class RBFSurface(object):
"""RBF surface
An RBF surface builds an interpolant on top of an RBF system object.
Attributes:
rbfs: RBF system object
coeff: interpolant coefficient vector
dirty: flag whether there have been changes since coeff computation
"""
def __init__(self, rbfs):
"""Initialize the surface
Args:
rbfs: RBF system object
"""
self._fxmgr = ArrayManager((rbfs.nsys,))
self.coeff = None
self.rbfs = rbfs
self.dirty = True
@property
def x(self):
"Array of points"
return self.rbfs.x
@property
def fx(self):
"Array of function values"
return self._fxmgr.view
@property
def dim(self):
"Dimension of the ambient space"
return self.rbfs.dim
@property
def npt(self):
"Number of centers"
return self.rbfs.npt
def set_fx(self, fx):
"Replace f with transformed function values"
self.fx[self.rbfs.mtail:] = fx
self.dirty = True
def fit(self):
"Compute the fit"
self.coeff = self.rbfs.solve(self.fx)
self.dirty = False
def update(self):
"Update the fit if stale"
if self.dirty:
self.fit()
def add_values(self, fx):
"""Add values to the system
Args:
fx: vector of values to append
"""
self._fxmgr.append(fx)
self.dirty = True
def add_centers(self, x):
"""Add a center to the system
Args:
x: center to add
"""
self.rbfs.add_centers(x)
def add_points(self, x, fx):
"""Add points to the system
NB: This function is only really safe if the RBF system is not
shared. To work with multiple surfaces associated with the same
RBF system, add centers to the system object and values to the
surfaces separately.
Args:
x: points to add to the RBF system
fx: vector of values to append
"""
self.add_centers(x)
self.add_values(fx)
def eval(self, y, d=None):
"""Evaluate the surface at y
Args:
y: Point or points at which to evaluate
Returns:
value of the surface at y
"""
self.update()
return self.rbfs.basis_dot(self.coeff, y, d)
def deriv(self, y, d=None):
"""Evaluate the derivative at y
Args:
y: Point or points at which to evaluate
Returns:
gradient of the surface at y
"""
self.update()
return self.rbfs.dbasis_dot(self.coeff, y, d)
def seminorm(self):
"Get the seminorm of the surface"
self.update()
c, M = self.coeff, self.rbfs.M
return np.dot(c, np.dot(M, c))
def dseminorm(self, y, tau):
"""Compute change in seminorm associated with adding center at y
Gutmann (2001) suggests choosing sample points in a global
optimization by finding where the smallest seminorm change
could be applied to the surface in order to achieve a given
target value.
Args:
y: Point at which to add a hypothetical new center
tau: Value of s(y) if augmented with hypothetical center
Returns:
g: Change in squared seminorm between original and updated surface
dg: Gradient of g with respect to y
"""
self.update()
rbfs = self.rbfs
d = rbfs.dist(y)
b = rbfs.basis(y, d)
w = rbfs.solve(b)
mu = -np.dot(w, b)
sy = np.dot(self.coeff, b)
eta = (sy-tau)/mu
g = (sy-tau)*eta
dg = (2*eta)*rbfs.dbasis_dot(self.coeff + eta*w, y, d)
return g, dg
def diseminorm(self, y, tau):
"""Compute 1/g(y) where g(y) is computed by dseminorm
The function h(y) = -1/g(y) is potentially nicer to work
with than just g(y), as it doesn't have any poles.
Args:
y: Point at which to add a hypothetical new center
tau: Value of s(y) if augmented with hypothetical center
Returns:
h: Value of h(y) = 1/g(y)
dh: Gradient of h with respect to y
"""
self.update()
rbfs = self.rbfs
d = rbfs.dist(y)
b = rbfs.basis(y, d)
w = rbfs.solve(b)
mu = -np.dot(w, b)
sy = np.dot(self.coeff, b)
gap = sy-tau
h = mu/gap**2
dh = -2/gap**2 * rbfs.dbasis_dot(mu/gap * self.coeff + w, y, d)
return h, dh
class RBFSurface(object):
"""RBF surface
An RBF surface builds an interpolant on top of an RBF system object.
