Create a Domain with Uniform Bounds

import minterpy as mp
import numpy as np

Approximating a function \(f: \Omega \rightarrow \mathbb{R}\) where \(\Omega\) is a custom hyper-rectangular domain in Minterpy involves constructing an (interpolating) polynomial \(Q\) such that:

\[ f(\boldsymbol{x}) \approx Q \circ \mathcal{T} (\boldsymbol{x}), \]

where \(Q: [-1, 1]^m \rightarrow \mathbb{R}\) is the polynomial defined on the internal (reference) domain and \(\mathcal{T}\) is the affine separable transformation from \(\Omega\) to \([-1, 1]^m\). Both the user-defined domain and the internal domain are encapsulated in the Domain class.

An instance of Domain can be constructed via different constructors:

• Domain: create an instance by specifying bounds for each dimension (see the example)

• Domain.uniform(): create an instance with uniform bounds (this page)

• Domain.identity(): create an instance with bounds identical to the internal domain (see the example)

In this guide, you will construct a Domain instance with uniform bounds using the factory method Domain.uniform().

About the factory method uniform()

The factory method Domain.uniform() constructs a Domain instance with identical bounds [a, b] applied across all dimensions, given a spatial dimension m, a lower bound a, and an upper bound b. The lower bound must be strictly less than the upper bound.

Example: Six-dimensional rectangular domain

Create a six-dimensional rectangular domain that corresponds to:

\[ \Omega = [1.0, 5.0]^6 \]

Domain instance

An instance of Domain of a given dimension with a uniform bound can be constructed using the factory method:

spatial_dimension = 6
lower = 1.0
upper = 5.0
dom = mp.Domain.uniform(spatial_dimension, lower, upper)

Spatial dimension

We can verify that the domain is six dimensional:

print(dom.spatial_dimension)

6

and with the appropriate bounds:

print(dom.bounds)

[[1. 5.]
 [1. 5.]
 [1. 5.]
 [1. 5.]
 [1. 5.]
 [1. 5.]]

Uniform domain

A domain is a uniform domain if it has identical bounds across all dimensions (\([a, b]^m\)). This can be verified for the current instance using the following property:

dom.is_uniform

True

Identity domain

A domain is an identity domain if the user-defined bounds match the internal reference domain (\([-1, 1]^m\)). This can be verified for the current instance using the following property:

dom.is_identity

False