class LowRankHessian:
"""
Operator that represents the action of the low rank approx
of the Hessian and of its inverse.
"""
def __init__(self, prior, d, U):
self.prior = prior
self.LowRankH = LowRankOperator(d, U)
dsolve = d / (np.ones(d.shape, dtype=d.dtype) + d)
self.LowRankHinv = LowRankOperator(dsolve, U)
self.help = Vector()
self.init_vector(self.help, 0)
self.help1 = Vector()
self.init_vector(self.help1, 0)
def init_vector(self, x, dim):
self.prior.init_vector(x, dim)
def inner(self, x, y):
Hx = Vector()
self.init_vector(Hx, 0)
self.mult(x, Hx)
return Hx.inner(y)
def mult(self, x, y):
self.prior.R.mult(x, y)
self.LowRankH.mult(y, self.help)
self.prior.R.mult(self.help, self.help1)
y.axpy(1, self.help1)
def solve(self, sol, rhs):
self.prior.Rsolver.solve(sol, rhs)
self.LowRankHinv.mult(rhs, self.help)
sol.axpy(-1, self.help)
class LowRankPosteriorSampler:
"""
Object to sample from the low-rank approximation
of the posterior.
y = ( I - U S U^TR) x,
where
S = I - (I + D)^{-1/2},
x ~ N(0, R^{-1})
"""
def __init__(self, prior, d, U):
self.prior = prior
ones = np.ones(d.shape, dtype=d.dtype)
self.d = ones - np.power(ones + d, -.5)
self.lrsqrt = LowRankOperator(self.d, U)
self.help = Vector()
self.init_vector(self.help, 0)
def init_vector(self, x, dim):
self.prior.init_vector(x, dim)
def sample(self, noise, s):
self.prior.R.mult(noise, self.help)
self.lrsqrt.mult(self.help, s)
s.axpy(-1, noise)
s *= -1.0
class LowRankHessianMisfit:
"""
Operator that represents the action of the low rank approx
of the Hessian misfit and of its inverse.
"""
def __init__(self, prior, d, U):
self.prior = prior
self.LowRankH = LowRankOperator(d, U)
self.help = Vector()
self.init_vector(self.help, 0)
def init_vector(self, x, dim):
self.prior.init_vector(x, dim)
def inner(self, x, y):
Hx = Vector()
self.init_vector(Hx, 0)
self.mult(x, Hx)
return Hx.inner(y)
def mult(self, x, y):
self.prior.R.mult(x, y)
self.LowRankH.mult(y, self.help)
self.prior.R.mult(self.help, y)