TransportMaps.Maps.Functionals.AnchoredIntegratedSquaredParametricFunctionalBase

Module Contents

Classes

AnchoredIntegratedSquaredParametricFunctional	Parameteric function \(f_{\bf a}({\bf x}) = \int_0^{x_d} h_{\bf a}^2(x_1,\ldots,x_{d-1},t) dt\)
IntegratedSquaredParametricFunctionApproximation	Parameteric function \(f_{\bf a}({\bf x}) = \int_0^{x_d} h_{\bf a}^2(x_1,\ldots,x_{d-1},t) dt\)

class TransportMaps.Maps.Functionals.AnchoredIntegratedSquaredParametricFunctionalBase.AnchoredIntegratedSquaredParametricFunctional(h, integ_ord_mult=6)

Bases: TransportMaps.Maps.Functionals.ParametricFunctionalBase.ParametricFunctional

Parameteric function \(f_{\bf a}({\bf x}) = \int_0^{x_d} h_{\bf a}^2(x_1,\ldots,x_{d-1},t) dt\)

Parameters:

• h (ParametricFunctionApproximation) – parametric function \(h\)

• integ_ord_mult (int) – multiplier for the number of Gauss points to be used in the approximation of \(\int_0^{{\bf x}_d}\). The resulting number of points is given by the product of the order in the \(d\) direction and integ_ord_mult.

property dim_in

property n_coeffs

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

Returns:

(int) – number of coefficients

property coeffs

Get the coefficients \({\bf a}\)

Returns:

(ndarray [\(N\)]) – coefficients

property multi_idxs

property semilattice

property basis_list

property full_basis_list

init_coeffs()

Initialize the coefficients \({\bf a}\)

get_multi_idxs()

Get the list of multi indices

Returns:

(list of tuple) – multi indices

set_multi_idxs(multi_idxs)

Set the list of multi indices

Parameters:

multi_idxs (list of tuple) – multi indices

precomp_evaluate(x, precomp=None, precomp_type='uni')

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

Parameters:

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

Returns:

(dict) – data structures

evaluate(x, precomp=None, idxs_slice=slice(None), cache=None)

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

precomp_grad_x(x, precomp, precomp_type='uni')

[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

grad_x(x, precomp=None, idxs_slice=slice(None), *args, **kwargs)

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].

Returns:

(ndarray [\(m,1,d\)]) –

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

grad_a_grad_x(x, precomp=None, idxs_slice=slice(None), *args, **kwargs)

Evaluate \(\nabla{\bf a} \nabla_{\bf x} f_{\bf a}\) 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].

Returns:

(ndarray [\(m,1,N,d\)]) –

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

precomp_hess_x(x, precomp, precomp_type='uni')

hess_x(x, precomp=None, idxs_slice=slice(None), *args, **kwargs)

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

Returns:

(ndarray [\(m,1,d,d\)]) –

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

grad_a_hess_x(x, precomp=None, idxs_slice=slice(None), *args, **kwargs)

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

Returns:

(ndarray [\(m,1,N,d,d\)]) –

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

grad_a(x, precomp=None, idxs_slice=slice(None), cache=None)

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, precomp=None, idxs_slice=slice(None), *args, **kwargs)

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].

Returns:

(ndarray [\(m,1,N,N\)]) –

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

precomp_partial_xd(x, precomp=None, precomp_type='uni')

[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

partial_xd(x, precomp=None, idxs_slice=slice(None), cache=None)

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 [\(m,1\)]) –

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

precomp_grad_x_partial_xd(x, precomp=None, precomp_type='uni')

[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

grad_x_partial_xd(x, precomp=None, idxs_slice=slice(None), cache=None)

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 [\(m,1,d\)]) –

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

grad_a_grad_x_partial_xd(x, precomp=None, idxs_slice=slice(None), cache=None)

Evaluate \(\nabla_{\bf a}\nabla_{\bf x}\partial_{x_d} f_{\bf a}\) 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 [\(m,1,N,d\)]) –

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

precomp_hess_x_partial_xd(x, precomp=None, precomp_type='uni')

hess_x_partial_xd(x, precomp=None, idxs_slice=slice(None), cache=None)

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_hess_x_partial_xd(x, precomp=None, idxs_slice=slice(None), cache=None)

Evaluate \(\nabla_{\bf a}\nabla^2_{\bf x}\partial_{x_d} f_{\bf a}\) 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 [\(m,1,N,d,d\)]) –

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

precomp_partial2_xd(x, precomp=None, precomp_type='uni')

[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

partial2_xd(x, precomp=None, idxs_slice=slice(None), cache=None)

Evaluate \(\partial^2_{x_d} f_{\bf a}\) 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 [\(m,1\)]) –

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

grad_a_partial_xd(x, precomp=None, idxs_slice=slice(None), cache=None)

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, precomp=None, idxs_slice=slice(None), cache=None)

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 [\(m,1,N,N\)]) –

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

class TransportMaps.Maps.Functionals.AnchoredIntegratedSquaredParametricFunctionalBase.IntegratedSquaredParametricFunctionApproximation(*args, **kwargs)

Bases: AnchoredIntegratedSquaredParametricFunctional

Parameteric function \(f_{\bf a}({\bf x}) = \int_0^{x_d} h_{\bf a}^2(x_1,\ldots,x_{d-1},t) dt\)

Parameters:

• h (ParametricFunctionApproximation) – parametric function \(h\)

• integ_ord_mult (int) – multiplier for the number of Gauss points to be used in the approximation of \(\int_0^{{\bf x}_d}\). The resulting number of points is given by the product of the order in the \(d\) direction and integ_ord_mult.