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import numpy
from numpy.linalg import norm
import jinja2
import pyopencl
import pyopencl.array
from pyopencl.elementwise import ElementwiseKernel
from pyopencl.reduction import ReductionKernel
import time
import fdfd_tools.operators
def type_to_C(float_type: numpy.float32 or numpy.float64) -> str:
"""
Returns a string corresponding to the C equivalent of a numpy type.
:param float_type: numpy type: float32, float64, complex64, complex128
:return: string containing the corresponding C type (eg. 'double')
"""
types = {
numpy.float32: 'float',
numpy.float64: 'double',
numpy.complex64: 'cfloat_t',
numpy.complex128: 'cdouble_t',
}
if float_type not in types:
raise Exception('Unsupported type')
return types[float_type]
def shape_source(shape) -> str:
"""
Defines sx, sy, sz C constants specifying the shape of the grid in each of the 3 dimensions.
:param shape: [sx, sy, sz] values.
:return: String containing C source.
"""
sxyz = """
// Field sizes
const int sx = {shape[0]};
const int sy = {shape[1]};
const int sz = {shape[2]};
""".format(shape=shape)
return sxyz
# Defines dix, diy, diz constants used for stepping in the x, y, z directions in a linear array
# (ie, given Ex[i] referring to position (x, y, z), Ex[i+diy] will refer to position (x, y+1, z))
dixyz_source = """
// Convert offset in field xyz to linear index offset
const int dix = 1;
const int diy = sx;
const int diz = sx * sy;
"""
# Given a linear index i and shape sx, sy, sz, defines x, y, and z
# as the 3D indices of the current element (i).
xyz_source = """
// Convert linear index to field index (xyz)
const int z = i / (sx * sy);
const int y = (i - z * sx * sy) / sx;
const int x = (i - y * sx - z * sx * sy);
"""
vec_source = """
if (i >= sx * sy * sz) {
PYOPENCL_ELWISE_CONTINUE;
}
//Pointers into the components of a vectorized vector-field
const int XX = 0;
const int YY = sx * sy * sz;
const int ZZ = sx * sy * sz * 2;
"""
E_ptrs = """
__global cdouble_t *Ex = E + XX;
__global cdouble_t *Ey = E + YY;
__global cdouble_t *Ez = E + ZZ;
"""
H_ptrs = """
__global cdouble_t *Hx = H + XX;
__global cdouble_t *Hy = H + YY;
__global cdouble_t *Hz = H + ZZ;
"""
# Source code for updating the E field; maxes use of dixyz_source.
maxwell_E_source = """
// E update equations
int imx, imy, imz;
if ( x == 0 ) {
imx = i + (sx - 1) * dix;
} else {
imx = i - dix;
}
if ( y == 0 ) {
imy = i + (sy - 1) * diy;
} else {
imy = i - diy;
}
if ( z == 0 ) {
imz = i + (sz - 1) * diz;
} else {
imz = i - diz;
}
// E update equations
{% if pec -%}
if (pec[XX + i]) {
Ex[i] = cdouble_new(0.0, 0.0);
} else
{%- endif -%}
{
cdouble_t tEx = cdouble_mul(Ex[i], oeps[XX + i]);
cdouble_t Dzy = cdouble_mul(cdouble_sub(Hz[i], Hz[imy]), inv_dhy[y]);
cdouble_t Dyz = cdouble_mul(cdouble_sub(Hy[i], Hy[imz]), inv_dhz[z]);
tEx = cdouble_add(tEx, cdouble_sub(Dzy, Dyz));
Ex[i] = cdouble_mul(tEx, Pl[XX + i]);
}
{% if pec -%}
if (pec[YY + i]) {
Ey[i] = cdouble_new(0.0, 0.0);
} else
{%- endif -%}
{