Attributes:
rbfs: RBF system object
coeff: interpolant coefficient vector
dirty: flag whether there have been changes since coeff computation
"""
def __init__(self, rbfs):
"""Initialize the surface
Args:
rbfs: RBF system object
"""
self._fxmgr = ArrayManager((rbfs.nsys,))
self.coeff = None
self.rbfs = rbfs
self.dirty = True
@property
def x(self):
"Array of points"
return self.rbfs.x
@property
def fx(self):
"Array of function values"
return self._fxmgr.view
@property
def dim(self):
"Dimension of the ambient space"
return self.rbfs.dim
@property
def npt(self):
"Number of centers"
return self.rbfs.npt
def set_fx(self, fx):
"Replace f with transformed function values"
self.fx[self.rbfs.mtail:] = fx
self.dirty = True
def fit(self):
"Compute the fit"
self.coeff = self.rbfs.solve(self.fx)
self.dirty = False
def update(self):
"Update the fit if stale"
if self.dirty:
self.fit()
def add_values(self, fx):
"""Add values to the system
Args:
fx: vector of values to append
"""
self._fxmgr.append(fx)
self.dirty = True
def add_centers(self, x):
"""Add a center to the system
Args:
x: center to add
"""
self.rbfs.add_centers(x)
def add_points(self, x, fx):
"""Add points to the system
NB: This function is only really safe if the RBF system is not
shared. To work with multiple surfaces associated with the same
RBF system, add centers to the system object and values to the
surfaces separately.
Args:
x: points to add to the RBF system
fx: vector of values to append
"""
self.add_centers(x)
self.add_values(fx)
def eval(self, y, d=None):
"""Evaluate the surface at y
Args:
y: Point or points at which to evaluate
Returns:
value of the surface at y
"""
self.update()
return self.rbfs.basis_dot(self.coeff, y, d)
def deriv(self, y, d=None):
"""Evaluate the derivative at y
Args:
y: Point or points at which to evaluate
Returns:
gradient of the surface at y
"""
self.update()
return self.rbfs.dbasis_dot(self.coeff, y, d)
def seminorm(self):
"Get the seminorm of the surface"
self.update()
c, M = self.coeff, self.rbfs.M
return np.dot(c, np.dot(M, c))
def dseminorm(self, y, tau):
"""Compute change in seminorm associated with adding center at y
Gutmann (2001) suggests choosing sample points in a global
optimization by finding where the smallest seminorm change
could be applied to the surface in order to achieve a given
target value.
Args:
y: Point at which to add a hypothetical new center
tau: Value of s(y) if augmented with hypothetical center
Returns:
g: Change in squared seminorm between original and updated surface
dg: Gradient of g with respect to y
"""
self.update()
rbfs = self.rbfs
d = rbfs.dist(y)
b = rbfs.basis(y, d)
w = rbfs.solve(b)
mu = -np.dot(w, b)
sy = np.dot(self.coeff, b)
eta = (sy-tau)/mu
g = (sy-tau)*eta
dg = (2*eta)*rbfs.dbasis_dot(self.coeff + eta*w, y, d)
return g, dg
def diseminorm(self, y, tau):
"""Compute 1/g(y) where g(y) is computed by dseminorm
The function h(y) = -1/g(y) is potentially nicer to work
with than just g(y), as it doesn't have any poles.
Args:
y: Point at which to add a hypothetical new center
tau: Value of s(y) if augmented with hypothetical center
Returns:
h: Value of h(y) = 1/g(y)
dh: Gradient of h with respect to y
"""
self.update()
rbfs = self.rbfs
d = rbfs.dist(y)
b = rbfs.basis(y, d)
w = rbfs.solve(b)
mu = -np.dot(w, b)
sy = np.dot(self.coeff, b)
gap = sy-tau
h = mu/gap**2
dh = -2/gap**2 * rbfs.dbasis_dot(mu/gap * self.coeff + w, y, d)
return h, dh
class BaseRBFSystem(object):
"""Base class for RBF system
The RBF system is the set of centers and basis functions
(polynomial and RBF), and the associated linear system.
