sht.py

# coding=utf-8

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import numpy as np
import torch
import torch.nn as nn
import torch.fft

from torch_harmonics.quadrature import legendre_gauss_weights, lobatto_weights, clenshaw_curtiss_weights
from torch_harmonics.legendre import _precompute_legpoly, _precompute_dlegpoly


class RealSHT(nn.Module):
    r"""
    Defines a module for computing the forward (real-valued) SHT.
    Precomputes Legendre Gauss nodes, weights and associated Legendre polynomials on these nodes.
    The SHT is applied to the last two dimensions of the input

    [1] Schaeffer, N. Efficient spherical harmonic transforms aimed at pseudospectral numerical simulations, G3: Geochemistry, Geophysics, Geosystems.
    [2] Wang, B., Wang, L., Xie, Z.; Accurate calculation of spherical and vector spherical harmonic expansions via spectral element grids; Adv Comput Math.
    """

    def __init__(self, nlat, nlon, lmax=None, mmax=None, grid="equiangular", norm="ortho", csphase=True):
        r"""
        Initializes the SHT Layer, precomputing the necessary quadrature weights

        Parameters:
        nlat: input grid resolution in the latitudinal direction
        nlon: input grid resolution in the longitudinal direction
        grid: grid in the latitude direction (for now only tensor product grids are supported)
        """

        super().__init__()

        self.nlat = nlat
        self.nlon = nlon
        self.grid = grid
        self.norm = norm
        self.csphase = csphase

        # TODO: include assertions regarding the dimensions

        # compute quadrature points and lmax based on the exactness of the quadrature
        if self.grid == "legendre-gauss":
            cost, w = legendre_gauss_weights(nlat, -1, 1)
            # maximum polynomial degree for Gauss Legendre is 2 * nlat - 1 >= 2 * lmax
            # and therefore lmax = nlat - 1 (inclusive)
            self.lmax = lmax or self.nlat
        elif self.grid == "lobatto":
            cost, w = lobatto_weights(nlat, -1, 1)
            # maximum polynomial degree for Gauss Legendre is 2 * nlat - 3 >= 2 * lmax
            # and therefore lmax = nlat - 2 (inclusive)
            self.lmax = lmax or self.nlat - 1
        elif self.grid == "equiangular":
            cost, w = clenshaw_curtiss_weights(nlat, -1, 1)
            # in principle, Clenshaw-Curtiss quadrature is only exact up to polynomial degrees of nlat
            # however, we observe that the quadrature is remarkably accurate for higher degress. This is why we do not
            # choose a lower lmax for now.
            self.lmax = lmax or self.nlat
        else:
            raise (ValueError("Unknown quadrature mode"))

        # apply cosine transform and flip them
        tq = np.flip(np.arccos(cost))

        # determine the dimensions
        self.mmax = mmax or self.nlon // 2 + 1

        # combine quadrature weights with the legendre weights
        weights = torch.from_numpy(w)
        pct = _precompute_legpoly(self.mmax, self.lmax, tq, norm=self.norm, csphase=self.csphase)
        pct = torch.from_numpy(pct)
        weights = torch.einsum("mlk,k->mlk", pct, weights)

        # remember quadrature weights
        self.register_buffer("weights", weights, persistent=False)

    def extra_repr(self):
        r"""
        Pretty print module
        """
        return f"nlat={self.nlat}, nlon={self.nlon},\n lmax={self.lmax}, mmax={self.mmax},\n grid={self.grid}, csphase={self.csphase}"

    def forward(self, x: torch.Tensor):

        if x.dim() < 2:
            raise ValueError(f"Expected tensor with at least 2 dimensions but got {x.dim()} instead")

        assert x.shape[-2] == self.nlat
        assert x.shape[-1] == self.nlon

        # apply real fft in the longitudinal direction
        x = 2.0 * torch.pi * torch.fft.rfft(x, dim=-1, norm="forward")

        # do the Legendre-Gauss quadrature
        x = torch.view_as_real(x)

        # distributed contraction: fork
        out_shape = list(x.size())
        out_shape[-3] = self.lmax
        out_shape[-2] = self.mmax
        xout = torch.zeros(out_shape, dtype=x.dtype, device=x.device)

