def make_derivative(expr, dim, fd_order, deriv_order, side, matvec, x0, symbolic):
# The stencil positions
indices, x0 = generate_indices(expr, dim, fd_order, side=side, x0=x0)
# Finite difference weights from Taylor approximation with these positions
if symbolic:
weights = symbolic_weights(expr, deriv_order, indices, x0)
else:
weights = numeric_weights(deriv_order, indices, x0)
return indices_weights_to_fd(expr, dim, indices, weights, matvec=matvec.val)
def generic_derivative(expr,
dim,
fd_order,
deriv_order,
stagger=None,
symbolic=False,
matvec=direct):
"""
Arbitrary-order derivative of a given expression.
Parameters
----------
expr : expr-like
Expression for which the derivative is produced.
dim : Dimension
The Dimension w.r.t. which to differentiate.
fd_order : int
Coefficient discretization order. Note: this impacts the width of
the resulting stencil.
deriv_order : int
Derivative order, e.g. 2 for a second-order derivative.
stagger : Side, optional
Shift of the finite-difference approximation.
Returns
-------
expr-like
``deriv-order`` derivative of ``expr``.
"""
diff = dim.spacing
adjoint_val = matvec.val**deriv_order
# Stencil positions
indices, x0 = generate_indices(expr, dim, diff, fd_order, stagger=stagger)
# Finite difference weights from Taylor approximation with these positions
if symbolic:
c = symbolic_weights(expr, deriv_order, indices, x0)
else:
c = finite_diff_weights(deriv_order, indices, x0)[-1][-1]
# Loop through positions
deriv = 0
all_dims = tuple(
set((dim, ) + tuple(i for i in expr.indices if i.root == dim)))
for i in range(len(indices)):
subs = dict((d, indices[i].subs({dim: d})) for d in all_dims)
deriv += expr.subs(subs) * c[i]
# Evaluate up to _PRECISION digits
deriv = (adjoint_val * deriv).evalf(_PRECISION)
return deriv
def generic_derivative(expr,
dim,
fd_order,
deriv_order,
symbolic=False,
matvec=direct,
x0=None):
"""
Arbitrary-order derivative of a given expression.
Parameters
----------
expr : expr-like
Expression for which the derivative is produced.
dim : Dimension
The Dimension w.r.t. which to differentiate.
fd_order : int
Coefficient discretization order. Note: this impacts the width of
the resulting stencil.
deriv_order : int
Derivative order, e.g. 2 for a second-order derivative.
stagger : Side, optional
Shift of the finite-difference approximation.
x0 : dict, optional
Origin of the finite-difference scheme as a map dim: origin_dim.
Returns
-------
expr-like
``deriv-order`` derivative of ``expr``.
"""
# First order derivative with 2nd order FD is highly non-recommended so taking
# first order fd that is a lot better
if deriv_order == 1 and fd_order == 2 and not symbolic:
fd_order = 1
# Stencil positions
indices, x0 = generate_indices(expr, dim, fd_order, x0=x0)
# Finite difference weights from Taylor approximation with these positions
if symbolic:
c = symbolic_weights(expr, deriv_order, indices, x0)
else:
c = numeric_weights(deriv_order, indices, x0)
return indices_weights_to_fd(expr, dim, indices, c, matvec=matvec.val)
def first_derivative(expr,
dim,
fd_order=None,
side=centered,
matvec=direct,
symbolic=False):
"""
First-order derivative of a given expression.
Parameters
----------
expr : expr-like
Expression for which the first-order derivative is produced.
dim : Dimension
The Dimension w.r.t. which to differentiate.
fd_order : int, optional
Coefficient discretization order. Note: this impacts the width of
the resulting stencil. Defaults to ``expr.space_order``
side : Side, optional
Side of the finite difference location, centered (at x), left (at x - 1)
or right (at x +1). Defaults to ``centered``.
matvec : Transpose, optional
Forward (matvec=direct) or transpose (matvec=transpose) mode of the
finite difference. Defaults to ``direct``.
Returns
-------
expr-like
First-order derivative of ``expr``.
Examples
--------
>>> from devito import Function, Grid, first_derivative, transpose
>>> grid = Grid(shape=(4, 4))
>>> x, _ = grid.dimensions
>>> f = Function(name='f', grid=grid)
>>> g = Function(name='g', grid=grid)
>>> first_derivative(f*g, dim=x)
-f(x, y)*g(x, y)/h_x + f(x + h_x, y)*g(x + h_x, y)/h_x
Semantically, this is equivalent to
>>> (f*g).dx
Derivative(f(x, y)*g(x, y), x)
The only difference is that in the latter case derivatives remain unevaluated.
