Name
Abstract | ||
|---|---|---|
This paper studies the convergence rate of randomized quasi-Monte Carlo (RQMC) for discontinuous functions, which are often of infinite variation in the sense of Hardy and Krause. It was previously known that the root mean square error (RMSE) of RQMC is only o(n(-1)/(2)) for discontinuous functions. For certain discontinuous functions in d dimensions, we prove that the RMSE of RQMC is O(n(-1/2-1/(4d-2)+epsilon)) for any epsilon > 0 and arbitrary n. If some discontinuity boundaries are parallel to some coordinate axes, the rate can be improved to O(n(-1/2-1/(4du-2)+epsilon)), where d(u) denotes the so-called irregular dimension, that is, the number of axes which are not parallel to the discontinuity boundaries. Moreover, this paper shows that the RMSE is O(n(-1/2-1/(2d))) for certain indicator functions. |
Year | DOI | Venue |
|---|---|---|
2015 | 10.1137/15M1007963 | SIAM JOURNAL ON NUMERICAL ANALYSIS |
Keywords | Field | DocType |
randomized quasi-Monte Carlo,numerical integration,discontinuous function,mean squared error,convex set | Continuous function,Mathematical optimization,Mathematical analysis,Discontinuity (linguistics),Numerical integration,Convex set,Mean squared error,Quasi-Monte Carlo method,Rate of convergence,Mathematics | Journal |
Volume | Issue | ISSN |
53 | 5 | 0036-1429 |
Citations | PageRank | References |
5 | 0.58 | 0 |
Authors | ||
2 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Zhijian He | 1 | 13 | 2.94 |
| Xiaoqun Wang | 2 | 404 | 34.39 |