        # contraction
        xout[..., 0] = torch.einsum("...km,mlk->...lm", x[..., : self.mmax, 0], self.weights.to(x.dtype))
        xout[..., 1] = torch.einsum("...km,mlk->...lm", x[..., : self.mmax, 1], self.weights.to(x.dtype))
        x = torch.view_as_complex(xout)

        return x


class InverseRealSHT(nn.Module):
    r"""
    Defines a module for computing the inverse (real-valued) SHT.
    Precomputes Legendre Gauss nodes, weights and associated Legendre polynomials on these nodes.
    nlat, nlon: Output dimensions
    lmax, mmax: Input dimensions (spherical coefficients). For convenience, these are inferred from the output dimensions

    [1] Schaeffer, N. Efficient spherical harmonic transforms aimed at pseudospectral numerical simulations, G3: Geochemistry, Geophysics, Geosystems.
    [2] Wang, B., Wang, L., Xie, Z.; Accurate calculation of spherical and vector spherical harmonic expansions via spectral element grids; Adv Comput Math.
    """

    def __init__(self, nlat, nlon, lmax=None, mmax=None, grid="equiangular", norm="ortho", csphase=True):

        super().__init__()

        self.nlat = nlat
        self.nlon = nlon
        self.grid = grid
        self.norm = norm
        self.csphase = csphase

        # compute quadrature points
        if self.grid == "legendre-gauss":
            cost, _ = legendre_gauss_weights(nlat, -1, 1)
            self.lmax = lmax or self.nlat
        elif self.grid == "lobatto":
            cost, _ = lobatto_weights(nlat, -1, 1)
            self.lmax = lmax or self.nlat - 1
        elif self.grid == "equiangular":
            cost, _ = clenshaw_curtiss_weights(nlat, -1, 1)
            self.lmax = lmax or self.nlat
        else:
            raise (ValueError("Unknown quadrature mode"))

        # apply cosine transform and flip them
        t = np.flip(np.arccos(cost))

        # determine the dimensions
        self.mmax = mmax or self.nlon // 2 + 1

        pct = _precompute_legpoly(self.mmax, self.lmax, t, norm=self.norm, inverse=True, csphase=self.csphase)
        pct = torch.from_numpy(pct)

        # register buffer
        self.register_buffer("pct", pct, persistent=False)

    def extra_repr(self):
        r"""
        Pretty print module
        """
        return f"nlat={self.nlat}, nlon={self.nlon},\n lmax={self.lmax}, mmax={self.mmax},\n grid={self.grid}, csphase={self.csphase}"

    def forward(self, x: torch.Tensor):

        if len(x.shape) < 2:
            raise ValueError(f"Expected tensor with at least 2 dimensions but got {len(x.shape)} instead")

        assert x.shape[-2] == self.lmax
        assert x.shape[-1] == self.mmax

        # Evaluate associated Legendre functions on the output nodes
        x = torch.view_as_real(x)

        rl = torch.einsum("...lm, mlk->...km", x[..., 0], self.pct.to(x.dtype))
        im = torch.einsum("...lm, mlk->...km", x[..., 1], self.pct.to(x.dtype))
        xs = torch.stack((rl, im), -1)

        # apply the inverse (real) FFT
        x = torch.view_as_complex(xs)
        x = torch.fft.irfft(x, n=self.nlon, dim=-1, norm="forward")

        return x


class RealVectorSHT(nn.Module):
    r"""
    Defines a module for computing the forward (real) vector SHT.
    Precomputes Legendre Gauss nodes, weights and associated Legendre polynomials on these nodes.
    The SHT is applied to the last three dimensions of the input.

