def __init__(self, method, pts, kernel, nugget=0.0, **kwargs):
self.method = method
self.kernel = kernel
self.pts = pts
try:
verbose = kwargs['verbose']
except KeyError:
verbose = False
self.verbose = verbose
if method == 'Dense':
from dense import GenerateDenseMatrix
self.mat = GenerateDenseMatrix(pts, kernel)
self.pts = pts
elif method == 'FFT':
from toeplitz import CreateRow, ToeplitzProduct
xmin = kwargs['xmin']
xmax = kwargs['xmax']
N = kwargs['N']
theta = kwargs['theta']
self.N = N
self.row, pts = CreateRow(xmin, xmax, N, kernel, theta)
self.pts = pts
elif method == 'Hmatrix':
from hmatrix import Hmatrix
n = np.size(pts, 0)
ind = np.arange(n)
rkmax = 32 if 'rkmax' not in kwargs else kwargs['rkmax']
eps = 1.e-9 if 'eps' not in kwargs else kwargs['eps']
self.H = Hmatrix(pts,
kernel,
ind,
verbose=verbose,
rkmax=rkmax,
eps=eps)
else:
raise NotImplementedError
self.P = None
n = pts.shape[0]
self.shape = (n, n)
self.nugget = nugget
self.dtype = 'd'
self.count = 0
self.solvmatvecs = 0
def __init__(self, method, pts, kernel, nugget = 0.0, **kwargs):
self.method = method
self.kernel = kernel
self.pts = pts
try:
verbose = kwargs['verbose']
except KeyError:
verbose = False
self.verbose = verbose
if method == 'Dense':
from dense import GenerateDenseMatrix
self.mat = GenerateDenseMatrix(pts, kernel)
self.pts = pts
elif method == 'FFT':
from toeplitz import CreateRow, ToeplitzProduct
xmin = kwargs['xmin']
xmax = kwargs['xmax']
N = kwargs['N']
theta = kwargs['theta']
self.N = N
self.row, pts = CreateRow(xmin,xmax,N,kernel,theta)
self.pts = pts
elif method == 'Hmatrix':
from hmatrix import Hmatrix
n = np.size(pts,0)
ind = np.arange(n)
rkmax = 32 if 'rkmax' not in kwargs else kwargs['rkmax']
eps = 1.e-9 if 'eps' not in kwargs else kwargs['eps']
self.H = Hmatrix(pts, kernel, ind, verbose = verbose, rkmax = rkmax, eps = eps)
else:
raise NotImplementedError
self.P = None
n = pts.shape[0]
self.shape = (n,n)
self.nugget = nugget
self.dtype = 'd'
self.count = 0
self.solvmatvecs = 0
class CovarianceMatrix:
"""
Implementation of covariance matrix corresponding to covariance kernel
Parameters:
-----------
method: string, {'Dense','FFT','Hmatrix}
decides what method to invoke
pts: n x dim
Location of the points
kernel: Kernel object
See covariance/kernel.py
nugget: double, optional. default = 0.
A hack to improve the convergence of the iterative solver when inverting for the covariance matrix. See solve() for more details.
Attributes:
-----------
shape: (N,N)
N is the size of matrix
P: NxN csr_matrix
corresponding to the preconditioner
Methods:
--------
matvec()
rmatvec()
reset()
itercount()
build_preconditioner()
solve()
Notes:
------
Implementation of three different methods
1. Dense Matrix
2. FFT based operations if kernel is stationary or translation invariant and points are on a regular grid
3. Hierarchical Matrix - works for arbitrary kernels on irregular grids
Details of this implementation (including errors and benchmarking) are provided in chapter 2 in [1]. For details on the algorithms see references within.
