SIGEST

Abstract

The SIGEST paper in this issue, “Hyperspherical Sparse Approximation Techniques for High-Dimensional Discontinuity Detection,” by G. Zhang, C. Webster, M. Gunzburger, and J. Burkardt, is an expanded version of their paper “A Hyperspherical Adaptive Sparse-Grid Method for High-Dimensional Discontinuity Detection” from the SIAM Journal on Numerical Analysis. This paper considers the problem of resolving discontinuities of functions on high-dimensional spaces. The paper uses sparse grid technology, which may not be familiar to many SIREV readers. Sparse grid methods were originally designed for quadrature in high dimensions. The obvious way to extend standard quadrature rules in one variable to several variables is a tensor product grid. This is a problem even in two dimensions because the number of nodes and weights of the rule is squared. In higher dimensions, tensor product grids are completely impractical because of the exponential growth in the number of nodes and weights. Sparse grids have high precision and only polynomial growth as the dimension increases. Since their invention by Smolyak in 1963, sparse grid methods have found applications in optimization, uncertainty quantification, chemistry, imaging, finance, machine learning, and several other areas. This paper applies sparse grid methods in a novel way to capture the discontinuity manifolds of functions of many variables. Section 3 of the paper reviews the literature in sparse grid methods for discontinuity detection. The problem with the prior methods is that mesh refinement near the discontinuity was needed for accuracy, and this refinement created a dense grid, thereby taking away much of the advantage of sparse grid methods. The authors solve this problem by observing that the surfaces across which functions are discontinuous are often smooth. In this case one can use sparse grids to approximate the surface itself. The authors employ a hyperspherical coordinate system which enables resolution of the discontinuity by solving several one-dimensional discontinuity detection problems. Sparse grid methods and their applications are necessarily technical, so the authors carefully guide the reader through the preliminary concepts in the first three sections while pointing to the broader literature in the field. A sparse grid novice should be able to navigate this part of the paper and get a good sense of how the methods work. The SIGEST paper contains the results from the original paper in sections 4 and 5, new results in section 6, and some very impressive examples in section 7.