    [1] Schaeffer, N. Efficient spherical harmonic transforms aimed at pseudospectral numerical simulations, G3: Geochemistry, Geophysics, Geosystems.
    [2] Wang, B., Wang, L., Xie, Z.; Accurate calculation of spherical and vector spherical harmonic expansions via spectral element grids; Adv Comput Math.
    """

    def __init__(self, nlat, nlon, lmax=None, mmax=None, grid="equiangular", norm="ortho", csphase=True):
        r"""
        Initializes the vector SHT Layer, precomputing the necessary quadrature weights

        Parameters:
        nlat: input grid resolution in the latitudinal direction
        nlon: input grid resolution in the longitudinal direction
        grid: type of grid the data lives on
        """

        super().__init__()

        self.nlat = nlat
        self.nlon = nlon
        self.grid = grid
        self.norm = norm
        self.csphase = csphase

        # compute quadrature points
        if self.grid == "legendre-gauss":
            cost, w = legendre_gauss_weights(nlat, -1, 1)
            self.lmax = lmax or self.nlat
        elif self.grid == "lobatto":
            cost, w = lobatto_weights(nlat, -1, 1)
            self.lmax = lmax or self.nlat - 1
        elif self.grid == "equiangular":
            cost, w = clenshaw_curtiss_weights(nlat, -1, 1)
            self.lmax = lmax or self.nlat
        else:
            raise (ValueError("Unknown quadrature mode"))

        # apply cosine transform and flip them
        tq = np.flip(np.arccos(cost))

        # determine the dimensions
        self.mmax = mmax or self.nlon // 2 + 1

        weights = torch.from_numpy(w)
        dpct = _precompute_dlegpoly(self.mmax, self.lmax, tq, norm=self.norm, csphase=self.csphase)
        dpct = torch.from_numpy(dpct)

        # combine integration weights, normalization factor in to one:
        l = torch.arange(0, self.lmax)
        norm_factor = 1.0 / l / (l + 1)
        norm_factor[0] = 1.0
        weights = torch.einsum("dmlk,k,l->dmlk", dpct, weights, norm_factor)
        # since the second component is imaginary, we need to take complex conjugation into account
        weights[1] = -1 * weights[1]

        # remember quadrature weights
        self.register_buffer("weights", weights, persistent=False)

    def extra_repr(self):
        r"""
        Pretty print module
        """
        return f"nlat={self.nlat}, nlon={self.nlon},\n lmax={self.lmax}, mmax={self.mmax},\n grid={self.grid}, csphase={self.csphase}"

    def forward(self, x: torch.Tensor):

        if x.dim() < 3:
            raise ValueError(f"Expected tensor with at least 3 dimensions but got {x.dim()} instead")

        assert x.shape[-2] == self.nlat
        assert x.shape[-1] == self.nlon

        # apply real fft in the longitudinal direction
        x = 2.0 * torch.pi * torch.fft.rfft(x, dim=-1, norm="forward")

        # do the Legendre-Gauss quadrature
        x = torch.view_as_real(x)

        # distributed contraction: fork
        out_shape = list(x.size())
        out_shape[-3] = self.lmax
        out_shape[-2] = self.mmax
        xout = torch.zeros(out_shape, dtype=x.dtype, device=x.device)

        # contraction - spheroidal component
        # real component
        xout[..., 0, :, :, 0] = torch.einsum("...km,mlk->...lm", x[..., 0, :, : self.mmax, 0], self.weights[0].to(x.dtype)) - torch.einsum(
            "...km,mlk->...lm", x[..., 1, :, : self.mmax, 1], self.weights[1].to(x.dtype)
        )

        # iamg component
        xout[..., 0, :, :, 1] = torch.einsum("...km,mlk->...lm", x[..., 0, :, : self.mmax, 1], self.weights[0].to(x.dtype)) + torch.einsum(
            "...km,mlk->...lm", x[..., 1, :, : self.mmax, 0], self.weights[1].to(x.dtype)
        )

        # contraction - toroidal component
        # real component
        xout[..., 1, :, :, 0] = -torch.einsum("...km,mlk->...lm", x[..., 0, :, : self.mmax, 1], self.weights[1].to(x.dtype)) - torch.einsum(
            "...km,mlk->...lm", x[..., 1, :, : self.mmax, 0], self.weights[0].to(x.dtype)
        )
        # imag component
        xout[..., 1, :, :, 1] = torch.einsum("...km,mlk->...lm", x[..., 0, :, : self.mmax, 0], self.weights[1].to(x.dtype)) - torch.einsum(
            "...km,mlk->...lm", x[..., 1, :, : self.mmax, 1], self.weights[0].to(x.dtype)
        )

        return torch.view_as_complex(xout)


class InverseRealVectorSHT(nn.Module):
    r"""
    Defines a module for computing the inverse (real-valued) vector SHT.
    Precomputes Legendre Gauss nodes, weights and associated Legendre polynomials on these nodes.

