UQ library for sparse grids and Bayesian inference
The Toolkit for Adaptive Stochastic Modeling and Non-Intrusive ApproximatioN is a collection of robust libraries for high dimensional integration and interpolation as well as parameter calibration. This documentation focuses on the libraries and the software API, refer to the PDF document on the project web-page for specifics about the mathematics of the implemented methods.
Sparse Grids is a family of algorithms for constructing multidimensional quadrature and interpolation rules using multiple tensor products of one dimensional rules with varying degree of precision. The Tasmanian Sparse Grids Module implements a variety of grids that fall into five major categories:
- Global grids use polynomials supported over the entire domain of integration or interpolation. Such grids are suitable for approximating functions with smooth response.
- Sequence grids work much like Global grids, but use optimized internal data-structures for rules that are based on sequences of points formed from solving a greedy optimization problem
- Local polynomial grids use hierarchical piece-wise polynomials with compact support. Such grids are suitable for approximating functions with localized sharp behavior.
- Wavelet grids use special functions that form a Riesz basis which allows for more accurate local error estimations. Compared to Local polynomial grids, the wavelet basis can provide similar accuracy with significantly fewer points, but the advantage of the Riesz basis could also be negated from the higher Lebesgue constant near the domain boundary.
- Fourier grids use trigonometric functions with varying frequency and rely on Discrete (complex) Fourier Transforms. Such grids are suitable for approximating periodic functions, since periodicity if preserved in the approximation.
The DiffeRential Evolution Adaptive Metropolis is a method to draw samples from an arbitrary probability distribution defined by an arbitrary non-negative function (not necessarily normalized to integrate to 1). The DREAM approach is similar to the classical Markov Chain Monte Carlo, but it evolves a large number of chains simultaneously which leads to better parallelization and (potentially) faster convergence. In addition, multiple chains allow for better exploration of the probability domain, which is often advantageous when working with multi-modal distributions.
One of the main applications of DREAM is in the field of Bayesian inference, where samples are drawn from a posterior distribution comprised from a data-informed likelihood and an arbitrary model. The DREAM module of Tasmanian can use Tasmanian Sparse Grids approximation to either the model or the likelihood.
Please cite us
If you use Tasmanian for your research, please cite the Manual and our work on global and locally adaptive grids.
--user install only, see the on-line documentation for details.
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