A Coarea Formulation for Grid-Based Evaluation of Volume Integrals

We present a new method for computing volume integrals based on data sampled on a regular Cartesian grid. We treat the case where the domain is defined implicitly by an inequality, and the input data include sampled values of the defining function and the integrand. The method employs Federer’s coarea formula (Federer, 1969, Geometric Measure Theory, Grundlehren der mathematischen Wissenschaften, Springer) to convert the volume integral to a one-dimensional quadrature over level set values where the integrand is an integral over a level set surface. Application of any standard quadrature method produces an approximation of the integral over the continuous range as a weighted sum of integrals over level sets corresponding to a discrete set of values. The integral over each level set is evaluated using the grid-based approach presented by Yurtoglu et al. (2018, “Treat All Integrals as Volume Integrals: A Unified, Parallel, Grid-Based Method for Evaluation of Volume, Surface, and Path Integrals on Implicitly Defined Domains,” J. Comput. Inf. Sci. Eng., 18, p. 3). The new coarea method fills a need for computing volume integrals whose integrand cannot be written in terms of a vector potential. We present examples with known results, specifically integration of polynomials over the unit sphere. We also present Saye’s (2015, “High-Order Quadrature Methods for Implicitly Defined Surfaces and Volumes in Hyperrectangles,” SIAM J. Sci. Comput., 37) example of integrating a logarithmic integrand over the intersection of a bounding box with an open domain implicitly defined by a trigonometric polynomial. For the final examples, the input data is a grid of mixture ratios from a direct numerical simulation of fluid mixing, and we demonstrate that the grid-based coarea method applies to computing volume integrals when no analytical form of the implicit defining function is given. The method is highly parallelizable, and the results presented are obtained using a parallel implementation capable of producing results at interactive rates.