TransportMaps.Maps.ODEs.DiscretizedODEsMaps

Module Contents

Classes

DiscretizedAutonomousODEsMap	Defines the map of discretized system of autonomous ODEs.
AutonomousForwardEulerMap	Defines the map of a forward Euler discretized system of autonomous ODEs.

class TransportMaps.Maps.ODEs.DiscretizedODEsMaps.DiscretizedAutonomousODEsMap(dt, rhs)

Bases: TransportMaps.Maps.Map

Defines the map of discretized system of autonomous ODEs.

Evaluates the map

\[{\bf u}_n \mapsto {\bf u}_{n+1}\]

that takes the state \({\bf u}_n\) at time \(t\) into the state \({\bf u}_{n+1}\) at time \(t+\Delta t\), thorugh the discretization of the ODE

\[\dot{\bf u} = f({\bf u}) \;.\]

__init__(dt, rhs)

Parameters:

• dt (float) – time step \(\Delta t\)

• rhs (Map) – the \(d\) dimensional map \(f\).

property dt

property rhs

abstract evaluate(u, *args, **kwargs)

[Abstract] Evaluate the map \(T\) at the points \({\bf x} \in \mathbb{R}^{m \times d_x}\).

Parameters:

• x (ndarray [\(m,d_x\)]) – evaluation points

• precomp (dict) – dictionary of precomputed values

• idxs_slice (slice) – if precomputed values are present, this parameter indicates at which of the points to evaluate. The number of indices represented by idxs_slice must match x.shape[0].

Returns:

(ndarray [\(m,d_y\)]) – transformed points

Raises:

NotImplementedError – to be implemented in sub-classes

abstract grad_x(u, *args, **kwargs)

[Abstract] Evaluate the gradient \(\nabla_{\bf x}T\) at the points \({\bf x} \in \mathbb{R}^{m \times d_x}\).

Parameters:

• x (ndarray [\(m,d_x\)]) – evaluation points

• precomp (dict) – dictionary of precomputed values

• idxs_slice (slice) – if precomputed values are present, this parameter indicates at which of the points to evaluate. The number of indices represented by idxs_slice must match x.shape[0].

Returns:

(ndarray [\(m,d_y,d_x\)]) – transformed points

Raises:

NotImplementedError – to be implemented in sub-classes

abstract tuple_grad_x(u, *args, **kwargs)

[Abstract] Evaluate the function and gradient.

Parameters:

• x (ndarray [\(m,d_x\)]) – evaluation points

• precomp (dict) – dictionary of precomputed values

• idxs_slice (slice) – if precomputed values are present, this parameter indicates at which of the points to evaluate. The number of indices represented by idxs_slice must match x.shape[0].

Returns:

(tuple) – function and gradient evaluation

Raises:

NotImplementedError – to be implemented in sub-classes

abstract hess_x(u, *args, **kwargs)

[Abstract] Evaluate the Hessian \(\nabla^2_{\bf x}T\) at the points \({\bf x} \in \mathbb{R}^{m \times d_x}\).

Parameters:

• x (ndarray [\(m,d_x\)]) – evaluation points

• precomp (dict) – dictionary of precomputed values

• idxs_slice (slice) – if precomputed values are present, this parameter indicates at which of the points to evaluate. The number of indices represented by idxs_slice must match x.shape[0].

Returns:

(ndarray [\(m,d_y,d_x,d_x\)]) – transformed points

Raises:

NotImplementedError – to be implemented in sub-classes

abstract action_hess_x(u, du, *args, **kwargs)

[Abstract] Evaluate the action of the Hessian \(\langle\nabla^2_{\bf x}T,\delta{\bf x}\rangle\) at the points \({\bf x} \in \mathbb{R}^{m \times d_x}\).

Parameters:

• x (ndarray [\(m,d_x\)]) – evaluation points

• dx (ndarray [\(m,d_x\)]) – direction on which to evaluate the Hessian

• precomp (dict) – dictionary of precomputed values

• idxs_slice (slice) – if precomputed values are present, this parameter indicates at which of the points to evaluate. The number of indices represented by idxs_slice must match x.shape[0].

