def plot_interior_displacement(self, soln):
nxy = 40
nz = 40
d = 0
xs = np.linspace(-10, 10, nxy)
zs = np.linspace(-0.1, -4.0, nz)
X, Y, Z = np.meshgrid(xs, xs, zs)
obs_pts = np.array([X.flatten(), Y.flatten(), Z.flatten()]).T.copy()
t = Timer(output_fnc=logger.debug)
interior_disp = -interior_integral(
obs_pts,
obs_pts,
self.all_mesh,
soln,
'elasticT3',
3,
8,
self.k_params,
self.float_type,
fmm_params=None #[100, 3.0, 3000, 25]
).reshape((nxy, nxy, nz, 3))
t.report('eval %.2E interior pts' % obs_pts.shape[0])
for i in range(nz):
plt.figure()
plt.pcolor(xs, xs, interior_disp[:, :, i, d])
plt.colorbar()
plt.title('at z = ' + ('%.3f' % zs[i]) + ' u' +
['x', 'y', 'z'][d])
plt.show()
def build_and_solve_T_regularized(data):
timer = Timer(output_fnc=logger.debug)
cs = build_constraints(data.surface_tris, data.fault_tris,
data.all_mesh[0], data.all_mesh[1], data.gauss_z)
timer.report("Constraints")
T_op = RegularizedSparseIntegralOp(
6,
6,
6,
2,
5,
2.5,
'elasticRT3',
'elasticRT3',
data.k_params,
data.all_mesh[0],
data.all_mesh[1],
data.float_type,
# farfield_op_type = TriToTriDirectFarfieldOp
farfield_op_type=FMMFarfieldOp(mac=2.5, pts_per_cell=100, order=2))
import tectosaur.fmm.tsfmm as tsfmm
tsfmm.report_interactions(T_op.farfield.fmm)
timer.report("Integrals")
mass_op = MultOp(MassOp(3, data.all_mesh[0], data.all_mesh[1]), 0.5)
iop = SumOp([T_op, mass_op])
timer.report('mass op/sum op')
soln = solve.iterative_solve(iop, cs, tol=1e-5)
timer.report("Solve")
return soln
def benchmark():
tct.logger.setLevel(logging.INFO)
m = 1
n = 50
K = 'elasticRH3'
m1 = mesh_gen.make_rect(n, n, [[-1, 0, 1], [-1, 0, -1], [1, 0, -1], [1, 0, 1]])
m2 = mesh_gen.make_rect(n, n, [[-3, 0, 1], [-3, 0, -1], [-2, 0, -1], [-2, 0, 1]])
x = np.random.rand(m2[1].shape[0] * 9)
t = Timer()
new_pts, new_tris = concat(m1, m2)
n_obs_tris = m1[1].shape[0]
sparse_op = RegularizedSparseIntegralOp(
8,8,8,2,5,2.5,K,K,[1.0, 0.25],new_pts,new_tris,
np.float32,TriToTriDirectFarfieldOp,
obs_subset = np.arange(0,n_obs_tris),
src_subset = np.arange(n_obs_tris,new_tris.shape[0])
)
t.report('assemble matrix free')
for i in range(m):
y1 = sparse_op.dot(x)
t.report('matrix free mv x10')
fmm = FMMFarfieldOp(mac = 4.5, pts_per_cell = 300)(
2, K, [1.0, 0.25], new_pts, new_tris, np.float32,
obs_subset = np.arange(0,n_obs_tris),
src_subset = np.arange(n_obs_tris,new_tris.shape[0])
)
report_interactions(fmm.fmm_obj)
t.report('setup fmm')
y2 = fmm.dot(x)
t.report('fmm mv x10')
print(y1, y2)
def benchmark_vert_adj():
from tectosaur.util.timer import Timer
import tectosaur.mesh.find_near_adj as find_near_adj
from tectosaur.nearfield.pairs_integrator import PairsIntegrator
kernel = 'elasticH3'
params = [1.0, 0.25]
float_type = np.float32
L = 5
nq_vert_adjacent = 7
nx = ny = int(2**L / np.sqrt(2))
t = Timer()
pts, tris = mesh_gen.make_rect(
nx, ny, [[-1, -1, 0], [-1, 1, 0], [1, 1, 0], [1, -1, 0]])
logger.debug('n_tris: ' + str(tris.shape[0]))
t.report('make rect')
close_or_touch_pairs = find_near_adj.find_close_or_touching(
pts, tris, 1.25)
nearfield_pairs, va, ea = find_near_adj.split_adjacent_close(
close_or_touch_pairs, tris)
t.report('find near')
pairs_int = PairsIntegrator(kernel, params, float_type, 1, 1, pts, tris)
t.report('setup integrator')
va_mat_rot = pairs_int.vert_adj(nq_vert_adjacent, va)
t.report('vert adj')
def build_and_solve_H_regularized(data):
timer = Timer(output_fnc=logger.debug)
cs = build_constraints(data.surface_tris, data.fault_tris,
data.all_mesh[0], data.all_mesh[1], data.gauss_z)
