A new class of equal-weight integration rules on the hypercube
The author considers the approximation problem of the integral on the \(n\)-dimensional hypercube (1) \(I[f] =\int_{[0,1]^n} f(x)dx\). He proposes to approximate the \(n\)-dimensional integral (1) with equal-weight integration rules of the type \(Q[f] = (1/N)\sum_{j=1}^n f(x^{(j)})\), where \(N\) is the number of points, as in quasi-Monte Carlo methods. However, the integration rules generate points cyclically, as with the method of good lattice points. Combining the appealing properties of high (modified) degree with the equally weighted points, one can obtain efficient integration rules in terms of the number of points required to attain a certain degree of the integration rule method. The paper is organized as follows: Section 2 introduces the new equal-weight integration rules. Section 3 defines the method degree and compares it with both algebraic and trigonometric degrees. The main theorem, in Section 4, provides conditions for the existence of the new integration rules for degrees up to 13. In Section 5, actual integration rules are provided for up to 7 million points and up to 500 dimensions.
scientific article; zbMATH DE number 2132463