creator Jimenez Rugama, Lluis Antoni advisor Hickernell, Fred J. In some definite integral problems the analytical solution is either unknown or hard to compute. As an alternative, one can approximate the solution with numerical methods that estimate the value of the integral. However, for high dimensional integrals many techniques suffer from the curse of dimensionality. This can be solved if we use quasi-Monte Carlo methods which do not suffer from this phenomenon. Section 2.2 describes digital sequences and rank-1 lattice node sequences, two of the most common points used in quasi-Monte Carlo. If one uses quasi-Monte Carlo, there is still another problem to address: how many points are needed to estimate the integral within a particular absolute error tolerance. In this dissertation, we propose two automatic cubatures based on digital sequences and rank-1 lattice node sequences that estimate high dimensional problems. These new algorithms are constructed in Chapter 3 and the user-specified absolute error tolerance is guaranteed to be satisfied for a specific set of integrands. In Chapter 4 we define a new estimator that satisfies a generalized tolerance function and includes a relative error tolerance option. An important property of quasi-Monte Carlo methods is that they are effective when the function has low effective dimension. In [1], Sobol’ defined the global sensitivity indices, which measure what part of the variance is explained by each dimension. We can use these indices to measure the effective dimensionality of a function. In Chapter 5 we extend our digital sequences cubature to estimate first order and total effect Sobol’ indices. Submitted by Erma Thomas (thomase@iit.edu) on 2017-05-15T19:54:18Z No. of bitstreams: 1 etdadmin_upload_471568.zip: 4226351 bytes, checksum: f7128a3d0385d34b82d6f415a785bc32 (MD5) Made available in DSpace on 2017-05-15T19:54:18Z (GMT). No. of bitstreams: 1 etdadmin_upload_471568.zip: 4226351 bytes, checksum: f7128a3d0385d34b82d6f415a785bc32 (MD5) Previous issue date: 2016-12 Ph.D. in Applied Mathematics, December 2016 2016 2016-12 http://hdl.handle.net/10560/4036 en adaptive algorithm error estimation high dimensional numerical integration quasi-Monte Carlo Dissertation born digital application/pdf In Copyright http://rightsstatements.org/page/InC/1.0/ Restricted Access MATH / Applied Mathematics Illinois Institute of Technology Affiliated department