# For H, we need to constrain the edges of the surface to have 0 displacement.
cs.extend(free_edge_constraints(data.surface_tris))
timer.report("Constraints")
H_op = RegularizedSparseIntegralOp(
6,
6,
6,
2,
5,
2.5,
'elasticRH3',
'elasticRH3',
data.k_params,
data.all_mesh[0],
data.all_mesh[1],
data.float_type,
# farfield_op_type = TriToTriDirectFarfieldOp
farfield_op_type=FMMFarfieldOp(mac=2.5, pts_per_cell=100, order=2))
iop = SumOp([H_op])
timer.report("Integrals")
soln = solve.iterative_solve(iop, cs, tol=1e-5)
timer.report("Solve")
return soln
def check_for_intersections(m):
pts, tris = m
close_or_touch_pairs = find_near_adj.find_close_or_touching(pts, tris, pts, tris, 2.0)
nearfield_pairs, va, ea = find_near_adj.split_adjacent_close(close_or_touch_pairs, tris, tris)
bad_pairs = []
t = Timer(output_fnc = logger.info)
# Three situations:
# 1) Nearfield pair is intersection
bad_pairs.extend(check_for_intersections_nearfield(pts, tris, nearfield_pairs))
t.report('nearfield')
# 2) Vertex adjacent pair actually intersects beyond just the vertex
# We can test for this by moving the shared vertex short distance
# along one of the edges of the first triangle. If there is still
# an intersection, then the triangles intersect at locations besides
# just the shared vertex.
bad_pairs.extend(check_for_intersections_va(pts, tris, va))
t.report('va')
# 3) Edge adjacent pair is actually coincident. <-- Easy!
bad_pairs.extend(check_for_intersections_ea(pts, tris, ea))
t.report('ea')
return np.array(bad_pairs, dtype = np.int64)
def main():
#python examples/okada.py 31 7 5.0 0.0 0.0 T
#python examples/okada.py 31 7 5.0 0.0 0.0 H
#python examples/okada.py 31 7 5.0 -0.5 -1.0 T
#python examples/okada.py 31 7 5.0 -0.5 -1.0 H
t = Timer(output_fnc=logger.info)
obj = Okada(int(sys.argv[1]),
n_fault=int(sys.argv[2]),
surface_w=float(sys.argv[3]),
top_depth=float(sys.argv[4]),
gauss_z=float(sys.argv[5]),
verbose=True)
# obj.plot_mesh()
if sys.argv[6] == 'T':
soln = obj.run(build_and_solve=build_and_solve_T_regularized)
else:
soln = obj.run(build_and_solve=build_and_solve_H_regularized)
t.report('tectosaur')
okada_soln = obj.okada_exact()
# np.save('okada_soln_for_plot.npy', [obj.all_mesh, obj.surface_tris, obj.fault_tris, soln, okada_soln])
t.report('okada')
obj.xsec_plot([soln], okada_soln)
# obj.plot_interior_displacement(soln)
obj.print_error(soln, okada_soln)
t.report('check')
def __init__(self, m, cfg):
cfg = calc_derived_constants(cfg)
self.cfg = cfg
if not 'Timer' in self.cfg or self.cfg['Timer'] is None:
self.cfg['Timer'] = lambda: Timer(output_fnc = lambda x: None)
self.setup_mesh(m)
self.setup_edge_bcs()
def make_meshes(surf_w, fault_L, top_depth, n_surf, n_fault):
t = Timer(output_fnc=logger.debug)
surface = make_free_surface(surf_w, n_surf)
t.report('make free surface')
fault = make_fault(fault_L, top_depth, n_fault)
