TransportMaps.Maps.Functionals.IdentityParametricMonotoneFunctionalBase

Module Contents

Classes

IdentityParametricMonotoneFunctional

Identity functional \(\mathbb{R}\rightarrow\mathbb{R}\).

class TransportMaps.Maps.Functionals.IdentityParametricMonotoneFunctionalBase.IdentityParametricMonotoneFunctional[source]

Bases: TransportMaps.Maps.Functionals.ParametricMonotoneFunctionalBase.ParametricMonotoneFunctional

Identity functional \(\mathbb{R}\rightarrow\mathbb{R}\).

property n_coeffs[source]

[Abstract] Get the number \(N\) of coefficients \({\bf a}\)

Returns:

(int) – number of coefficients

property coeffs[source]

[Abstract] Get the coefficients \({\bf a}\)

Returns:

(ndarray [\(N\)]) – coefficients

precomp_evaluate(*args, **kwargs)[source]

[Abstract] Precompute necessary structures for the evaluation of \(f_{\bf a}\) at x.

Parameters:

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

Returns:

(dict) – data structures

precomp_Vandermonde_evaluate(*args, **kwargs)[source]
precomp_grad_x(*args, **kwargs)[source]

[Abstract] Precompute necessary structures for the evaluation of \(\nabla_{\bf x} f_{\bf a}\) at x

Parameters:

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

Returns:

(dict) – data structures

precomp_Vandermonde_grad_x(*args, **kwargs)[source]
precomp_hess_x(*args, **kwargs)[source]
precomp_Vandermonde_hess_x(*args, **kwargs)[source]
precomp_partial_xd(*args, **kwargs)[source]

[Abstract] Precompute necessary structures for the evaluation of \(\partial_{x_d} f_{\bf a}\) at x.

Parameters:

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

Returns:

(dict) – data structures

precomp_Vandermonde_partial_xd(*args, **kwargs)[source]
precomp_grad_x_partial_xd(*args, **kwargs)[source]

[Abstract] Precompute necessary structures for the evaluation of \(\nabla_{\bf x}\partial_{x_d} f_{\bf a}\) at x.

Parameters:

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

Returns:

(dict) – data structures

precomp_Vandermonde_grad_x_partial_xd(*args, **kwargs)[source]
precomp_hess_x_partial_xd(*args, **kwargs)[source]
precomp_Vandermonde_hess_x_partial_xd(*args, **kwargs)[source]
precomp_partial2_xd(*args, **kwargs)[source]

[Abstract] Precompute necessary structures for the evaluation of \(\partial^2_{x_d} f_{\bf a}\) at x.

Parameters:

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

Returns:

(dict) – data structures

precomp_Vandermonde_partial2_xd(*args, **kwargs)[source]
evaluate(x, *args, **kwargs)[source]

[Abstract] Evaluate \(f_{\bf a}\) at x.

Parameters:
  • x (ndarray [\(m,d\)]) – 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].

  • cache (dict) – cache

Returns:

(ndarray [\(m,1\)]) – function evaluations

grad_x(x, *args, **kwargs)[source]

[Abstract] Evaluate \(\nabla_{\bf x} f_{\bf a}\) at x.

Parameters:
  • x (ndarray [\(m,d\)]) – 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].

  • cache (dict) – cache

Returns:

(ndarray [\(m,1,d\)]) –

\(\nabla_{\bf x} f_{\bf a}({\bf x})\)

hess_x(x, *args, **kwargs)[source]

[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(x, dx, *args, **kwargs)[source]

[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

partial_xd(x, *args, **kwargs)[source]

[Abstract] Evaluate \(\partial_{x_d} f_{\bf a}\) at x.

Parameters:
  • x (ndarray [\(m,d\)]) – 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].

  • cache (dict) – cache

Returns:

(ndarray [\(m,1\)]) –

\(\partial_{x_d} f_{\bf a}({\bf x})\)

grad_x_partial_xd(x, *args, **kwargs)[source]

[Abstract] Evaluate \(\nabla_{\bf x}\partial_{x_d} f_{\bf a}\) at x.

Parameters:
  • x (ndarray [\(m,d\)]) – 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].

  • cache (dict) – cache

Returns:

(ndarray [\(m,1,d\)]) –

\(\nabla_{\bf x}\partial_{x_d} f_{\bf a}({\bf x})\)

hess_x_partial_xd(x, *args, **kwargs)[source]

[Abstract] Evaluate \(\nabla^2_{\bf x}\partial_{x_d} f_{\bf a}\) at x.

Parameters:
  • x (ndarray [\(m,d\)]) – 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].

  • cache (dict) – cache

Returns:

(ndarray [\(m,1,d,d\)]) –

\(\nabla^2_{\bf x}\partial_{x_d} f_{\bf a}({\bf x})\)

grad_a(x, *args, **kwargs)[source]

[Abstract] Evaluate \(\nabla_{\bf a} f_{\bf a}\) at x.

Parameters:
  • x (ndarray [\(m,d\)]) – 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].

  • cache (dict) – cache

Returns:

(ndarray [\(m,1,N\)]) –

\(\nabla_{\bf a} f_{\bf a}({\bf x})\)

hess_a(x, *args, **kwargs)[source]

[Abstract] Evaluate \(\nabla^2_{\bf a} f_{\bf a}\) at x.

Parameters:
  • x (ndarray [\(m,d\)]) – 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].

  • cache (dict) – cache

Returns:

(ndarray [\(m,1,N,N\)]) –

\(\nabla^2_{\bf a} f_{\bf a}({\bf x})\)

action_hess_a(x, *args, **kwargs)[source]
grad_a_partial_xd(x, *args, **kwargs)[source]

[Abstract] Evaluate \(\nabla_{\bf a}\partial_{x_d} f_{\bf a}\) at x.

Parameters:
  • x (ndarray [\(m,d\)]) – 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].

  • cache (dict) – cache

Returns:

(ndarray [\(m,1,N\)]) –

\(\nabla_{\bf a}\partial_{x_d} f_{\bf a}({\bf x})\)

hess_a_partial_xd(x, *args, **kwargs)[source]

[Abstract] Evaluate \(\nabla^2_{\bf a}\partial_{x_d} f_{\bf a}\) at x.

Parameters:
  • x (ndarray [\(m,d\)]) – 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].

  • cache (dict) – cache

Returns:

(ndarray [\(m,1,N,N\)]) –

\(\nabla^2_{\bf a}\partial_{x_d} f_{\bf a}({\bf x})\)