Alvise Sommariva

• Software: GitHub (L. Rinaldi homepage) 

Object:

ORTHOCUB: integral and differential cubature rules by orthogonal moments.

In what follows, we mention the routines that are stored in this project.

• cheap_startup: given the algebraic degree of precision ade, it computes a low cardinality cubature rule rule_ref on the reference square [−1, 1]^2 or cube [−1, 1]^3 w.r.t. tensorial-Chebyshev weight (17) or (18). Next, calling the subroutine dCHEBVAND_orthn, it evaluates the tensorial-orthonormal Chebyshev basis {ψ_j}_j of total degree ade at the nodes, storing the results in the Van- dermonde matrix V_ref. The sequence of indices (h, k) or (h, k, l), defining the ordering of the basis elements, is listed in the matrix basis_indices. We stress that this routine in independent of the functional L to be used and is only related to the ADE of the reference cubature rule.

• cheap_rule: given the matrix dbox of dimension 2 × d in which the i column con- tains the extrema of the bounding box w.r.t. the i-th variable, the moments m storing L(ψ j({Λ−1}(· − C))), as well as rule_ref, V_ref computed by cheap_startup, it determines the nodes X and the weights w of the ORTHOCUB cubature rule.

• dCHEBVAND_orthn evaluates the Vandermonde matrix on the set X ⊂ Ω ⊆ B, relatively to the scaled tensorial-Chebyshev {φ_j}_j. As input, it is requires the variable dbox describing B and the sequence of degree indices basis_indices.

• vandermonde_jacobi allows the evaluation of the tensorial-Jacobi polynomials. This general purpose function takes into account a specific normalization of the basis, e.g. monic, orthonormal as well classical ones depending on the weight.

• cub_square_mpx, cub_cube_mpx are low cardinality cubature rules, respectively on the unit-square [−1, 1]^2 and on the cube [−1, 1]^3 relatively to the tensorial-Chebyshev measure.

• mom_derivative_2D, mom_derivative_3D produce the derivatives of order with re- spect to a variable specified by the user of the polynomial basis {φ_k} on a specific 2D or 3D mesh. Depending on the mesh, it is possible to define a suitable bounding box B via the variable dbox.

As for the examples regarding numerical cubature, the algorithms require some additional functions.

• compute_spline_boundary is useful in the case of the spline-curvilinear domains. It provides the description of its boundary, given as inputs the abscissae of the vertices XV and the relative ordinates YV , a variable spline_parms storing the order of the piecewise splines and the vertices involved. In case the spline is cubic, spline_type fixes the additional conditions (e.g. natural, periodic). As output, the routine supplies the vector of structures Sx and Sy that determine the piecewise splines x = x(t), y = y(t) parametrizing the boundary.

• spline_chebmom, given the polynomial degree n, the vectors of structures Sx, Sy, the basis ordering basis_indices, computes the moments of the basis {ψ_j}_j and, if required, of {φ_j}_j. This purpose is obtained by applying Green theorem, requiring the numerical integration of a certain piecewise polynomial function on the boundary.

• QMC_union_balls, given the vectors of centers and radii defining the balls Bi, i = 1, 2, . . ., computes card Halton points XB in the bounding box B of the integration domain Ω = ∪_i B_i. Next, after the application of an in-domain function, provides as output the nodes and the weights of a QMC rule in Ω.

5.2 Description of the demos

In order to replicate the examples, we have implemented the following demos, used for the numerical experiments above.

• demo_cubature_splines illustrates the basic usage of the ORTHOCUB technique to approximate definite integrals on a spline-curvilinear domain Ω1 (see Figure 1). • demo_cubature_ade_splines, fixed a degree of precision n equal to 2, 4, . . . , 16, approximates via the rules proposed in this work 100 integrals of random polynomials of the form p_n(x, y) = (c_0 + c_1 x + c_2 y)^n. Each exact integral is computed via Green theorem. Finally it plots the single relative errors and the value of their geometric mean.

• demo_cubature_sumweights_spline first computes ORTHOCUB rules with ADE equal to 2, 4, . . . , 16 for integration on Ω1 and then their stability ratios as in Table 2.

• demo_cubature_weights_spline, first computes ORTHOCUB rules with ADE equal to 2, 4, . . . , 16 for integration on Ω1, and then sorts the weights, illustrating their distribution as in Figure 1.

• demo_cubature_QMC shows the basic usage of the ORTHOCUB technique to approximate definite integrals on the domain Ω2 that is union of 5 balls (see Figure 2).