t.report('make fault')
all_mesh = mesh_modify.concat(surface, fault)
t.report('concat meshes')
surface_tris = all_mesh[1][:surface[1].shape[0]]
fault_tris = all_mesh[1][surface[1].shape[0]:]
return all_mesh, surface_tris, fault_tris
def compile_module(tmpl, tmpl_name, save_code, tmpl_args):
if tmpl_args is None:
tmpl_args = dict()
code = template_with_mako(tmpl, tmpl_args)
t = Timer(output_fnc = logger.debug)
logger.debug('start compiling ' + tmpl_name)
if save_code:
save_code_to_tmp(code)
module_info = dict()
module_info['tmpl_args'] = tmpl_args
module_info['module'] = compile(code)
t.report('compile')
gpu_module[tmpl_name] = gpu_module.get(tmpl_name, []) + [module_info]
return module_info['module']
def __init__(self, m, cfg): #, which_side):
cfg = calc_derived_constants(cfg)
# self.which_side = which_side
self.cfg = cfg
self.cfg['Timer'] = self.cfg.get(
'Timer', lambda: Timer(output_fnc=lambda x: None))
self.setup_mesh(m)
self.setup_edge_bcs()
self.disp_deficit = np.zeros(m.n_dofs('surf'))
self.old_slip_deficit = np.zeros(m.n_dofs('fault'))
def benchmark_adjacency():
from tectosaur.mesh.mesh_gen import make_rect
from tectosaur.util.timer import Timer
L = 8
nx = ny = int(2**L / np.sqrt(2))
t = Timer()
m = make_rect(nx, ny, [[-1, -1, 0], [-1, 1, 0], [1, 1, 0], [1, -1, 0]])
logger.debug('n_tris: ' + str(m[1].shape[0]))
t.report('make')
close_pairs = find_close_or_touching(m[0], m[1], 1.25)
t.report('close or touching')
close, va, ea = split_adjacent_close(close_pairs, m[1])
t.report('find adj')
def displacement_bie(sm, pr, m, input, solve_for, op_type):
selfop = MassOp(3, m[0], m[1])
t = Timer()
Uop = op_type([],
1,
1,
5,
3,
4,
4.0,
'U',
sm,
pr,
m[0],
m[1],
use_tables=True)
Uop2 = SparseIntegralOp([],
1,
1,
5,
3,
4,
4.0,
'U',
sm,
pr,
m[0],
m[1],
use_tables=True)
v = np.random.rand(Uop2.shape[1])
a = Uop2.dot(v)
b = Uop.dot(v)
import ipdb
ipdb.set_trace()
t.report('U')
Top = SparseIntegralOp([],
1,
1,
5,
3,
4,
4.0,
'T',
sm,
pr,
m[0],
m[1],
use_tables=True)
t.report('T')
if solve_for is 't':
lhs = Uop
rhs = Top.dot(input) + selfop.dot(input)
else:
lhs = SumOp([Top, selfop])
rhs = Uop.dot(input)
return lhs, rhs
def test_c2e(request):
n = 10
m1 = mesh_gen.make_rect(n, n, [[-1, 0, 1], [-1, 0, -1], [1, 0, -1], [1, 0, 1]])
m2 = mesh_gen.make_rect(n, n, [[-3, 0, 1], [-3, 0, -1], [-2, 0, -1], [-2, 0, 1]])
K = 'elasticRH3'
t = Timer()
cfg = make_config(K, [1.0, 0.25], 1.1, 2.5, 2, np.float64, treecode = True)
tree1 = make_tree(m1, cfg, 100)
tree2 = make_tree(m2, cfg, 100)
print(len(tree1.nodes))
fmm = FMM(tree1, m1, tree2, m2, cfg)
u2e_ops = []
for n in tree2.nodes:
UT, eig, V = fmm.u2e_ops
alpha = cfg.alpha
R = n.bounds.R
inv_eig = R * eig / ((R * eig) ** 2 + alpha ** 2)
u2e_ops.append((V * inv_eig).dot(UT))
return np.array(u2e_ops)
def traction_bie(sm, pr, m, input, solve_for, op_type):