Compatible with scipy.sparse.LinearOperator. For example, Q = CovarianceMatrix(...); Qop = aslineroperator(Q)
References:
----------
.. [1] A.K. Saibaba, Fast solvers for geostatistical inverse problems and uncertainty quantification, PhD Thesis 2013, Stanford University
Examples:
---------
import numpy as np
pts = np.random.randn(1000,2)
def kernel(r): return np.exp(-r)
Qd = CovarianceMatrix('Dense', pts, kernel)
Qh = CovarianceMatrix('Hmatrix', pts, kernel, rkmax = 32, eps = 1.e-6)
"""
def __init__(self, method, pts, kernel, nugget=0.0, **kwargs):
self.method = method
self.kernel = kernel
self.pts = pts
try:
verbose = kwargs['verbose']
except KeyError:
verbose = False
self.verbose = verbose
if method == 'Dense':
from dense import GenerateDenseMatrix
self.mat = GenerateDenseMatrix(pts, kernel)
self.pts = pts
elif method == 'FFT':
from toeplitz import CreateRow, ToeplitzProduct
xmin = kwargs['xmin']
xmax = kwargs['xmax']
N = kwargs['N']
theta = kwargs['theta']
self.N = N
self.row, pts = CreateRow(xmin, xmax, N, kernel, theta)
self.pts = pts
elif method == 'Hmatrix':
from hmatrix import Hmatrix
n = np.size(pts, 0)
ind = np.arange(n)
rkmax = 32 if 'rkmax' not in kwargs else kwargs['rkmax']
eps = 1.e-9 if 'eps' not in kwargs else kwargs['eps']
self.H = Hmatrix(pts,
kernel,
ind,
verbose=verbose,
rkmax=rkmax,
eps=eps)
else:
raise NotImplementedError
self.P = None
n = pts.shape[0]
self.shape = (n, n)
self.nugget = nugget
self.dtype = 'd'
self.count = 0
self.solvmatvecs = 0
def matvec(self, x):
"""
Computes the matrix-vector product
Parameters:
-----------
x: (n,) ndarray
a vector of size n
Returns:
-------
y: (n,) ndarray
Notes:
------
The result of this calculation are dependent on the method chosen. All methods except 'Hmatrix' are exact.
"""
method = self.method
if method == 'Dense':
y = np.dot(self.mat, x)
elif method == 'FFT':
y = ToeplitzProduct(x, self.row, self.N)
elif method == 'Hmatrix':
y = np.zeros_like(x, dtype='d')
self.H.mult(x, y, self.verbose)
y += self.nugget * y
self.count += 1
return y
def rmatvec(self, x):
"""
Computes the matrix transpose-vector product
Parameters:
-----------
x: (n,) ndarray
a vector of size n
Returns:
-------
y: (n,) ndarray
Notes:
------
the result of this calculation are dependent on the method chosen. All methods except 'Hmatrix' are exact.
Because of symmetry it is almost the same as matvec, except 'Hmatrix' which is numerically different
"""
method = self.method
if method == 'Dense':
y = np.dot(self.mat.T, x)
elif method == 'FFT':
y = ToeplitzProduct(x, self.row, self.N)
elif method == 'Hmatrix':
y = np.zeros_like(x, dtype='d')
self.H.transpmult(x, y, self.verbose)
y += self.nugget * y
self.count += 1
return y
def reset(self):
"""
Resets the counter of matvecs and solves
"""
self.count = 0
self.solvmatvecs = 0
return
def itercount(self):
"""
Returns the counter of matvecs
"""
return self.count
def build_preconditioner(self, k=100):
"""
Implementation of the preconditioner based on changing basis.
Parameters:
-----------
k: int, optional. default = 100
Number of local centers in the preconditioner. Controls the sparity of the preconditioner.
Notes:
------
Implementation of the preconditioner based on local centers.
The parameter k controls the sparsity and the effectiveness of the preconditioner.
Larger k is more expensive but results in fewer iterations.
For large ill-conditioned systems, it was best to use a nugget effect to make the problem better conditioned.
To Do: implementation based on local centers and additional points. Will remove the hack of using nugget effect.
