform along specified axis.

class FFTWDCT(*args, rigor=None, **kw)

Fast cosine transform using FFTW.

backward(cdata, gdata, axis): Apply backward transform along specified axis.

forward(gdata, cdata, axis): Apply forward transform along specified axis.

class FFTWFastChebyshevTransform(grid_size, coeff_size, a, b, a0, b0, **kw): Fast ultraspherical transform using scipy.fft and spectral conversion.

class FFTWHalfComplexFFT(*args, rigor=None, **kw)

Real-to-real FFT using FFTW half-complex DFT.

backward(cdata, gdata, axis): Apply backward transform along specified axis.

forward(gdata, cdata, axis): Apply forward transform along specified axis.

repack(cdata, temp, axis): Repack into complex coefficients and rescale for unit-amplitude normalization.

unpack_rescale(temp, cdata, axis, rescale): Unpack halfcomplex coefficients and rescale for unit-amplitude normalization.

class FFTWRealFFT(*args, rigor=None, **kw)

Real-to-real FFT using FFTW.

backward(cdata, gdata, axis): Apply backward transform along specified axis.

forward(gdata, cdata, axis): Apply forward transform along specified axis.

class FastChebyshevTransform(grid_size, coeff_size, a, b, a0, b0, **kw)

Abstract base class for fast Chebyshev transforms including ultraspherical conversion. Subclasses should inherit from this class, then a FastCosineTransform subclass.

KM_orig

Kmax_orig

M_orig

class FastCosineTransform(*args, **kw)

Abstract base class for fast cosine transforms.

resize_rescale_backward(data_in, data_out, axis, Kmax): Resize by padding/trunction and rescale to unit amplitude.

resize_rescale_forward(data_in, data_out, axis, Kmax): Resize by padding/trunction and rescale to unit amplitude.

backward_rescale_pos = 0.5

backward_rescale_zero = 1

forward_rescale_pos

forward_rescale_zero

class JacobiMMT(grid_size, coeff_size, a, b, a0, b0, dealias_before_converting=None)

Jacobi polynomial MMTs.

backward_matrix(): Build backward transform matrix.

forward_matrix(): Build forward transform matrix.

class JacobiTransform(grid_size, coeff_size, a, b, a0, b0, dealias_before_converting=None)

Abstract base class for Jacobi polynomial transforms.

Parameters:

grid_size (int) – Grid size (N) along transform dimension.
coeff_size (int) – Coefficient size (M) along transform dimension.
a (int) – Jacobi “a” parameter for polynomials.
b (int) – Jacobi “b” parameters for polynomials.
a0 (int) – Jacobi “a” parameter for the quadrature grid.
b0 (int) – Jacobi “b” parameter for the quadrature grid.

Notes

TODO: We need to define the normalization we use here.

M

N

a

a0

b

b0

dealias_before_converting = None

class NonSeparableTransform(grid_shape, coeff_size, axis, dtype)

Abstract base class for all transforms.

backward(cdata, gdata, axis)

forward(gdata, cdata, axis)

N2c

N2g

class PolynomialTransform(grid_size, coeff_size)

Abstract base class for all transforms.

backward(cdata, gdata, axis)

forward(gdata, cdata, axis)

static resize_reduced(data_in, data_out): Resize data by padding/truncation.

class RealFFT(grid_size, coeff_size)

Abstract base class for real-to-real FFTs using real-to-complex algorithms.

repack_rescale(cdata, temp, axis, rescale): Repack into complex coefficients and rescale for unit-amplitude normalization.

unpack_rescale(temp, cdata, axis, rescale): Unpack complex coefficients and rescale for unit-amplitude normalization.

class RealFourierMMT(grid_size, coeff_size)

Real-to-real Fourier MMT.

backward_matrix(): Build backward transform matrix.

forward_matrix(): Build forward transform matrix.

class RealFourierTransform(grid_size, coeff_size)

Abstract base class for real-to-real Fourier transforms.