# Why am I not able to get the same accuracy with the traction BIE as I am
# with the displacement BIE? Is there something wrong with how I calculate things?
# Maybe with the traction inputs?
# TODO: MAYBE THIS IS RELATED TO HIGH ERROR IN THE H OPERATOR?
selfop = MassOp(3, m[0], m[1])
selfop.mat *= -1
nqv = 15
t = Timer()
Hop = SparseIntegralOp([],
1,
1, (nqv, nqv * 2, nqv),
3,
6,
4.0,
'H',
sm,
pr,
m[0],
m[1],
use_tables=True)
t.report('H')
Aop = SparseIntegralOp([],
1,
1,
7,
3,
6,
4.0,
'A',
sm,
pr,
m[0],
m[1],
use_tables=True)
t.report('A')
if solve_for is 't':
lhs = SumOp([Aop, selfop])
rhs = Hop.dot(input)
else:
lhs = Hop
rhs = Aop.dot(input) + selfop.dot(input)
return lhs, rhs
def build_c2e(tree, check_r, equiv_r, cfg):
t = Timer()
assembler = FarfieldTriMatrix(cfg.K.name, cfg.params, 4, np.float64)
check_surf = inscribe_surf(Ball(center = (0,0,0), R = 1), check_r, cfg.surf)
equiv_surf = inscribe_surf(Ball(center = (0,0,0), R = 1), equiv_r, cfg.surf)
new_pts, new_tris = concat(check_surf, equiv_surf)
n_check_tris = check_surf[1].shape[0]
check_tris = new_tris[:n_check_tris]
equiv_tris = new_tris[n_check_tris:]
mat = assembler.assemble(new_pts, check_tris, equiv_tris)
nrows = mat.shape[0] * 9
ncols = mat.shape[3] * 9
equiv_to_check = mat.reshape((nrows, ncols))
t.report('build e2cs')
U, eig, VT = np.linalg.svd(equiv_to_check)
out = (U.T.copy(), eig.copy(), VT.T.copy())
t.report('svd')
return out
async def farfield_dot(self, v):
t = Timer(output_fnc=logger.debug)
logger.debug("start farfield_dot")
out = await self.farfield.async_dot(v)
t.report('farfield_dot')
return out
async def nearfield_dot(self, v):
t = Timer(output_fnc=logger.debug)
logger.debug("start nearfield_dot")
out = self.nearfield.dot(v)
t.report("nearfield_dot")
return out
def benchmark_bsrmv():
from tectosaur.util.timer import Timer
float_type = np.float32
blocksize = 9
nb = 100000
nnzbs = 5000000
t = Timer()
rows = np.random.randint(nb, size=(nnzbs))
cols = np.random.randint(nb, size=(nnzbs))
data = np.ones((nnzbs, blocksize, blocksize))
x = np.random.rand(nb * blocksize).astype(float_type)
t.report('random')
mat_bcoo = sparse.BCOOMatrix(rows, cols, data,
(nb * blocksize, nb * blocksize))
t.report('make bcoo')
mat_bsr = mat_bcoo.to_bsr()
t.report('make bsr')
mat_sp = mat_bsr.to_scipy()
t.report('make scipy')
for i in range(3):
y = mat_bsr.dot(x)
t.report('bsr mv')
y2 = mat_bcoo.dot(x)
t.report('bcoo mv')
correct = mat_sp.dot(x)
t.report('scipy mv')
np.testing.assert_almost_equal(y, correct, 2)
np.testing.assert_almost_equal(y2, correct, 2)
def __init__(self, pts, tris, obs_subset, src_subset, nq_coincident,
nq_edge_adj, nq_vert_adjacent, nq_far, nq_near,
near_threshold, K_near_name, K_far_name, params, float_type):
n_obs_dofs = obs_subset.shape[0] * 9
n_src_dofs = src_subset.shape[0] * 9
self.shape = (n_obs_dofs, n_src_dofs)
timer = Timer(output_fnc=logger.debug, tabs=1)
pairs_int = PairsIntegrator(K_near_name, params, float_type, nq_far,
nq_near, pts, tris)
correction_pairs_int = PairsIntegrator(K_far_name, params, float_type,
nq_far, nq_near, pts, tris)
timer.report('setup pairs integrator')
co_tris = np.intersect1d(obs_subset, src_subset)
