IBM¶
-
class
probnum.diffeq.
IBM
(ordint, spatialdim, diffconst, precond_step=1.0)[source]¶ Bases:
probnum.diffeq.odefiltsmooth.prior.ODEPrior
Integrated Brownian motion of order \(q\) prior.
The integrated Brownian motion prior is represented through the LTI SDE
\[dX(t) = F X(t) dt + L dB(t)\]where for readibility reasons we did not write the preconditioner matrix \(P\); see
ODEPrior
for explanations.It has driftmatrix \(F\) given by
\[\begin{split}F = I_d \otimes \tilde F, \quad \tilde F = \begin{pmatrix} 0 & I_q \\ 0 & 0 \end{pmatrix}\end{split}\]where the top left zero-vector has \(q\) rows and 1 column.
It has dispersion matrix \(L\) given by
\[L = I_d \otimes \tilde L, \quad \tilde L = \sigma \, e_{q+1}\]where \(\sigma\) is the diffusion constant, that is, \(\sigma^2\) is the intensity of each dimension of the \(d\)-dimensional Brownian motion driving the SDE and \(e_{q+1}=(0, ..., 0, 1)\) is the \((q+1)\)-st unit vector.
The Brownian motion \(B=B(t)\) driving the SDE has unit diffusion \(Q = I_d\).
- Parameters
ordint (
int
) – Order of integration \(q\). The higher \(q\), the higher the order of the ODE filter.spatialdim (
int
) – Spatial dimension \(d\) of the ordinary differential equation that is to be modelled.diffconst (
float
) – Diffusion constant \(sigma\) of the stochastic process.precond_step (
float
) – Expected step size \(h\) used in the ODE filter. This quantity is used for preconditioning, seeODEPrior
for a clear explanation. Default is \(h=1\).
Attributes Summary
Spatial dimension (utility attribute).
Convenience property to return the readily-computed inverse preconditioner without having to remember abbreviations.
Convenience property to return the readily-computed preconditioner without having to remember abbreviations.
Methods Summary
__call__
(arr_or_rv[, start, stop])Transition a random variable or a realization of one.
discretise
(step)Returns a discrete transition model (i.e.
drift
(time, state, **kwargs)jacobian
(time, state, **kwargs)precond2nordsieck
(step)Computes preconditioner inspired by Nordsieck.
proj2coord
(coord)Projection matrix to \(i\)-th coordinates.
transition_realization
(real, start, stop, …)Transition a realization of a random variable from time \(t\) to time \(t+\Delta t\).
transition_rv
(rv, start, stop, **kwargs)Transition a random variable from time \(t\) to time \(t+\Delta t\).
Attributes Documentation
-
dimension
¶ Spatial dimension (utility attribute).
-
dispersionmatrix
¶
-
driftmatrix
¶
-
forcevec
¶
-
inverse_preconditioner
¶ Convenience property to return the readily-computed inverse preconditioner without having to remember abbreviations.
- Returns
Inverse preconditioner matrix \(P^{-1}\)
- Return type
np.ndarray, shape=(d(q+1), d(q+1))
-
preconditioner
¶ Convenience property to return the readily-computed preconditioner without having to remember abbreviations.
- Returns
Preconditioner matrix \(P\)
- Return type
np.ndarray, shape=(d(q+1), d(q+1))
Methods Documentation
-
__call__
(arr_or_rv, start=None, stop=None, **kwargs)¶ Transition a random variable or a realization of one.
The input is either interpreted as a random variable or as a realization. Accordingly, the respective methods are called:
transition_realization()
ortransition_rv()
.- Return type
(‘RandomVariable’, typing.Dict)
-
discretise
(step)¶ Returns a discrete transition model (i.e. mild solution to SDE) using matrix fraction decomposition.
That is, matrices A(h) and Q(h) and vector s(h) such that the transition is
\[x | x_\text{old} \sim \mathcal{N}(A(h) x_\text{old} + s(h), Q(h)) ,\]which is the transition of the mild solution to the LTI SDE.
-
drift
(time, state, **kwargs)¶
-
jacobian
(time, state, **kwargs)¶
-
precond2nordsieck
(step)¶ Computes preconditioner inspired by Nordsieck.
Computes the matrix \(P\) given by
\[P = I_d \otimes diag (1, h, h^2, ..., h^q)\]as well as its inverse \(P^{-1}\).
- Parameters
step (float) – Step size \(h\) used for preconditioning. If \(h\) is so small that \(h^q! < 10^{-15}\), it is being set to \(h = (\cdot 10^{-15})^{1/q}\).
- Return type
(<class ‘numpy.ndarray’>, <class ‘numpy.ndarray’>)
- Returns
precond (np.ndarray, shape=(d(q+1), d(q+1))) – Preconditioner matrix \(P\).
invprecond (np.ndarray, shape=(d(q+1), d(q+1))) – Inverse preconditioner matrix \(P^{-1}\).
-
proj2coord
(coord)¶ Projection matrix to \(i\)-th coordinates.
Computes the matrix
\[H_i = \left[ I_d \otimes e_i \right] P^{-1},\]where \(e_i\) is the \(i\)-th unit vector, that projects to the \(i\)-th coordinate of a vector. If the ODE is multidimensional, it projects to each of the \(i\)-th coordinates of each ODE dimension.
- Parameters
coord (int) – Coordinate index \(i\) which to project to. Expected to be in range \(0 \leq i \leq q + 1\).
- Returns
Projection matrix \(H_i\).
- Return type
np.ndarray, shape=(d, d*(q+1))
-
transition_realization
(real, start, stop, **kwargs)¶ Transition a realization of a random variable from time \(t\) to time \(t+\Delta t\).
For random variable \(x_t\), it returns the random variable defined by
\[x_{t + \Delta t} \sim p(x_{t + \Delta t} | x_t = r) .\]This is different to
transition_rv()
which computes the parametrization of \(x_{t + \Delta t}\) based on the parametrization of \(x_t\).Nb: Think of transition as a verb, i.e. this method “transitions” a realization of a random variable.
- Parameters
real – Realization of the random variable.
start – Starting point \(t\).
stop – End point \(t + \Delta t\).
- Returns
RandomVariable – Random variable, describing the state at time \(t + \Delta t\) based on realization at time \(t\).
dict – Additional information in form of a dictionary, for instance the cross-covariance in the prediction step, access to which is useful in smoothing.
See also
transition_rv()
Apply transition to a random variable.
-
transition_rv
(rv, start, stop, **kwargs)¶ Transition a random variable from time \(t\) to time \(t+\Delta t\).
For random variable \(x_t\), it returns the random variable defined by
\[x_{t + \Delta t} \sim p(x_{t + \Delta t} | x_t) .\]This returns a random variable where the parametrization depends on the paramtrization of \(x_t\). This is different to
transition_rv()
which computes the parametrization of \(x_{t + \Delta t}\) based on a realization of \(x_t\).Nb: Think of transition as a verb, i.e. this method “transitions” a random variable.
- Parameters
rv – Realization of the random variable.
start – Starting point \(t\).
stop – End point \(t + \Delta t\).
- Returns
RandomVariable – Random variable, describing the state at time \(t + \Delta t\) based on realization at time \(t\).
dict – Additional information in form of a dictionary, for instance the cross-covariance in the prediction step, access to which is useful in smoothing.
See also
transition_realization()
Apply transition to a realization of a random variable.