It does not involve actual function values or coefficients
for interpolants. One RBF system object may be shared among
multiple interpolants.
Derived classes should overload:
- fill_basis - evaluate basis at points
- basis_dot - compute linear combination of basis values
- dbasis_dot - compute linear combination of basis gradients
Derived classes may also overload the factor and solve routines.
Attributes:
new_points: Number of points since the last update
"""
def __init__(self, dim, mtail, maxpts):
"""Initialize the system
Args:
dim: Dimension of the system
mtail: Dimension of the polynomial tail
maxpts: Initial estimate of maximum number of centers
"""
self._mtail = mtail
self._xmgr = ArrayManager((0, dim), (maxpts, dim))
self._Mmgr = ArrayManager((self.nsys, self.nsys))
self.new_points = 0
@property
def x(self):
"Array of points"
return self._xmgr.view
@property
def M(self):
"System matrix"
return self._Mmgr.view
@property
def dim(self):
"Dimension of the ambient space"
return self._xmgr.shape[1]
@property
def npt(self):
"Number of centers"
return self._xmgr.shape[0]
@property
def mtail(self):
"Dimension of the polynomial tail space"
return self._mtail
@property
def nsys(self):
"System size"
return self.npt + self.mtail
@property
def dirty(self):
"Return true if there are new points"
return self.new_points > 0
def add_centers(self, x):
"""Add centers to the RBF interpolant
Args:
x: One or more new center locations (one per row if multiple)
"""
if len(x.shape) == 1:
self._xmgr.append(np.array([x]))
self.new_points += 1
else:
self._xmgr.append(x)
self.new_points += x.shape[0]
def dist(self, y):
"""Compute distances of all points to y
Args:
y: One or more points (one per row if multiple)
"""
if len(y.shape) == 1:
return dist.cdist(self.x, np.expand_dims(y, axis=0)).ravel()
else:
return dist.cdist(self.x, y)
def basis(self, y, d=None):
"""Compute the vector of basis functions at y
Args:
y: Point at which to evaluate basis functions
d: Distance from y to RBF (not required)
"""
b = np.zeros((self.nsys, 1))
self.fill_basis(b, np.expand_dims(y, axis=0), d=None)
b.shape = (self.nsys,)
return b
def compliance(self, y):
"""Get the compliance of the surface at y
Args:
y: Point at which to compute surface compliance
"""
self.update()
b = self.basis(y)
return -np.dot(b, self.solve(b))
def update(self):
"""Update the system matrix if points have been added
"""
if self.new_points == 0:
return
x, nnew, Mmgr = self.x, self.new_points, self._Mmgr
Mmgr.resize((self.nsys, self.nsys))
M = Mmgr.view
self.fill_basis(M[:, -nnew:], x[-nnew:, :])
M[-nnew:, :] = M[:, -nnew:].T
self.refactor()
self.new_points = 0
def refactor(self):
"""Compute factorization
"""
eta = 1e-16 * np.eye(self.M.shape[0]) * la.norm(self.M, 1) * la.norm(self.M, np.inf)
self.lupiv = la.lu_factor(self.M + eta)
def solve(self, rhs):
"""Solve with the system matrix
Args:
rhs: Right hand side in system
Returns:
solution to M*x = rhs
"""
self.update()
return la.lu_solve(self.lupiv, rhs)
def fill_basis(self, B, y, d=None):
"""Compute a matrix of basis functions at y
Args:
B: Matrix to fill with basis function values
y: Points at which to evaluate basis functions
d: Distance from y to centers (optional)
"""
pass
def basis_dot(self, v, y, d=None):
"""Form a linear combination of basis functions
Args:
v: Coefficient vector in linear combination
y: Point at which to evaluate basis functions
d: Distance from y to centers (optional)
"""
pass
def dbasis_dot(self, v, y, d=None):
"""Form a linear combination of the basis gradients at y
Args:
v: Coefficient vector in linear combination
y: Point at which to evaluate basis function gradients
d: Distance from y to centers (optional)
"""
pass
class BaseRBFSystem(object):
"""Base class for RBF system
The RBF system is the set of centers and basis functions
(polynomial and RBF), and the associated linear system.