co_indices = np.array([co_tris, co_tris]).T.copy()
co_dofs = to_dof_space(co_indices, obs_subset, src_subset)
co_mat = pairs_int.coincident(nq_coincident, co_indices)
timer.report("Coincident")
co_mat_correction = correction_pairs_int.correction(co_indices, True)
timer.report("Coincident correction")
close_or_touch_pairs = find_near_adj.find_close_or_touching(
pts, tris[obs_subset], pts, tris[src_subset], near_threshold)
nearfield_pairs_dofs, va_dofs, ea_dofs = find_near_adj.split_adjacent_close(
close_or_touch_pairs, tris[obs_subset], tris[src_subset])
nearfield_pairs = to_tri_space(nearfield_pairs_dofs, obs_subset,
src_subset)
va = to_tri_space(va_dofs, obs_subset, src_subset)
va = np.hstack((va, np.zeros((va.shape[0], 1))))
ea = resolve_ea_rotation(tris,
to_tri_space(ea_dofs, obs_subset, src_subset))
timer.report("Find nearfield/adjacency")
ea_mat_rot = pairs_int.edge_adj(nq_edge_adj, ea)
timer.report("Edge adjacent")
if ea.shape[0] == 0:
ea_mat_correction = 0 * ea_mat_rot
else:
ea_mat_correction = correction_pairs_int.correction(
ea[:, :2], False)
timer.report("Edge adjacent correction")
va_mat_rot = pairs_int.vert_adj(nq_vert_adjacent, va)
timer.report("Vert adjacent")
va_mat_correction = correction_pairs_int.correction(va[:, :2], False)
timer.report("Vert adjacent correction")
nearfield_mat = pairs_int.nearfield(nearfield_pairs)
timer.report("Nearfield")
nearfield_correction = correction_pairs_int.correction(
nearfield_pairs, False)
timer.report("Nearfield correction")
self.mat = build_nearfield(
self.shape, (co_mat - co_mat_correction, co_dofs),
(ea_mat_rot - ea_mat_correction, ea_dofs[:, :2]),
(va_mat_rot - va_mat_correction, va_dofs[:, :2]),
(nearfield_mat - nearfield_correction, nearfield_pairs_dofs))
timer.report("Assemble matrix")
self.mat_no_correction = build_nearfield(
self.shape,
(co_mat, co_dofs),
(ea_mat_rot, ea_dofs[:, :2]),
(va_mat_rot, va_dofs[:, :2]),
(nearfield_mat, nearfield_pairs_dofs),
)
timer.report("Assemble uncorrected matrix")
def test_tri_fmm_full():
np.random.seed(100)
n = 40
K = 'elasticRA3'
m1 = mesh_gen.make_rect(n, n, [[-1, 0, 1], [-1, 0, -1], [1, 0, -1], [1, 0, 1]])
m2 = mesh_gen.make_rect(n, n, [[-3, 0, 1], [-3, 0, -1], [-2, 0, -1], [-2, 0, 1]])
x = np.random.rand(m2[1].shape[0] * 9)
t = Timer()
cfg = make_config(K, [1.0, 0.25], 1.1, 4.5, 2, np.float32)
tree1 = make_tree(m1, cfg, 100)
tree2 = make_tree(m2, cfg, 100)
print('n_nodes: ', str(len(tree1.nodes)))
fmm = FMM(tree1, m1, tree2, m2, cfg)
report_interactions(fmm)
fmmeval = FMMEvaluator(fmm)
t.report('setup fmm')
full = op(m1, m2, K = K)
t.report('setup dense')
t.report('gen x')
y1 = full.dot(x)
t.report('dense mv')
x_tree = fmm.to_tree(x)
import taskloaf as tsk
async def call_fmm(tsk_w):
return (await fmmeval.eval(tsk_w, x_tree))
fmm_res = tsk.run(call_fmm)
y2 = fmm.to_orig(fmm_res)
t.report('fmm mv')
np.testing.assert_almost_equal(y1, y2)
def mv(v):
t = Timer(output_fnc = logger.debug)
mv.iter += 1
out = cm.T.dot(iop.dot(cm.dot(v)))
t.report('iteration # ' + str(mv.iter))
return out
def runner(solve_for, use_bie, refine, use_fmm):
if use_fmm == 'fmm':
op_type = FMMIntegralOp
else:
op_type = SparseIntegralOp