The expanded form is obtained via ``evaluate``
>>> (f*g).dx.evaluate
-f(x, y)*g(x, y)/h_x + f(x + h_x, y)*g(x + h_x, y)/h_x
For the adjoint mode of the first derivative, pass ``matvec=transpose``
>>> g = Function(name='g', grid=grid)
>>> first_derivative(f*g, dim=x, matvec=transpose)
f(x, y)*g(x, y)/h_x - f(x + h_x, y)*g(x + h_x, y)/h_x
This is also accessible via the .T shortcut
>>> (f*g).dx.T.evaluate
f(x, y)*g(x, y)/h_x - f(x + h_x, y)*g(x + h_x, y)/h_x
"""
side = side.adjoint(matvec)
diff = dim.spacing
adjoint_val = matvec.val
order = fd_order or expr.space_order
# Stencil positions for non-symmetric cross-derivatives with symmetric averaging
ind = generate_indices(expr, dim, diff, order, side=side)[0]
# Finite difference weights from Taylor approximation with these positions
if symbolic:
c = symbolic_weights(expr, 1, ind, dim)
else:
c = finite_diff_weights(1, ind, dim)[-1][-1]
# Loop through positions
deriv = 0
all_dims = tuple(
set((dim, ) + tuple([i for i in expr.indices if i.root == dim])))
for i in range(len(ind)):
subs = dict([(d, ind[i].subs({dim: d})) for d in all_dims])
deriv += expr.subs(subs) * c[i]
# Evaluate up to _PRECISION digits
deriv = (adjoint_val * deriv).evalf(_PRECISION)
return deriv
def first_derivative(expr, dim, fd_order=None, side=centered, matvec=direct,
symbolic=False, x0=None):
"""
First-order derivative of a given expression.
Parameters
----------
expr : expr-like
Expression for which the first-order derivative is produced.
dim : Dimension
The Dimension w.r.t. which to differentiate.
fd_order : int, optional
Coefficient discretization order. Note: this impacts the width of
the resulting stencil. Defaults to ``expr.space_order``
side : Side, optional
Side of the finite difference location, centered (at x), left (at x - 1)
or right (at x +1). Defaults to ``centered``.
matvec : Transpose, optional
Forward (matvec=direct) or transpose (matvec=transpose) mode of the
finite difference. Defaults to ``direct``.
x0 : dict, optional
Origin of the finite-difference scheme as a map dim: origin_dim.
Returns
-------
expr-like
First-order derivative of ``expr``.
Examples
--------
>>> from devito import Function, Grid, first_derivative, transpose
>>> grid = Grid(shape=(4, 4))
>>> x, _ = grid.dimensions
>>> f = Function(name='f', grid=grid)
>>> g = Function(name='g', grid=grid)
>>> first_derivative(f*g, dim=x)
-f(x, y)*g(x, y)/h_x + f(x + h_x, y)*g(x + h_x, y)/h_x
Semantically, this is equivalent to
>>> (f*g).dx
Derivative(f(x, y)*g(x, y), x)
The only difference is that in the latter case derivatives remain unevaluated.
The expanded form is obtained via ``evaluate``
>>> (f*g).dx.evaluate
-f(x, y)*g(x, y)/h_x + f(x + h_x, y)*g(x + h_x, y)/h_x
For the adjoint mode of the first derivative, pass ``matvec=transpose``
>>> g = Function(name='g', grid=grid)
>>> first_derivative(f*g, dim=x, matvec=transpose)
-f(x, y)*g(x, y)/h_x + f(x - h_x, y)*g(x - h_x, y)/h_x
This is also accessible via the .T shortcut
>>> (f*g).dx.T.evaluate
-f(x, y)*g(x, y)/h_x + f(x - h_x, y)*g(x - h_x, y)/h_x
Finally the x0 argument allows to choose the origin of the finite-difference
>>> first_derivative(f, dim=x, x0={x: 1})
-f(1, y)/h_x + f(h_x + 1, y)/h_x
"""
side = side
order = fd_order or expr.space_order
# Stencil positions for non-symmetric cross-derivatives with symmetric averaging
ind = generate_indices(expr, dim, order, side=side, x0=x0)[0]
# Finite difference weights from Taylor approximation with these positions
if symbolic:
c = symbolic_weights(expr, 1, ind, dim)
else:
c = finite_diff_weights(1, ind, dim)[-1][-1]
return indices_weights_to_fd(expr, dim, ind, c, matvec=matvec.val)
