TransportMaps.Maps.Functionals.MonotoneFunctionalBase

Module Contents

Classes

MonotoneFunctional

Abstract class for the functional \(f \approx f_{\bf a} = \sum_{{\bf i} \in \mathcal{I}} {\bf a}_{\bf i} \Phi_{\bf i}\) assumed to be monotonic in \(x_d\)

class TransportMaps.Maps.Functionals.MonotoneFunctionalBase.MonotoneFunctional(dim)

Bases: TransportMaps.Maps.Functionals.FunctionalBase.Functional

Abstract class for the functional \(f \approx f_{\bf a} = \sum_{{\bf i} \in \mathcal{I}} {\bf a}_{\bf i} \Phi_{\bf i}\) assumed to be monotonic in \(x_d\)

The class defines a series of methods (like the inverse) specific to monotone functions.

xd_misfit(x, args)

Compute \(f_{\bf a}({\bf x}) - y\)

Given the fixed coordinates \({\bf x}_{1:d-1}\), the value \(y\), and the last coordinate \({\bf x}_d\), compute:

\[f_{\bf a}({\bf x}_{1:d-1},{\bf x}_d) - y\]

Parameters:

• x (float) – evaluation point \({\bf x}_d\)

• args (tuple) – containing \(({\bc x}_{1:d-1},y)\)

Returns:

(float) – misfit.

partial_xd_misfit(x, args)

Compute \(\partial_{x_d} f_{\bf a}({\bf x}) - y = \partial_{x_d} f_{\bf a}({\bf x})\)

Given the fixed coordinates \({\bf x}_{1:d-1}\), the value \(y\), and the last coordinate \({\bf x}_d\), compute:

\[\partial f_{\bf a}({\bf x}_{1:d-1},{\bf x}_d)\]

Parameters:

• x (float) – evaluation point \({\bf x}_d\)

• args (tuple) – containing \(({\bc x}_{1:d-1},y)\)

Returns:

(float) – misfit derivative.

inverse(xmd, y, xtol=1e-12, rtol=1e-15)

Compute \({\bf x}_d\) s.t. \(f_{\bf a}({\bf x}_{1:d-1},{\bf x}_d) - y = 0\).

Given the fixed coordinates \({\bf x}_{1:d-1}\), the value \(y\), find the last coordinate \({\bf x}_d\) such that:

\[f_{\bf a}({\bf x}_{1:d-1},{\bf x}_d) - y = 0\]

We will define this value the inverse of \(f_{\bf a}({\bf x})\) and denote it by \(f_{\bf a}^{-1}({\bf x}_{1:d-1})(y)\).

Parameters:

• xmd (ndarray [\(d-1\)]) – fixed coordinates \({\bf x}_{1:d-1}\)

• y (float) – value \(y\)

• xtol (float) – absolute tolerance

• rtol (float) – relative tolerance

Returns:

(float) – inverse value \(x\).

partial_xd_inverse(xmd, y)

Compute \(\partial_y f_{\bf a}^{-1}({\bf x}_{1:d-1})(y)\).

Parameters:

• xmd (ndarray [\(d-1\)]) – fixed coordinates \({\bf x}_{1:d-1}\)

• y (float) – value \(y\)

Returns:

(float) – derivative of the inverse value \(x\).