Returns:

(ndarray [\(m,d_y,d_x\)]) – transformed points

Raises:

NotImplementedError – to be implemented in sub-classes

class TransportMaps.Maps.ODEs.DiscretizedODEsMaps.AutonomousForwardEulerMap(dt, rhs)

Bases: DiscretizedAutonomousODEsMap

Defines the map of a forward Euler discretized system of autonomous ODEs.

Evaluates the Euler step:

\[{\bf u}_{n+1} = {\bf u}_n + \Delta t \cdot f({\bf u}_n)\]

where \(f:\mathbb{R}^d \rightarrow \mathbb{R}^d\) is the right hand side of the ODE system.

__init__(dt, rhs)

Parameters:

• dt (float) – time step \(\Delta t\)

• rhs (Map) – the \(d\) dimensional map \(f\).

evaluate(u, *args, **kwargs)

[Abstract] Evaluate the map \(T\) at the points \({\bf x} \in \mathbb{R}^{m \times d_x}\).

Parameters:

• x (ndarray [\(m,d_x\)]) – evaluation points

• precomp (dict) – dictionary of precomputed values

• idxs_slice (slice) – if precomputed values are present, this parameter indicates at which of the points to evaluate. The number of indices represented by idxs_slice must match x.shape[0].

Returns:

(ndarray [\(m,d_y\)]) – transformed points

Raises:

NotImplementedError – to be implemented in sub-classes

grad_x(u, *args, **kwargs)

[Abstract] Evaluate the gradient \(\nabla_{\bf x}T\) at the points \({\bf x} \in \mathbb{R}^{m \times d_x}\).

Parameters:

• x (ndarray [\(m,d_x\)]) – evaluation points

• precomp (dict) – dictionary of precomputed values

• idxs_slice (slice) – if precomputed values are present, this parameter indicates at which of the points to evaluate. The number of indices represented by idxs_slice must match x.shape[0].

Returns:

(ndarray [\(m,d_y,d_x\)]) – transformed points

Raises:

NotImplementedError – to be implemented in sub-classes

tuple_grad_x(u, *args, **kwargs)

[Abstract] Evaluate the function and gradient.

Parameters:

• x (ndarray [\(m,d_x\)]) – evaluation points

• precomp (dict) – dictionary of precomputed values

• idxs_slice (slice) – if precomputed values are present, this parameter indicates at which of the points to evaluate. The number of indices represented by idxs_slice must match x.shape[0].

Returns:

(tuple) – function and gradient evaluation

Raises:

NotImplementedError – to be implemented in sub-classes

hess_x(u, *args, **kwargs)

[Abstract] Evaluate the Hessian \(\nabla^2_{\bf x}T\) at the points \({\bf x} \in \mathbb{R}^{m \times d_x}\).

Parameters:

• x (ndarray [\(m,d_x\)]) – evaluation points

• precomp (dict) – dictionary of precomputed values

• idxs_slice (slice) – if precomputed values are present, this parameter indicates at which of the points to evaluate. The number of indices represented by idxs_slice must match x.shape[0].

Returns:

(ndarray [\(m,d_y,d_x,d_x\)]) – transformed points

Raises:

NotImplementedError – to be implemented in sub-classes

action_hess_x(u, du, *args, **kwargs)

[Abstract] Evaluate the action of the Hessian \(\langle\nabla^2_{\bf x}T,\delta{\bf x}\rangle\) at the points \({\bf x} \in \mathbb{R}^{m \times d_x}\).

Parameters:

• x (ndarray [\(m,d_x\)]) – evaluation points

• dx (ndarray [\(m,d_x\)]) – direction on which to evaluate the Hessian

• precomp (dict) – dictionary of precomputed values

• idxs_slice (slice) – if precomputed values are present, this parameter indicates at which of the points to evaluate. The number of indices represented by idxs_slice must match x.shape[0].

Returns:

(ndarray [\(m,d_y,d_x\)]) – transformed points

Raises:

NotImplementedError – to be implemented in sub-classes