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.
- Working Group I: Parallel Monte Carlo Neutronics Simulation (Leaders: T.K. Kelley, T.M. Evans and S.P. Hamilton)
- Working Group II: Probabilistic Numerics (Leaders: C.Oates, T.J. Sullivan)
- Working Group III: Sampling by Interacting Particle Systems (Leaders: Jianfeng Lu and Jonathan Mattingly)
- Working Group IV: Representative Points for Small-data and Big-data Problems (Leaders: V. R Joseph and Simon Mak)
- Working Group V: Sampling and Analysis in High Dimensions When Samples Are Expensive (Leaders: Fred Hickernell, Mac Hyman and Paul Constantine)
- Working Group VI: Adaptive Choice of Sobolev Space Weights (Leader: Art Owen)
- Working Group VII: Multivariate Decomposition Method (MDM) and Applications (Leaders: Dirk Nuyens and Alec Gilbert)
- Working Group VIII: Application of QMC to PDEs with random coefficients (Leaders: Frances Kuo and Alec Gilbert)
- Working Group IX: Multivariate Integration and Approximation in the Context of IBC (Leader: Peter Kritzer)
- Working Group X: Simpson Plays Billiards? (Leader: Dirk Nuyens)
Additional information for Working Group Leaders and Web Masters is provided HERE
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