Cubature Formulas for Multisymmetric Functions

A multisymmetric function is a real-valued function satisfying the following definition.

For example, the function f(x,y,z,w) = x+y+z+w is (4,1)-multisymmetric while the function g(x,y,z,w) = exp(10*x - y + 10*z - w) is (2,2)-multisymmetric.

This repository contains the julia code used in this paper to calculate cubature rules for multisymmetric functions as well as the formulas obtained using this code. The formulas can be found in the folder formulas and are labelled in the following way: The file (n,m,d)-nodes.txt contains the nodes of a cubature formula of degree d for (n,m)-multisymmetric functions. The corresponding weights can be found in the file (n,m,d)-weights.txt. Note that our approach only yields formulas with odd degrees.

Should you calculate a formula not yet listed here, please submit a pull request so we can make it available to the public. This code is published under a GNU general public license.

Requirements

This code is written in julia v0.5. The following packages are used:

• Combinatorics
• DriftDiffusionPoissonSystems
• FastGaussQuadrature
• Gurobi
• Iterators
• JuMP
• Sobol
• SymPy

Note that the Gurobi Optimization Software is not Open Source, however free academic licenses are available.

How to use the code

To load or calculate cubature formulas with parameters (n,m,d), execute the following lines in the julia console

julia> include("LPcubature.jl")
julia> nodes,weights = return_cubature_formula(n,m,d)

The function return_cubature_formula(...) expects three positive integers as arguments, where the third argument d should be odd. The function checks whether the formula for the parameters (n,m,d) is in the folder formulas. If the formula is not found, it is computed and stored there. The function returns two Arrays with Float64-entries, the nodes and weights of the respective formula.

Examples

Expectation of a stochastic partial differential equation - comparison to quasi-Monte Carlo.

The file SDEexample.jl contains the code used to make the plots below. To compare the efficiency of our formulas to a common integration technique, we compared our results to results obtained by using a quasi-Monte Carlo method (the Sobol sequence).

The first plots show the absolute error of the exact expectation to the approximation obtained by using a quasi-Monte Carlo method with N samples. The exact expectation is displayed in the last plot.

The first plots above show the absolute error of the exact expectation o to the approximation obtained by using a multisymmetric cubature rule of degree d. The exact expectation is displayed in the last plot. The amount of function evaluations is 4 for d=3 and 48 for d=11. It can be seen that our formulas obtain much better results needing far less function evaluations compared to quasi-Monte Carlo.

Authors

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• Gudmund Pammer ([email protected])
• Stefan Rigger ([email protected])