cdouble_t tEy = cdouble_mul(Ey[i], oeps[YY + i]);
cdouble_t Dxz = cdouble_mul(cdouble_sub(Hx[i], Hx[imz]), inv_dhz[z]);
cdouble_t Dzx = cdouble_mul(cdouble_sub(Hz[i], Hz[imx]), inv_dhx[x]);
tEy = cdouble_add(tEy, cdouble_sub(Dxz, Dzx));
Ey[i] = cdouble_mul(tEy, Pl[YY + i]);
}
{% if pec -%}
if (pec[ZZ + i]) {
Ez[i] = cdouble_new(0.0, 0.0);
} else
{%- endif -%}
{
cdouble_t tEz = cdouble_mul(Ez[i], oeps[ZZ + i]);
cdouble_t Dyx = cdouble_mul(cdouble_sub(Hy[i], Hy[imx]), inv_dhx[x]);
cdouble_t Dxy = cdouble_mul(cdouble_sub(Hx[i], Hx[imy]), inv_dhy[y]);
tEz = cdouble_add(tEz, cdouble_sub(Dyx, Dxy));
Ez[i] = cdouble_mul(tEz, Pl[ZZ + i]);
}
"""
# Source code for updating the H field; maxes use of dixyz_source and assumes mu=0
maxwell_H_source = """
// H update equations
int ipx, ipy, ipz;
if ( x == sx - 1 ) {
ipx = i - (sx - 1) * dix;
} else {
ipx = i + dix;
}
if ( y == sy - 1 ) {
ipy = i - (sy - 1) * diy;
} else {
ipy = i + diy;
}
if ( z == sz - 1 ) {
ipz = i - (sz - 1) * diz;
} else {
ipz = i + diz;
}
{% if pmc -%}
if (pmc[XX + i]) {
Hx[i] = cdouble_new(0.0, 0.0);
} else
{%- endif -%}
{
cdouble_t Dzy = cdouble_mul(cdouble_sub(Ez[ipy], Ez[i]), inv_dey[y]);
cdouble_t Dyz = cdouble_mul(cdouble_sub(Ey[ipz], Ey[i]), inv_dez[z]);
{%- if mu -%}
Hx[i] = cdouble_mul(inv_mu[XX + i], cdouble_sub(Dzy, Dyz));
{%- else -%}
Hx[i] = cdouble_sub(Dzy, Dyz);
{%- endif %}
}
{% if pmc -%}
if (pmc[YY + i]) {
Hy[i] = cdouble_new(0.0, 0.0);
} else
{%- endif -%}
{
cdouble_t Dxz = cdouble_mul(cdouble_sub(Ex[ipz], Ex[i]), inv_dez[z]);
cdouble_t Dzx = cdouble_mul(cdouble_sub(Ez[ipx], Ez[i]), inv_dex[x]);
{%- if mu -%}
Hy[i] = cdouble_mul(inv_mu[YY + i], cdouble_sub(Dxz, Dzx));
{%- else -%}
Hy[i] = cdouble_sub(Dxz, Dzx);
{%- endif %}
}
{% if pmc -%}
if (pmc[XX + i]) {
Hx[i] = cdouble_new(0.0, 0.0);
} else
{%- endif -%}
{
cdouble_t Dyx = cdouble_mul(cdouble_sub(Ey[ipx], Ey[i]), inv_dex[x]);
cdouble_t Dxy = cdouble_mul(cdouble_sub(Ex[ipy], Ex[i]), inv_dey[y]);
{%- if mu -%}
Hz[i] = cdouble_mul(inv_mu[ZZ + i], cdouble_sub(Dyx, Dxy));
{%- else -%}
Hz[i] = cdouble_sub(Dyx, Dxy);
{%- endif %}
}
"""
p2e_source = '''
Ex[i] = cdouble_mul(Pr[XX + i], p[XX + i]);
Ey[i] = cdouble_mul(Pr[YY + i], p[YY + i]);
Ez[i] = cdouble_mul(Pr[ZZ + i], p[ZZ + i]);
'''
preamble = '''
#define PYOPENCL_DEFINE_CDOUBLE
#include <pyopencl-complex.h>
'''
ctype = type_to_C(numpy.complex128)
def ptrs(*args):
return [ctype + ' *' + s for s in args]
def create_a(context, shape, mu=False, pec=False, pmc=False):
dhs = [ctype + ' *inv_dh' + a for a in 'xyz']
des = [ctype + ' *inv_de' + a for a in 'xyz']
header = shape_source(shape) + dixyz_source + xyz_source + vec_source + E_ptrs
P2E_kernel = ElementwiseKernel(context,
name='P2E',
preamble=preamble,
operation=header + p2e_source,
arguments=', '.join(ptrs('E', 'p', 'Pr')))
pmc_arg = ['int *pmc']
e2h_source = header + H_ptrs + jinja2.Template(maxwell_H_source).render(mu=mu, pmc=pmc)
E2H_kernel = ElementwiseKernel(context,
name='E2H',
preamble=preamble,
operation=e2h_source,
arguments=', '.join(ptrs('E', 'H', 'inv_mu') + pmc_arg + des))
