archimedes.approximation

Linear basis-expansion / function approximation infrastructure.

• Basis - a basis family (e.g. OrthogonalPolynomialBasis)

• FunctionSpace - a Basis on a domain along with a quadrature-based inner product and associated operations (e.g. FunctionSpace.project()).

• BasisMatrix - a basis evaluated at a set of quadrature nodes; a generalized Vandermonde matrix.

• Function - a FunctionSpace plus a coefficient vector.

Bases are univariate by default. TensorBasis combines one per dimension into a multivariate basis but otherwise works similarly to the univariate bases.

Two composition primitives build custom bases out of existing ones. ConstrainedBasis recombines one basis’s functions by a fixed matrix (e.g. the nullspace of a boundary condition constraint). ConcatBasis stacks functions from several bases side by side (e.g. spectral-element vertex + bubble functions).

Examples

Project a function onto a degree-5 Legendre space and evaluate it:

>>> import numpy as np
>>> from archimedes import approximation as approx
>>> space = approx.FunctionSpace.legendre(n_basis=6)
>>> f = space.project(lambda x: np.sin(np.pi * x))
>>> round(float(f(np.array([0.5]))[0]), 3)
1.005

Classes

BSplineBasis(degree, knots)	Univariate B-spline basis on a general knot vector.
Basis()	A finite-dimensional family of basis functions \(\{\phi_i\}_{i=1}^n\).
BasisMatrix(matrix, weights, nodes)	A Basis evaluated at a set of quadrature nodes.
ConcatBasis(pieces, quad_rule)	A basis made of combining other bases.
ConstrainedBasis(base, matrix)	A basis whose functions are fixed linear combinations of another's
CubicHermiteBasis()	A basis of cubic Hermite shape functions
FourierBasis(n_basis[, kind, density])	Trigonometric basis on a periodic interval.
Function(coefficients, space)	A specific element of a FunctionSpace:
FunctionSpace(basis, domain[, ...])	The linear span of a Basis on a fixed target domain.
LagrangeBasis(reference_nodes[, node_family])	Lagrange (nodal) interpolating polynomial basis
MonomialBasis(n_basis)	Monomial (power series) basis on an interval.
OrthogonalPolynomialBasis(measure, n_basis)	A classical orthonormal polynomial basis
PiecewiseBasis(element_basis, breakpoints[, ...])	A Basis constructed by tiling a local basis across elements.
ProductParameters([dims])	Target-domain parameters for a TensorBasis
TensorBasis(bases)	Tensor product of univariate bases, one per dimension.