Complexity of numerical integration over spherical caps in a Sobolev space setting

• Authors: Kerstin Hesse
• Venue: Journal of Complexity
• Year: 2011

Let r=2, let S^r be the unit sphere in R^r^+^1, and let C(z;@c):={x@?S^r:x@?z=cos@c} be the spherical cap with center z@?S^r and radius @c@?(0,@p]. Let H^s(S^r) be the Sobolev (Hilbert) space of order s of functions on the sphere S^r, and let Q"m be a rule for numerical integration over C(z;@c) with m nodes in C(z;@c). Then the worst-case error of the rule Q"m in H^s(S^r), with sr/2, is bounded below by c"r","s","@cm^-^s^/^r. The worst-case error in H^s(S^r) of any rule Q"m"("n") that has m(n) nodes in C(z;@c), positive weights, and is exact for all spherical polynomials of degree @?n is bounded above by c@?"r","s","@cn^-^s. If positive weight rules Q"m"("n") with m(n) nodes in C(z;@c) and polynomial degree of exactness n have m(n)~n^r nodes, then the worst-case error is bounded above by c@?"r","s","@c(m(n))^-^s^/^r, giving the same order m^-^s^/^r as in the lower bound. Thus the complexity in H^s(S^r) of numerical integration over C(z;@c) with m nodes is of the order m^-^s^/^r. The constants c"r","s","@c and c@?"r","s","@c in the lower and upper bounds do not depend in the same way on the area |C(z;@c)|~@c^r of the cap. A possible explanation for this discrepancy in the behavior of the constants is given. We also explain how the lower and upper bounds on the worst-case error in a Sobolev space setting can be extended to numerical integration over a general non-empty closed and connected measurable subset @W of S^r that is the closure of an open set.