Quasi-Monte Carlo methods in non-cubical spaces
Abstract/Contents
- Abstract
- Monte Carlo integration is a widely used technique for approximating high dimensional integrals. However, due to the inherent randomness of this method, the convergence is typically slow. Quasi-Monte Carlo (QMC) on the other hand gives a much better rate of convergence by using low-discrepancy sequences. These point sets are much more uniformly distributed than random samples. Most QMC research focuses on sampling from the unit cube. However, many problems in real-world applications are defined over much more general spaces, such as triangle, spheres, spherical triangles and discs. This dissertation deals with solving such problems of numerical integration defined over non-cubical domains. We introduce two QMC constructions in the triangle with a vanishing discrepancy. The first is a version of the van der Corput sequence customized to the unit triangle. The second construction rotates an integer lattice through an angle whose tangent is a quadratic irrational number. We then generalize the van der Corput construction to study the problem of numerical integration over the Cartesian products of s spaces of dimension d via scrambled geometric nets. We also show the asymptotic normality of the scrambled geometric net estimator. We further discuss some of the issues of why transformations of the unit cube to the domain of interest fails to give good results. Since products of simplices is throughout of special interest to us, we end the dissertation with few results of QMC tractability on that domain.
Description
| Type of resource | text |
|---|---|
| Form | electronic; electronic resource; remote |
| Extent | 1 online resource. |
| Publication date | 2016 |
| Issuance | monographic |
| Language | English |
Creators/Contributors
| Associated with | Basu, Kinjal |
|---|---|
| Associated with | Stanford University, Department of Statistics. |
| Primary advisor | Owen, Art B |
| Thesis advisor | Owen, Art B |
| Thesis advisor | Candès, Emmanuel J. (Emmanuel Jean) |
| Thesis advisor | Chatterjee, Sourav |
| Advisor | Candès, Emmanuel J. (Emmanuel Jean) |
| Advisor | Chatterjee, Sourav |
Subjects
| Genre | Theses |
|---|
Bibliographic information
| Statement of responsibility | Kinjal Basu. |
|---|---|
| Note | Submitted to the Department of Statistics. |
| Thesis | Thesis (Ph.D.)--Stanford University, 2016. |
| Location | electronic resource |
Access conditions
- Copyright
- © 2016 by Kinjal Basu
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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