References:
-----------
"""
from time import time
from scipy.spatial import cKDTree
from scipy.spatial.distance import pdist, cdist
from scipy.linalg import solve
from scipy.sparse import csr_matrix
#If preconditioner already exists, then nothing to do
if self.P != None: return
pts = self.pts
kernel = self.kernel
N = pts.shape[0]
#Build the tree
start = time()
tree = cKDTree(pts, leafsize=32)
end = time()
if self.verbose:
print "Tree building time = %g" % (end - start)
#Find the nearest neighbors of all the points
start = time()
dist, ind = tree.query(pts, k=k)
end = time()
if self.verbose:
print "Nearest neighbor computation time = %g" % (end - start)
Q = np.zeros((k, k), dtype='d')
y = np.zeros((k, 1), dtype='d')
row = np.tile(np.arange(N), (k, 1)).transpose()
col = np.copy(ind)
nu = np.zeros((N, k), dtype='d')
y[0] = 1.
start = time()
for i in np.arange(N):
Q = kernel(cdist(pts[ind[i, :], :], pts[ind[i, :], :]))
nui = np.linalg.solve(Q, y)
nu[i, :] = np.copy(nui.transpose())
end = time()
if self.verbose:
print "Building preconditioner took = %g" % (end - start)
ij = np.zeros((N * k, 2), dtype='i')
ij[:, 0] = np.copy(np.reshape(row, N * k, order='F').transpose())
ij[:, 1] = np.copy(np.reshape(col, N * k, order='F').transpose())
data = np.copy(np.reshape(nu, N * k, order='F').transpose())
self.P = csr_matrix((data, ij.transpose()), shape=(N, N), dtype='d')
return
def solve(self, b, maxiter=1000, tol=1.e-10):
"""
Compute Q^{-1}b
Parameters:
-----------
b: (n,) ndarray
given right hand side
maxiter: int, optional. default = 1000
Maximum number of iterations for the linear solver
tol: float, optional. default = 1.e-10
Residual stoppingtolerance for the iterative solver
Notes:
------
If 'Dense' then inverts using LU factorization otherwise uses iterative solver
- MINRES if used without preconditioner or GMRES with preconditioner.
Preconditioner is not guaranteed to be positive definite.
"""
if self.method == 'Dense':
from scipy.linalg import solve
x = solve(self.mat, b)
else:
from scipy.sparse.linalg import gmres, aslinearoperator, minres
P = self.P
Aop = aslinearoperator(self)
residual = _Residual()
if P != None:
x, info = gmres(Aop,
b,
tol=tol,
restart=30,
maxiter=1000,
callback=residual,
M=P)
else:
x, info = minres(Aop,
b,
tol=tol,
maxiter=maxiter,
callback=residual)
self.solvmatvecs += residual.itercount()
if self.verbose:
print "Number of iterations is %g and status is %g" % (
residual.itercount(), info)
return x
def kernel(R): return (1+np.sqrt(3)*R)*np.exp(-np.sqrt(3)*R) elif n == '52': def kernel(R): return (1+np.sqrt(5)*R + 5.*R/3.)*np.exp(-np.sqrt(5)*R) multtime = [] setuptime = [] for N in sizes: indx = np.arange(N) pts = np.random.rand(N,2) start = time() Q = Hmatrix(pts, kernel, indx, eps = 1.e-9, rkmax = 32) _stime = time()-start setuptime.append(_stime) print "Hmatrix setup of size %g took %g secs" %(N,_stime) x = np.random.randn(N,1) y = 0*x _time = 0. for i in np.arange(3): start = time() Q.mult(x,y) _time += time()-start multtime.append(_time/3.)
class CovarianceMatrix:
"""
Implementation of covariance matrix corresponding to covariance kernel
Parameters:
-----------
method: string, {'Dense','FFT','Hmatrix}
decides what method to invoke
pts: n x dim
Location of the points
kernel: Kernel object
See covariance/kernel.py
nugget: double, optional. default = 0.