pec_arg = ['int *pec']
h2e_source = header + H_ptrs + jinja2.Template(maxwell_E_source).render(pec=pec)
H2E_kernel = ElementwiseKernel(context,
name='H2E',
preamble=preamble,
operation=h2e_source,
arguments=', '.join(ptrs('E', 'H', 'oeps', 'Pl') + pec_arg + dhs))
def spmv(E, H, p, idxes, oeps, inv_mu, pec, pmc, Pl, Pr, e):
e2 = P2E_kernel(E, p, Pr, wait_for=e)
e2 = E2H_kernel(E, H, inv_mu, pmc, *idxes[0], wait_for=[e2])
e2 = H2E_kernel(E, H, oeps, Pl, pec, *idxes[1], wait_for=[e2])
return [e2]
return spmv
def create_xr_step(context):
update_xr_source = '''
x[i] = cdouble_add(x[i], cdouble_mul(alpha, p[i]));
r[i] = cdouble_sub(r[i], cdouble_mul(alpha, v[i]));
'''
xr_args = ', '.join(ptrs('x', 'p', 'r', 'v') + [ctype + ' alpha'])
xr_kernel = ElementwiseKernel(context,
name='XR',
preamble=preamble,
operation=update_xr_source,
arguments=xr_args)
def xr_update(x, p, r, v, alpha, e):
return [xr_kernel(x, p, r, v, alpha, wait_for=e)]
return xr_update
def create_rhoerr_step(context):
update_ri_source = '''
(double3)(r[i].real * r[i].real, \
r[i].real * r[i].imag, \
r[i].imag * r[i].imag)
'''
ri_dtype = pyopencl.array.vec.double3
ri_kernel = ReductionKernel(context,
name='RHOERR',
preamble=preamble,
dtype_out=ri_dtype,
neutral='(double3)(0.0, 0.0, 0.0)',
map_expr=update_ri_source,
reduce_expr='a+b',
arguments=ctype + ' *r')
def ri_update(r, e):
g = ri_kernel(r, wait_for=e).astype(ri_dtype).get()
rr, ri, ii = [g[q] for q in 'xyz']
rho = rr + 2j * ri - ii
err = rr + ii
return rho, err
return ri_update
def create_p_step(context):
update_p_source = '''
p[i] = cdouble_add(r[i], cdouble_mul(beta, p[i]));
'''
p_args = ptrs('p', 'r') + [ctype + ' beta']
p_kernel = ElementwiseKernel(context,
name='P',
preamble=preamble,
operation=update_p_source,
arguments=', '.join(p_args))
def p_update(p, r, beta, e):
return [p_kernel(p, r, beta, wait_for=e)]
return p_update
def create_dot(context):
dot_dtype = numpy.complex128
dot_kernel = ReductionKernel(context,
name='dot',
preamble=preamble,
dtype_out=dot_dtype,
neutral='cdouble_new(0.0, 0.0)',
map_expr='cdouble_mul(p[i], v[i])',
reduce_expr='cdouble_add(a, b)',
arguments=ptrs('p', 'v'))
def ri_update(p, v, e):
g = dot_kernel(p, v, wait_for=e)
return g.get()
return ri_update
def cg_solver(omega, dxes, J, epsilon, mu=None, pec=None, pmc=None, adjoint=False,
max_iters=40000, err_thresh=1e-6, context=None, verbose=False):
start_time = time.perf_counter()
b = -1j * omega * J
shape = [d.size for d in dxes[0]]
'''
** In this comment, I use the notation M* = conj(M),
M.T = transpose(M), M' = ctranspose(M), M N = dot(M, N)
This solver uses a symmetrized wave operator M = (L A R) = (L A R).T
(where L = inv(R) are diagonal preconditioner matrices) when
solving the wave equation; therefore, it solves the problem
M y = d
=> (L A R) (inv(R) x) = (L b)
=> A x = b
with x = R y
From the fact that M is symmetric, we can write
(L A R)* = M* = M' = (L A R)' = R' A' L' = R* A' L*
We obtain M* by conjugating all of our arguments (except J).
