biosspheres.formulations.mtf.mtf#

This module contains routines for obtaining linear operators arising in multiple traces formulations when using spherical harmonics basis as test and trial functions.

Extended summary#

The routines: - x_j_diagonal - x_j_diagonal_inv - x_diagonal_with_its_inv are for obtaining the operators denoted as X in the documentation. x_diagonal and x_j_diagonal_inv are for one sphere. x_diagonal_with_its_inv is for n spheres.
The routines: - mtf_1_matrix - mtf_1_linear_operator - mtf_1_reduced_matrix_laplace - mtf_1_reduced_linear_operator - mtf_1_reduced_matrix_helmholtz - mtf_1_reduced_linear_operator_helmholtz are for one sphere. Each one has its matrix and linear operator (with scipy) counterpart. The reduced versions is a modification of the mtf using the Schur’s complement.
The routines: - mtf_n_matrix - mtf_n_linear_operator_v1 - mtf_n_reduced_matrix are for multiple spheres.

Routine listings#

x_j_diagonal x_j_diagonal_inv x_diagonal_with_its_inv mtf_1_matrix mtf_1_linear_operator mtf_n_matrix mtf_n_linear_operator_v1 mtf_1_reduced_matrix_laplace mtf_1_reduced_linear_operator mtf_1_reduced_matrix_helmholtz mtf_1_reduced_linear_operator_helmholtz mtf_n_reduced_matrix

Functions

`mtf_1_linear_operator`(a_0j, a_j, x_j, x_j_inv)	Returns a scipy linear operator, it is equivalent to the matrix M = [2 A_{1,1}^0 , -X^{-1} ] [ -X , 2A_{1,1} ] = [ -2K_0 , 2V_0 , -I , 0 ] [ 2W_0 , 2K_0 , 0 , pi I ] [ -I , 0 , -2K_1 , 2V_1 ] [ 0 , pi^(-1) I , 2W_1 , 2K_1 ] All blocks are diagonal matrices.
`mtf_1_matrix`(r, pi, a_0j, a_j)	Returns the matrix M = [2 A_{1,1}^0 , -X^{-1} ] [ -X , 2A_{1,1} ] = [ -2K_0 , 2V_0 , -I , 0 ] [ 2W_0 , 2K_0 , 0 , pi I ] [ -I , 0 , -2K_1 , 2V_1 ] [ 0 , pi^(-1) I , 2W_1 , 2K_1 ] All blocks are diagonal matrices.
`mtf_1_reduced_linear_operator`(big_l, r, pi)	Returns a scipy linear operator equivalent to the matrix M_red
`mtf_1_reduced_linear_operator_helmholtz`(...)	Returns a scipy linear operator equivalent to the matrix M_red = 2 (A_{1,1}^0 - X^{-1} A_{1,1} X)
`mtf_1_reduced_matrix_helmholtz`(big_l, r, k0, ...)	Returns the matrix M_red = 2 (A_{1,1}^0 - X^{-1} A_{1,1} X)
`mtf_1_reduced_matrix_laplace`(pi, a_0)	Returns the matrix M_red
`mtf_n_linear_operator_v1`(big_a_0_cross, ...)
`mtf_n_matrix`(big_a_0_cross, ...)
`mtf_n_reduced_matrix`(big_a_0_cross, ...)
`x_diagonal_with_its_inv`(n, big_l, radii, pii)	Returns a numpy array with the diagonals of the matrices given in the functions x_j_diagonal and x_j_diagonal_inv for n spheres.
`x_j_diagonal`(big_l, r, pi[, azimuthal])	Returns a numpy array with the diagonal of the matrix: X_j = [I , 0 ] [0 , -(pi)^{-1} I ] with I[l(2l+1) + m] = < I Y_l,m ; Y_l,m >_L^2(S). = r*2 for each l such that 0 <= l <= big_l, and with S: surface sphere radius r Y_l,m: spherical harmonic degree l, order m.
`x_j_diagonal_inv`(big_l, r, pi[, azimuthal])	Returns a numpy array with the diagonal of the matrix: X_j = [I , 0 ] [0 , -(pi)^{-1} I ] with I[l(2l+1) + m] = < I Y_l,m ; Y_l,m >_L^2(S). = r*2 for each l such that 0 <= l <= big_l, and with S: surface sphere radius r Y_l,m: spherical harmonic degree l, order m.

biosspheres.formulations.mtf.mtf.mtf_1_linear_operator(a_0j: LinearOperator, a_j: LinearOperator, x_j: ndarray, x_j_inv: ndarray) → LinearOperator[source]#

Returns a scipy linear operator, it is equivalent to the matrix M = [2 A_{1,1}^0 , -X^{-1} ]

[ -X , 2A_{1,1} ]

= [ -2K_0 , 2V_0 , -I , 0 ]
[ 2W_0 , 2K*_0 , 0 , pi I ] [ -I , 0 , -2K_1 , 2V_1 ] [ 0 , pi^(-1) I , 2W_1 , 2K*_1 ]

All blocks are diagonal matrices.

Parameters:

a_0j (sparse.linalg.LinearOperator,)
a_j (sparse.linalg.LinearOperator,)
x_j (np.ndarray)
x_j_inv (np.ndarray)

Returns:

linear_operator – a scipy linear operator.

Return type:

sparse.linalg.LinearOperator