    [1] Schaeffer, N. Efficient spherical harmonic transforms aimed at pseudospectral numerical simulations, G3: Geochemistry, Geophysics, Geosystems.
    [2] Wang, B., Wang, L., Xie, Z.; Accurate calculation of spherical and vector spherical harmonic expansions via spectral element grids; Adv Comput Math.
    """

    def __init__(self, nlat, nlon, lmax=None, mmax=None, grid="equiangular", norm="ortho", csphase=True):

        super().__init__()

        self.nlat = nlat
        self.nlon = nlon
        self.grid = grid
        self.norm = norm
        self.csphase = csphase

        # compute quadrature points
        if self.grid == "legendre-gauss":
            cost, _ = legendre_gauss_weights(nlat, -1, 1)
            self.lmax = lmax or self.nlat
        elif self.grid == "lobatto":
            cost, _ = lobatto_weights(nlat, -1, 1)
            self.lmax = lmax or self.nlat - 1
        elif self.grid == "equiangular":
            cost, _ = clenshaw_curtiss_weights(nlat, -1, 1)
            self.lmax = lmax or self.nlat
        else:
            raise (ValueError("Unknown quadrature mode"))

        # apply cosine transform and flip them
        t = np.flip(np.arccos(cost))

        # determine the dimensions
        self.mmax = mmax or self.nlon // 2 + 1

        dpct = _precompute_dlegpoly(self.mmax, self.lmax, t, norm=self.norm, inverse=True, csphase=self.csphase)
        dpct = torch.from_numpy(dpct)

        # register weights
        self.register_buffer("dpct", dpct, persistent=False)

    def extra_repr(self):
        r"""
        Pretty print module
        """
        return f"nlat={self.nlat}, nlon={self.nlon},\n lmax={self.lmax}, mmax={self.mmax},\n grid={self.grid}, csphase={self.csphase}"

    def forward(self, x: torch.Tensor):

        if x.dim() < 3:
            raise ValueError(f"Expected tensor with at least 3 dimensions but got {x.dim()} instead")

        assert x.shape[-2] == self.lmax
        assert x.shape[-1] == self.mmax

        # Evaluate associated Legendre functions on the output nodes
        x = torch.view_as_real(x)

        # contraction - spheroidal component
        # real component
        srl = torch.einsum("...lm,mlk->...km", x[..., 0, :, :, 0], self.dpct[0].to(x.dtype)) - torch.einsum("...lm,mlk->...km", x[..., 1, :, :, 1], self.dpct[1].to(x.dtype))
        # iamg component
        sim = torch.einsum("...lm,mlk->...km", x[..., 0, :, :, 1], self.dpct[0].to(x.dtype)) + torch.einsum("...lm,mlk->...km", x[..., 1, :, :, 0], self.dpct[1].to(x.dtype))

        # contraction - toroidal component
        # real component
        trl = -torch.einsum("...lm,mlk->...km", x[..., 0, :, :, 1], self.dpct[1].to(x.dtype)) - torch.einsum("...lm,mlk->...km", x[..., 1, :, :, 0], self.dpct[0].to(x.dtype))
        # imag component
        tim = torch.einsum("...lm,mlk->...km", x[..., 0, :, :, 0], self.dpct[1].to(x.dtype)) - torch.einsum("...lm,mlk->...km", x[..., 1, :, :, 1], self.dpct[0].to(x.dtype))

        # reassemble
        s = torch.stack((srl, sim), -1)
        t = torch.stack((trl, tim), -1)
        xs = torch.stack((s, t), -4)

        # apply the inverse (real) FFT
        x = torch.view_as_complex(xs)
        x = torch.fft.irfft(x, n=self.nlon, dim=-1, norm="forward")

        return x