Then we solve
(R* A' L*) v = (R* b)
and obtain x:
x = L* v
We can accomplish all this simply by conjugating everything (except J) and
reversing the order of L and R
'''
if adjoint:
# Conjugate everything
dxes = [[numpy.conj(d) for d in dd] for dd in dxes]
omega = numpy.conj(omega)
epsilon = numpy.conj(epsilon)
if mu is not None:
mu = numpy.conj(mu)
L, R = fdfd_tools.operators.e_full_preconditioners(dxes)
if adjoint:
b_preconditioned = R @ b
else:
b_preconditioned = L @ b
'''
Allocate GPU memory and load in data
'''
if context is None:
context = pyopencl.create_some_context(False)
queue = pyopencl.CommandQueue(context)
def load_field(v, dtype=numpy.complex128):
return pyopencl.array.to_device(queue, v.astype(dtype))
r = load_field(b_preconditioned) # load preconditioned b into r
H = pyopencl.array.zeros_like(r)
x = pyopencl.array.zeros_like(r)
v = pyopencl.array.zeros_like(r)
p = pyopencl.array.zeros_like(r)
alpha = 1.0 + 0j
rho = 1.0 + 0j
errs = []
inv_dxes = [[load_field(1 / d) for d in dd] for dd in dxes]
oeps = load_field(-omega ** 2 * epsilon)
Pl = load_field(L.diagonal())
Pr = load_field(R.diagonal())
mu = numpy.ones_like(epsilon)
# pec = numpy.zeros_like(epsilon)
# pmc = numpy.zeros_like(epsilon)
if mu is None:
invm = load_field(numpy.array([]))
else:
invm = load_field(1 / mu)
if pec is None:
gpec = load_field(numpy.array([]), dtype=int)
else:
gpec = load_field(pec, dtype=int)
if pmc is None:
gpmc = load_field(numpy.array([]), dtype=int)
else:
gpmc = load_field(pmc, dtype=int)
'''
Generate OpenCL kernels
'''
has_mu, has_pec, has_pmc = [q is not None for q in (mu, pec, pmc)]
a_step_full = create_a(context, shape, has_mu, has_pec, has_pmc)
xr_step = create_xr_step(context)
rhoerr_step = create_rhoerr_step(context)
p_step = create_p_step(context)
dot = create_dot(context)
def a_step(E, H, p, events):
return a_step_full(E, H, p, inv_dxes, oeps, invm, gpec, gpmc, Pl, Pr, events)
'''
Start the solve
'''
start_time2 = time.perf_counter()
_, err2 = rhoerr_step(r, [])
b_norm = numpy.sqrt(err2)
print('b_norm check: ', b_norm)
success = False
for k in range(max_iters):
if verbose:
print('[{:06d}] rho {:.4} alpha {:4.4}'.format(k, rho, alpha), end=' ')
rho_prev = rho
e = xr_step(x, p, r, v, alpha, [])
rho, err2 = rhoerr_step(r, e)
errs += [numpy.sqrt(err2) / b_norm]
if verbose:
print('err', errs[-1])
if errs[-1] < err_thresh:
success = True
break
e = p_step(p, r, rho/rho_prev, [])
e = a_step(v, H, p, e)
alpha = rho / dot(p, v, e)
if k % 1000 == 0:
print(k)
'''
Done solving
'''
time_elapsed = time.perf_counter() - start_time
# Undo preconditioners
if adjoint:
x = (Pl * x).get()
else:
x = (Pr * x).get()
if success:
print('Success', end='')
else:
print('Failure', end=', ')
print(', {} iterations in {} sec: {} iterations/sec \
'.format(k, time_elapsed, k / time_elapsed))
print('final error', errs[-1])
print('overhead {} sec'.format(start_time2 - start_time))
A0 = fdfd_tools.operators.e_full(omega, dxes, epsilon, mu).tocsr()
if adjoint:
# Remember we conjugated all the contents of A earlier
A0 = A0.T
print('Post-everything residual:', norm(A0 @ x - b) / norm(b))
return x