A hack to improve the convergence of the iterative solver when inverting for the covariance matrix. See solve() for more details.
Attributes:
-----------
shape: (N,N)
N is the size of matrix
P: NxN csr_matrix
corresponding to the preconditioner
Methods:
--------
matvec()
rmatvec()
reset()
itercount()
build_preconditioner()
solve()
Notes:
------
Implementation of three different methods
1. Dense Matrix
2. FFT based operations if kernel is stationary or translation invariant and points are on a regular grid
3. Hierarchical Matrix - works for arbitrary kernels on irregular grids
Details of this implementation (including errors and benchmarking) are provided in chapter 2 in [1]. For details on the algorithms see references within.
Compatible with scipy.sparse.LinearOperator. For example, Q = CovarianceMatrix(...); Qop = aslineroperator(Q)
References:
----------
.. [1] A.K. Saibaba, Fast solvers for geostatistical inverse problems and uncertainty quantification, PhD Thesis 2013, Stanford University
Examples:
---------
import numpy as np
pts = np.random.randn(1000,2)
def kernel(r): return np.exp(-r)
Qd = CovarianceMatrix('Dense', pts, kernel)
Qh = CovarianceMatrix('Hmatrix', pts, kernel, rkmax = 32, eps = 1.e-6)
"""
def __init__(self, method, pts, kernel, nugget = 0.0, **kwargs):
self.method = method
self.kernel = kernel
self.pts = pts
try:
verbose = kwargs['verbose']
except KeyError:
verbose = False
self.verbose = verbose
if method == 'Dense':
from dense import GenerateDenseMatrix
self.mat = GenerateDenseMatrix(pts, kernel)
self.pts = pts
elif method == 'FFT':
from toeplitz import CreateRow, ToeplitzProduct
xmin = kwargs['xmin']
xmax = kwargs['xmax']
N = kwargs['N']
theta = kwargs['theta']
self.N = N
self.row, pts = CreateRow(xmin,xmax,N,kernel,theta)
self.pts = pts
elif method == 'Hmatrix':
from hmatrix import Hmatrix
n = np.size(pts,0)
ind = np.arange(n)
rkmax = 32 if 'rkmax' not in kwargs else kwargs['rkmax']
eps = 1.e-9 if 'eps' not in kwargs else kwargs['eps']
self.H = Hmatrix(pts, kernel, ind, verbose = verbose, rkmax = rkmax, eps = eps)
else:
raise NotImplementedError
self.P = None
n = pts.shape[0]
self.shape = (n,n)
self.nugget = nugget
self.dtype = 'd'
self.count = 0
self.solvmatvecs = 0
def matvec(self, x):
"""
Computes the matrix-vector product
Parameters:
-----------
x: (n,) ndarray
a vector of size n
Returns:
-------
y: (n,) ndarray
Notes:
------
The result of this calculation are dependent on the method chosen. All methods except 'Hmatrix' are exact.
"""
method = self.method
if method == 'Dense':
y = np.dot(self.mat,x)
elif method == 'FFT':
y = ToeplitzProduct(x, self.row, self.N)
elif method == 'Hmatrix':
y = np.zeros_like(x, dtype = 'd')
self.H.mult(x, y, self.verbose)
y += self.nugget*y
self.count += 1
return y
def rmatvec(self, x):
"""
Computes the matrix transpose-vector product
Parameters:
-----------
x: (n,) ndarray
a vector of size n
Returns:
-------
y: (n,) ndarray
Notes:
------
the result of this calculation are dependent on the method chosen. All methods except 'Hmatrix' are exact.
