A numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions

AbstractWe present a numerical algorithm for the construction of efficient, high-order quadratures in two and higher dimensions. Quadrature rules constructed via this algorithm possess positive weights and interior nodes, resembling the Gaussian quadratures in one dimension. In addition, rules can be generated with varying degrees of symmetry, adaptable to individual domains. We illustrate the performance of our method with numerical examples, and report quadrature rules for polynomials on triangles, squares, and cubes, up to degree 50. These formulae are near optimal in the number of nodes used, and many of them appear to be new

Efficient adaptive integration of functions with sharp gradients and cusps in n-dimensional parallelepipeds

In this paper, we study the efficient numerical integration of functions with sharp gradients and cusps. An adaptive integration algorithm is presented that systematically improves the accuracy of the integration of a set of functions. The algorithm is based on a divide and conquer strategy and is independent of the location of the sharp gradient or cusp. The error analysis reveals that for a $C^0$ function (derivative-discontinuity at a point), a rate of convergence of $n+1$ is obtained in $R^n$. Two applications of the adaptive integration scheme are studied. First, we use the adaptive quadratures for the integration of the regularized Heaviside function---a strongly localized function that is used for modeling sharp gradients. Then, the adaptive quadratures are employed in the enriched finite element solution of the all-electron Coulomb problem in crystalline diamond. The source term and enrichment functions of this problem have sharp gradients and cusps at the nuclei. We show that the optimal rate of convergence is obtained with only a marginal increase in the number of integration points with respect to the pure finite element solution with the same number of elements. The adaptive integration scheme is simple, robust, and directly applicable to any generalized finite element method employing enrichments with sharp local variations or cusps in $n$-dimensional parallelepiped elements.Comment: 22 page

Sparse Quadrature for High-Dimensional Integration with Gaussian Measure

In this work we analyze the dimension-independent convergence property of an abstract sparse quadrature scheme for numerical integration of functions of high-dimensional parameters with Gaussian measure. Under certain assumptions of the exactness and the boundedness of univariate quadrature rules as well as the regularity of the parametric functions with respect to the parameters, we obtain the convergence rate $O(N^{-s})$, where $N$ is the number of indices, and $s$ is independent of the number of the parameter dimensions. Moreover, we propose both an a-priori and an a-posteriori schemes for the construction of a practical sparse quadrature rule and perform numerical experiments to demonstrate their dimension-independent convergence rates

Efficient calculation of the worst-case error and (fast) component-by-component construction of higher order polynomial lattice rules

We show how to obtain a fast component-by-component construction algorithm for higher order polynomial lattice rules. Such rules are useful for multivariate quadrature of high-dimensional smooth functions over the unit cube as they achieve the near optimal order of convergence. The main problem addressed in this paper is to find an efficient way of computing the worst-case error. A general algorithm is presented and explicit expressions for base~2 are given. To obtain an efficient component-by-component construction algorithm we exploit the structure of the underlying cyclic group. We compare our new higher order multivariate quadrature rules to existing quadrature rules based on higher order digital nets by computing their worst-case error. These numerical results show that the higher order polynomial lattice rules improve upon the known constructions of quasi-Monte Carlo rules based on higher order digital nets

Smoothing the payoff for efficient computation of Basket option prices

We consider the problem of pricing basket options in a multivariate Black Scholes or Variance Gamma model. From a numerical point of view, pricing such options corresponds to moderate and high dimensional numerical integration problems with non-smooth integrands. Due to this lack of regularity, higher order numerical integration techniques may not be directly available, requiring the use of methods like Monte Carlo specifically designed to work for non-regular problems. We propose to use the inherent smoothing property of the density of the underlying in the above models to mollify the payoff function by means of an exact conditional expectation. The resulting conditional expectation is unbiased and yields a smooth integrand, which is amenable to the efficient use of adaptive sparse grid cubature. Numerical examples indicate that the high-order method may perform orders of magnitude faster compared to Monte Carlo or Quasi Monte Carlo in dimensions up to 35