It does not involve actual function values or coefficients
for interpolants. One RBF system object may be shared among
multiple interpolants.
Derived classes should overload:
- fill_basis - evaluate basis at points
- basis_dot - compute linear combination of basis values
- dbasis_dot - compute linear combination of basis gradients
Derived classes may also overload the factor and solve routines.
Attributes:
new_points: Number of points since the last update
"""
def __init__(self, dim, mtail, maxpts):
"""Initialize the system
Args:
dim: Dimension of the system
mtail: Dimension of the polynomial tail
maxpts: Initial estimate of maximum number of centers
"""
self._mtail = mtail
self._xmgr = ArrayManager((0, dim), (maxpts, dim))
self._Mmgr = ArrayManager((self.nsys, self.nsys))
self.new_points = 0
@property
def x(self):
"Array of points"
return self._xmgr.view
@property
def M(self):
"System matrix"
return self._Mmgr.view
@property
def dim(self):
"Dimension of the ambient space"
return self._xmgr.shape[1]
@property
def npt(self):
"Number of centers"
return self._xmgr.shape[0]
@property
def mtail(self):
"Dimension of the polynomial tail space"
return self._mtail
@property
def nsys(self):
"System size"
return self.npt + self.mtail
@property
def dirty(self):
"Return true if there are new points"
return self.new_points > 0
def add_centers(self, x):
"""Add centers to the RBF interpolant
Args:
x: One or more new center locations (one per row if multiple)
"""
if len(x.shape) == 1:
self._xmgr.append(np.array([x]))
self.new_points += 1
else:
self._xmgr.append(x)
self.new_points += x.shape[0]
def dist(self, y):
"""Compute distances of all points to y
Args:
y: One or more points (one per row if multiple)
"""
if len(y.shape) == 1:
return dist.cdist(self.x, np.expand_dims(y, axis=0)).ravel()
else:
return dist.cdist(self.x, y)
def basis(self, y, d=None):
"""Compute the vector of basis functions at y
Args:
y: Point at which to evaluate basis functions
d: Distance from y to RBF (not required)
"""
b = np.zeros((self.nsys, 1))
self.fill_basis(b, np.expand_dims(y, axis=0), d=None)
b.shape = (self.nsys,)
return b
def compliance(self, y):
"""Get the compliance of the surface at y
Args:
y: Point at which to compute surface compliance
"""
self.update()
b = self.basis(y)
return -np.dot(b, self.solve(b))
def update(self):
"""Update the system matrix if points have been added
"""
if self.new_points == 0:
return
x, nnew, Mmgr = self.x, self.new_points, self._Mmgr
Mmgr.resize((self.nsys, self.nsys))
M = Mmgr.view
self.fill_basis(M[:, -nnew:], x[-nnew:, :])
M[-nnew:, :] = M[:, -nnew:].T
self.refactor()
self.new_points = 0
def refactor(self):
"""Compute factorization
"""
eta = min(1e-5, 1e-16 * np.sqrt(la.norm(self.M, 1) * la.norm(self.M, np.inf))) * np.eye(self.M.shape[0])
self.lupiv = la.lu_factor(self.M + eta)
def solve(self, rhs):
"""Solve with the system matrix
Args:
rhs: Right hand side in system
Returns:
solution to M*x = rhs
"""
self.update()
return la.lu_solve(self.lupiv, rhs)
def fill_basis(self, B, y, d=None):
"""Compute a matrix of basis functions at y
Args:
B: Matrix to fill with basis function values
y: Points at which to evaluate basis functions
d: Distance from y to centers (optional)
"""
pass
def basis_dot(self, v, y, d=None):
"""Form a linear combination of basis functions
Args:
v: Coefficient vector in linear combination
y: Point at which to evaluate basis functions
d: Distance from y to centers (optional)
"""
pass
def dbasis_dot(self, v, y, d=None):
"""Form a linear combination of the basis gradients at y
Args:
v: Coefficient vector in linear combination
y: Point at which to evaluate basis function gradients
d: Distance from y to centers (optional)
"""
pass