Parameters:

grid_size (int) – Grid size (N) along transform dimension.
coeff_size (int) – Coefficient size (M) along transform dimension.

Notes

Let KN = (N - 1) // 2 be the maximum fully resolved (non-Nyquist) mode on the grid. Let KM = (M - 1) // 2 be the maximum retained mode in coeff space. Then K = min(KN, KM) is the maximum wavenumber used in the transforms. A unit-amplitude normalization is used.

Forward transform:

if k == 0:: a(k) = (1/N) sum_{x=0}^{N-1} f(x) b(k) = 0
elif k <= K:: a(k) = (2/N) sum_{x=0}^{N-1} f(x) cos(-2 pi k x / N) b(k) = -(2/N) sum_{x=0}^{N-1} f(x) sin(-2 pi k x / N)
else:: a(k) = 0 b(k) = 0

Backward transform:

f(x) = sum_{k=0}^{K} a(k) cos(2 pi k x / N) - b(k) sin(2 pi k x / N)

Coefficient ordering:

The cosine and minus-sine coefficients are interleaved as [a(0), b(0), a(1), b(1), a(2), b(2), …, a(KM), b(KM)] where the k = 0 minus-sine mode is zeroed in both directions.

KM

KN

Kmax

M

N

property wavenumbers: One-dimensional global wavenumber array.

class SWSHColatitudeTransform(Ntheta, Lmax, m_maps, s)

Abstract base class for all transforms.

backward_reduced(cdata, gdata)

forward_reduced(gdata, cdata)

Lmax

Ntheta

m_maps

s

class ScipyComplexFFT(grid_size, coeff_size)

Complex-to-complex FFT using scipy.fft.

backward(cdata, gdata, axis): Apply backward transform along specified axis.

forward(gdata, cdata, axis): Apply forward transform along specified axis.

class ScipyDCT(*args, **kw)

Fast cosine transform using scipy.fft.

backward(cdata, gdata, axis): Apply backward transform along specified axis.

forward(gdata, cdata, axis): Apply forward transform along specified axis.

class ScipyFastChebyshevTransform(grid_size, coeff_size, a, b, a0, b0, **kw): Fast ultraspherical transform using scipy.fft and spectral conversion.

class ScipyRealFFT(grid_size, coeff_size)

Real-to-real FFT using scipy.fft.

backward(cdata, gdata, axis): Apply backward transform along specified axis.

forward(gdata, cdata, axis): Apply forward transform along specified axis.

class SeparableMatrixTransform

Abstract base class for separable matrix-multiplication transforms.

backward(cdata, gdata, axis): Apply backward transform along specified axis.

abstractmethod backward_matrix(): Build backward transform matrix.

forward(gdata, cdata, axis): Apply forward transform along specified axis.

abstractmethod forward_matrix(): Build forward transform matrix.

class SeparableTransform

Abstract base class for transforms that only apply to one dimension, independent of all others.

abstractmethod backward(cdata, gdata, axis): Apply backward transform along specified axis.

abstractmethod forward(gdata, cdata, axis): Apply forward transform along specified axis.

class Transform: Abstract base class for all transforms.

backward_DFT(cdata, gdata, axis)

backward_disk(cdata, gdata, axis, k, s, local_m): Apply bakward radial transform to data with fixed s and varying m.

forward_DFT(gdata, cdata, axis)

forward_disk(gdata, cdata, axis, k0, k, s, local_m): Apply forward radial transform to data with fixed s and varying m.

reduce_array(data, axis): Return reduced 3D view of array collapsed above and below specified axis.

reduced_view_3(data, axis)

reduced_view_4(data, axis)

reduced_view_5(data, axis)

register_transform(basis, name): Decorator to add transform to basis class dictionary.

GET_DEALIAS_BEFORE_CONVERTING

GET_FFTW_RIGOR

logger