# 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)[source]

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)[source]

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)[source]

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)[source]

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:
Returns:

(float) – inverse value $$x$$.

partial_xd_inverse(xmd, y)[source]

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

Parameters:
Returns:

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