• demo_cubature_ade_QMC determines via QMC_union_balls two QMC rules SL, SH on Ω2, based respectively on 105 and 106 in the bounding box B. Next, fixed a degree of precision n equal to 2, 4, . . . , 16, approximates via the rules proposed in this work 100 integrals of random polynomials of the form p_n(x, y, z) = (c_0 + c_1 x + c_2 y * c_3 z)^n. In particular, takes into account SH (p_n) as reference value, while as approximation of the moments m uses the vector whose i-th component is SL(φ_i). Finally it plots the single relative errors and the value of their geometric mean.

• demo_cubature_sumweights_QMC computes ORTHOCUB rules with ADE equal to 2, 4, . . . , 16 for QMC integration on Ω1 and their stability ratios as in Table 2.

• demo_cubature_weights_QMC computes ORTHOCUB rules with ADE equal to 2, 4, . . . , 16 for QMC integration on Ω2, sorts the weights and illustrates their distribution as in Figure 2.

• demo_derivative_ade_2D computes by the ORTHOCUB technique described above the first and second order derivatives of random polynomials of the form p_n(x, y) = (c_0 + c_1 x + c_2 y)^n on a mesh of the first 10000 Halton points {P_k} in [−1, 1]^2, for n = 2, 4, . . . , 16. For any exponent n and point of mesh Pk, we determine by an ORTHOCUB rule on [−1, 1]^2 defined by mom_derivative_2D the numerical approximation of these derivatives on P_k. Since the exact value is known analytically, we can easily establish the relative error in norm 2. Integration and differentiation by orthogonal moments

• demo_derivative_ade_3D computes by the ORTHOCUB technique described above the first and second order derivatives of random polynomials of the form p_n(x, y, z) = (c_0 + c_1 x + c_2 y + c_3 z)^n on a mesh of the first 10000 Halton points {P_k} in [−1, 1]^3, for n = 2, 4, . . . , 16. For any exponent n and point of mesh P_k, we determine by an ORTHOCUB rule on [−1, 1]^3 defined by mom_derivative_3D the numerical approximation of these derivatives on P_k. Since the exact value is known analytically, we can easily establish the relative error in norm 2.

• PDF: ORTHOCUB: integral and differential cubature rules by orthogonal moments, L. Rinaldi, A. Sommariva and M. Vianello

• Author of the routines: L. Rinaldi

• Software: GitHub (L. Rinaldi homepage) 

• Relevant subroutines/folders: 

         The .zip files contain two folder with demos that perform experiments respectively over spline curvilinear polygons and 3D complex domains.

Each folder contains in particular

• • demo_ADE: quadrature of polynomial integrands (to establish numerically the properties regarding the degree of exactness), 

  • demo_CUB: quadrature of polynomial integrands,

  • demo_CPU: average cputimes of the rules,

  • demo_WEIGHTS: distribution of the weights,

  • demo_SUMWEIGHTS: stability of the rules.

     Furthermore, a routine shows the geometry of the domain and the distribution of the nodes in a parallelepiped containing the integration domain.

• PDF: Effective numerical integration on complex shaped elements by discrete signed measures , L. Rinaldi, A. Sommariva and M. Vianello

• Author of the routines: L. Rinaldi

• Software: https://github.com/DimitriKenne/CPOLYMESH

• Relevant subroutines/folders: discrete_extremal_sets_constructor, discrete_orthogonal_polynomials_constructor_and_evaluator, domains_structure, demo_cpom,lebesgue_constant_evaluator,polynomial_mesh_constructor,stabilized_vandermonde_matrix_constructor,demo_cdes.py,demo_cleb.py.

• PDF: CPOLYMESH, Python codes for complex  polynomial  approximation  by  Chebyshev  admissible  meshes

Author: D.J. Kenne

• Software: https://github.com/mattiastorgato/TRIGAUSS

• Relevant subroutines: gauss.py, gqcircsect.py, gqcircsegm.py, trigauss.py

• PDF: Python package for subperiodic trigonometric quadrature and multivariate applications

Author: M. Storgato

• Software: https://github.com/2002232/Squaring-polygonal-elements-with-one-curved-side-in-Python.git

• Subroutines: auto_rotation.py, chebvand.py, chebyshev.py, circtrap.py, comprexcub.py, cubature_rules_1d.py, demo_polygcirc.py, gauss.py, gqellblend.py, points2distances.py, polygauss.py, polygcirc.py, r_jacobi.py, tridisolve.py, trigauss.py

• PDF: G. Traversin, Python package for numerical cubature on polygonal elements with one curved edge.

Author: G. Traversin