Parallel algorithm for computing points on a computation front hyperplane

TL;DR: A parallel algorithm for computing points on a computation front hyperplane is described and the computation of the coordinates of points lying in the intersection of the domain with the hyperplane at a given step in the course of hyperplane motion is addressed.

Abstract: A parallel algorithm for computing points on a computation front hyperplane is described. This task arises in the computation of a quantity defined on a multidimensional rectangular domain. Three-dimensional domains are usually discussed, but the material is given in the general form when the number of measurements is at least two. When the values of a quantity at different points are internally independent (which is frequently the case), the corresponding computations are independent as well and can be performed in parallel. However, if there are internal dependences (as, for example, in the Gauss-Seidel method for systems of linear equations), then the order of scanning points of the domain is an important issue. A conventional approach in this case is to form a computation front hyperplane (a usual plane in the three-dimensional case and a line in the two-dimensional case) that moves linearly across the domain at a certain angle. At every step in the course of motion of this hyperplane, its intersection points with the domain can be treated independently and, hence, in parallel, but the steps themselves are executed sequentially. At different steps, the intersection of the hyperplane with the entire domain can have a rather complex geometry and the search for all points of the domain lying on the hyperplane at a given step is a nontrivial problem. This problem (i.e., the computation of the coordinates of points lying in the intersection of the domain with the hyperplane at a given step in the course of hyperplane motion) is addressed below. The computations over the points of the hyperplane can be executed in parallel.