Much of applied mathematics has to do with computable approximations for continuous problems, that is, replacing continuous phenomena by finite sets of values.
The fundamental problem is `placing points in space’. This can be accomplished by random sampling methods, like Monte Carlo (MC); or deterministic sampling methods, like quasi-Monte Carlo (QMC), lattice rules, or digital nets; or, yet again, randomized versions of these. It can also be accomplished by sparse grids, such as Smolyak points; or blue noise points from computer graphics.
Once the points have been selected, one can compute integrals, construct function approximations, follow trajectories, measure variable importance, or solve PDEs.
The core problem in QMC is numerical integration, and it is hardest when the spaces are awkward, either high-dimensional or non-cubical. Of course, cost is always an issue for the number of points one can afford to place.
The mathematical tools underlying QMC come in two flavors:
- Classical discrepancy theory, going back almost 100 years to work of Weil.
- Complexity theory, studying the limits of what can be done, under a huge array of scenarios involving costs, dimension, smoothness and other factors.
The goal of the SAMSI program is to explore the potential of QMC and other deterministic, randomized and hybrid sampling methods for a wide range of applications, including the numerical solution of PDEs; machine learning; computer graphics; Markov chain sampling, like MCMC and MCQMC; sequential Monte Carlo; and uncertainty quantification.
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