Tent-transformed lattice rules for integration and approximation of multivariate non-periodic functions
This work provides efficient algorithms for high-dimensional non-periodic function approximation and integration, which is important for applications in computational science and engineering.
The authors develop algorithms for multivariate integration and approximation of smooth non-periodic functions using tent-transformed rank-1 lattice points, achieving constructive worst-case error bounds with good convergence rates by connecting to the periodic Korobov space.
We develop algorithms for multivariate integration and approximation in the weighted half-period cosine space of smooth non-periodic functions. We use specially constructed tent-transformed rank-1 lattice points as cubature nodes for integration and as sampling points for approximation. For both integration and approximation, we study the connection between the worst-case errors of our algorithms in the cosine space and the worst-case errors of some related algorithms in the well-known weighted Korobov space of smooth periodic functions. By exploiting this connection, we are able to obtain constructive worst-case error bounds with good convergence rates for the cosine space.