a = 1.0
b = 2.0
sm = 1.0
pr = 0.25
pa = 3.0
pb = 1.0
center = [0, 0, 0]
ua, ub = correct_displacement(a, b, pa, -pb, sm, pr, np.array([a, b]))
print(correct_displacement(a, b, pa, -pb, sm, pr, np.array([a, b])))
print(correct_displacement(a, b, -pa, -pb, sm, pr, np.array([a, b])))
print(correct_displacement(a, b, -pa, pb, sm, pr, np.array([a, b])))
print(correct_displacement(a, b, pa, pb, sm, pr, np.array([a, b])))
t = Timer()
m_inner = mesh.make_sphere(center, a, refine)
m_outer = mesh.make_sphere(center, b, refine)
m_outer = mesh.flip_normals(m_outer)
t.report('make meshes')
m = mesh.concat(m_inner, m_outer)
t.report('concat meshes')
print(m[1].shape)
n_inner = m_inner[1].shape[0]
inner_tris = m[1][:n_inner]
outer_tris = m[1][n_inner:]
tri_pts = m[0][m[1]]
# mesh.plot_mesh3d(*m)
cs = build_constraints(inner_tris, outer_tris, m, solve_for)
t.report('build constraints')
# solving: u(x) + int(T*u) = int(U*t)
# values in radial direction because the sphere is centered at (0,0,0)
r = tri_pts - np.array(center)[np.newaxis, np.newaxis, :]
input_nd = r / np.linalg.norm(r, axis=2)[:, :, np.newaxis]
if solve_for is 't':
input_nd[:n_inner] *= ua
input_nd[n_inner:] *= ub
else:
input_nd[:n_inner] *= pa
input_nd[n_inner:] *= pb
input = input_nd.reshape(-1)
if use_bie == 'u':
lhs, rhs = displacement_bie(sm, pr, m, input, solve_for, op_type)
elif use_bie == 't':
lhs, rhs = traction_bie(sm, pr, m, input, solve_for, op_type)
# soln = direct_solve(lhs, cs, rhs)
soln = iterative_solve(lhs, cs, rhs)
soln = np.array(soln).reshape((int(lhs.shape[0] / 9), 3, 3))
soln_mag = np.sqrt(np.sum(soln**2, axis=2))
inner_soln_mag = soln_mag[:n_inner]
outer_soln_mag = soln_mag[n_inner:]
mean_inner = np.mean(inner_soln_mag)
mean_outer = np.mean(outer_soln_mag)
print(np.std(inner_soln_mag))
print(np.std(outer_soln_mag))
if solve_for is 't':
print(mean_inner, pa)
print(mean_outer, pb)
np.testing.assert_almost_equal(mean_inner, pa, 1)
np.testing.assert_almost_equal(mean_outer, pb, 1)
else:
print(mean_inner, ua)
print(mean_outer, ub)
np.testing.assert_almost_equal(mean_inner, ua, 1)
np.testing.assert_almost_equal(mean_outer, ub, 1)
def iterative_solve(iop, constraints, rhs=None, tol=1e-8):
timer = Timer(output_fnc=logger.debug)
cm, c_rhs, _ = build_constraint_matrix(constraints, iop.shape[1])
timer.report('Build constraint matrix')
cm = cm.tocsr()
timer.report('constraint matrix tocsr')
cmT = cm.T
if rhs is None:
rhs_constrained = cmT.dot(-iop.dot(c_rhs))
else:
rhs_constrained = cmT.dot(rhs - iop.dot(c_rhs))
timer.report('constrain rhs')
n = rhs_constrained.shape[0]
iter = [0]
def mv(v):
iter[0] += 1
logger.debug('iteration # ' + str(iter[0]))
return cmT.dot(iop.dot(cm.dot(v)))
# P = sparse.linalg.spilu(cmT.dot(iop.nearfield_no_correction_dot(cm)))
timer.report("Build preconditioner")
def prec_f(x):
# return P.solve(x)
return x
M = sparse.linalg.LinearOperator((n, n), matvec=prec_f)
A = sparse.linalg.LinearOperator((n, n), matvec=mv)
def report_res(R):
logger.debug('residual: ' + str(R))
pass
soln = sparse.linalg.gmres(A,
rhs_constrained,
M=M,
tol=tol,
callback=report_res,
restart=200)
timer.report("GMRES")
return cm.dot(soln[0]) + c_rhs