Because of symmetry it is almost the same as matvec, except 'Hmatrix' which is numerically different
"""
method = self.method
if method == 'Dense':
y = np.dot(self.mat.T,x)
elif method == 'FFT':
y = ToeplitzProduct(x, self.row, self.N)
elif method == 'Hmatrix':
y = np.zeros_like(x, dtype = 'd')
self.H.transpmult(x, y, self.verbose)
y += self.nugget*y
self.count += 1
return y
def reset(self):
"""
Resets the counter of matvecs and solves
"""
self.count = 0
self.solvmatvecs = 0
return
def itercount(self):
"""
Returns the counter of matvecs
"""
return self.count
def build_preconditioner(self, k = 100):
"""
Implementation of the preconditioner based on changing basis.
Parameters:
-----------
k: int, optional. default = 100
Number of local centers in the preconditioner. Controls the sparity of the preconditioner.
Notes:
------
Implementation of the preconditioner based on local centers.
The parameter k controls the sparsity and the effectiveness of the preconditioner.
Larger k is more expensive but results in fewer iterations.
For large ill-conditioned systems, it was best to use a nugget effect to make the problem better conditioned.
To Do: implementation based on local centers and additional points. Will remove the hack of using nugget effect.
References:
-----------
"""
from time import time
from scipy.spatial import cKDTree
from scipy.spatial.distance import pdist, cdist
from scipy.linalg import solve
from scipy.sparse import csr_matrix
#If preconditioner already exists, then nothing to do
if self.P != None: return
pts = self.pts
kernel = self.kernel
N = pts.shape[0]
#Build the tree
start = time()
tree = cKDTree(pts, leafsize = 32)
end = time()
if self.verbose:
print "Tree building time = %g" % (end-start)
#Find the nearest neighbors of all the points
start = time()
dist, ind = tree.query(pts,k = k)
end = time()
if self.verbose:
print "Nearest neighbor computation time = %g" % (end-start)
Q = np.zeros((k,k),dtype='d')
y = np.zeros((k,1),dtype='d')
row = np.tile(np.arange(N), (k,1)).transpose()
col = np.copy(ind)
nu = np.zeros((N,k),dtype='d')
y[0] = 1.
start = time()
for i in np.arange(N):
Q = kernel(cdist(pts[ind[i,:],:],pts[ind[i,:],:]))
nui = np.linalg.solve(Q,y)
nu[i,:] = np.copy(nui.transpose())
end = time()
if self.verbose: print "Building preconditioner took = %g" % (end-start)
ij = np.zeros((N*k,2), dtype = 'i')
ij[:,0] = np.copy(np.reshape(row,N*k,order='F').transpose() )
ij[:,1] = np.copy(np.reshape(col,N*k,order='F').transpose() )
data = np.copy(np.reshape(nu,N*k,order='F').transpose())
self.P = csr_matrix((data,ij.transpose()),shape=(N,N), dtype = 'd')
return
def solve(self, b, maxiter = 1000, tol = 1.e-10):
"""
Compute Q^{-1}b
Parameters:
-----------
b: (n,) ndarray
given right hand side
maxiter: int, optional. default = 1000
Maximum number of iterations for the linear solver
tol: float, optional. default = 1.e-10
Residual stoppingtolerance for the iterative solver
Notes:
------
If 'Dense' then inverts using LU factorization otherwise uses iterative solver
- MINRES if used without preconditioner or GMRES with preconditioner.
Preconditioner is not guaranteed to be positive definite.
"""
if self.method == 'Dense':
from scipy.linalg import solve
x = solve(self.mat, b)
else:
from scipy.sparse.linalg import gmres, aslinearoperator, minres
P = self.P
Aop = aslinearoperator(self)
residual = _Residual()
if P != None:
x, info = gmres(Aop, b, tol = tol, restart = 30, maxiter = 1000, callback = residual, M = P)
else:
x, info = minres(Aop, b, tol = tol, maxiter = maxiter, callback = residual )
self.solvmatvecs += residual.itercount()
if self.verbose:
print "Number of iterations is %g and status is %g"% (residual.itercount(), info)
return x